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Extensions of Group Actions and the Hilbert-Smith Conjecture

Permanent Link: http://ufdc.ufl.edu/UFE0045160/00001

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Title: Extensions of Group Actions and the Hilbert-Smith Conjecture
Physical Description: 1 online resource (85 p.)
Language: english
Creator: Maissen, James R
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: action -- conjecture -- group -- hilbert -- smith -- topology
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: The Hilbert-Smith Conjecture proposes that every effective compact group action on a compact manifold is a Lie group. The conjecture is the generalization of Hilbert’s fifth problem, and it is still open for dimensions 4 and higher. It is well known that the conjecture is equivalent to postulating that there is no effective action of a p-adic group on a compact manifold. We explore several well-known examples of free and effective p-adic group actions on spaces that are not manifolds. We provide constructions for other actions on similar spaces by the group of p-adic numbers and other pro-finite groups. We prove that for any Peano continuum admitting an effective p-adic action, the continuum can be equivariantly partitioned. While the quotient map of the action is not generally a covering map, we extend many standard results on covering maps to it. We also present a different approach to the Hilbert-Smith conjecture by looking internally at the space under the action of the group. We show that free p-adic actions on the space of irrationals are unique up to conjugation. We also show that should a counter-example to the Hilbert-Smith conjecture exist, the counter-example would be an extension of this unique free p-adic group action on the space of irrationals. This motivates the investigation of compact extensions of compact metric group actions on separable metric spaces. While we show that there is always an extension of a p-adic action to some metric compactification, a free action does not necessarily extend to a free action. We give sufficient conditions to guarantee an extension of a group action to a given compactification. We present examples of group actions failing to extend without those conditions. The conjecture is translated into terms of the ring of continuous functions on the space of irrationals. We give an equivalent version of the conjecture in these terms, and explore how the new setting facilitates our investigation of extending actions and answering the conjecture. We give sufficient conditions for ensuring that extending each homeomorphisms of a group action to a compactification will have the group action extend.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by James R Maissen.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Keesling, James E.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045160:00001

Permanent Link: http://ufdc.ufl.edu/UFE0045160/00001

Material Information

Title: Extensions of Group Actions and the Hilbert-Smith Conjecture
Physical Description: 1 online resource (85 p.)
Language: english
Creator: Maissen, James R
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2013

Subjects

Subjects / Keywords: action -- conjecture -- group -- hilbert -- smith -- topology
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The Hilbert-Smith Conjecture proposes that every effective compact group action on a compact manifold is a Lie group. The conjecture is the generalization of Hilbert’s fifth problem, and it is still open for dimensions 4 and higher. It is well known that the conjecture is equivalent to postulating that there is no effective action of a p-adic group on a compact manifold. We explore several well-known examples of free and effective p-adic group actions on spaces that are not manifolds. We provide constructions for other actions on similar spaces by the group of p-adic numbers and other pro-finite groups. We prove that for any Peano continuum admitting an effective p-adic action, the continuum can be equivariantly partitioned. While the quotient map of the action is not generally a covering map, we extend many standard results on covering maps to it. We also present a different approach to the Hilbert-Smith conjecture by looking internally at the space under the action of the group. We show that free p-adic actions on the space of irrationals are unique up to conjugation. We also show that should a counter-example to the Hilbert-Smith conjecture exist, the counter-example would be an extension of this unique free p-adic group action on the space of irrationals. This motivates the investigation of compact extensions of compact metric group actions on separable metric spaces. While we show that there is always an extension of a p-adic action to some metric compactification, a free action does not necessarily extend to a free action. We give sufficient conditions to guarantee an extension of a group action to a given compactification. We present examples of group actions failing to extend without those conditions. The conjecture is translated into terms of the ring of continuous functions on the space of irrationals. We give an equivalent version of the conjecture in these terms, and explore how the new setting facilitates our investigation of extending actions and answering the conjecture. We give sufficient conditions for ensuring that extending each homeomorphisms of a group action to a compactification will have the group action extend.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by James R Maissen.
Thesis: Thesis (Ph.D.)--University of Florida, 2013.
Local: Adviser: Keesling, James E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2013
System ID: UFE0045160:00001


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EXTENSIONSOFGROUPACTIONSANDTHEHILBERT-SMITHCONJECTUREByJAMESR.MAISSENADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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2013JamesR.Maissen 2

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ToMarie,theloveofmylife 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadvisor,Dr.JamesKeesling,forhiswonderfulinsightandsupport.Iamgratefultohavehadthepleasureofworkingwithhim.Aftereachandeverymeeting,Ileavewiththeenergyandenthusiasmthatheprojectstoeveryonearoundhim.IamalsogratefulforallthewonderfulsupportthatIhavefoundintheMathematicsDepartmentattheUniversityofFlorida.Iwouldliketothankallthemembersofmycommitteeforallofthehelpthattheyhavegiventomeovertheyears.Theyarepartofawonderfultraditionofnurturingmathematicians,andIcanonlyhopetosoonjointheranksofsuchalovingfamily.Inparticular,IwouldliketothankDr.DavidC.WilsonforjoiningDr.Keeslingandmeindiscussions,asIhavefoundthatthreeisaperfectnumberforsuchtalks.But,moreover,Iwouldliketothankhimfornotonlysharinghismathematicalacumen,butalsoinkeepingmywritingontrack.ToDr.BeverlyBrechner,Ioweagreatdealofthanks,aswithoutherIwouldnothavefoundthelovefortopologythatIhavetoday.Imustalsothankherforallofthecountlessmeetingswehadinyearsgoneby,andforintroducingmetotheHilbert-Smithconjectureintherstplace.IwouldalsoliketothankDr.AlexDranishnikov.HewassuchawonderfulteacherwhenIrststartedasagraduatestudentinthedepartmentandsolidlyreafrmedmyloveofthesubject.Iwishtoespeciallythankhimfordirectingandmaintainingthestudenttopologyseminar.Ialwaysndmyselflookingforwardtotheseminarandespeciallytheproblemsessionassociatedwithit.Beyondthat,Iwishtothankhimforbeingsuchawonderfulresourceandwealthofmathematicalknowledge;heisatreasureIwillsorelymisshavingelsewhere.Iwouldnallyliketothankthetwofacultymembersthathaveserved,sequentially,asmyoutsidecommitteemember.Servingasanoutsidememberhastobethemostdauntingofrolestollonacommittee,andIappreciatebothofthemfordoingsoonmybehalf.ToDr.MalayGhosh,Iwishtoextendmythanksforhistimeandkindness,andIamjustsorrythatschedulingissuespreventedhimfromseeingmeto 4

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theveryendwiththis.ThanksalsogoouttoDr.PaulGaderforhiswillingnesstojoinmycommitteesolateintheprocess,andforbeingsoeagerandabletogetuptospeedonit.Ihavefoundmyselfcalmedintwootherwiseverystressfultimesasaresultofthemannerinwhichtheyconductthemselves,andIsincerelywishtothankbothofthemforit.Iwouldliketothankmylatemotherforfosteringaloveoflearninginme.Iwishthatshecouldhaveseenthisday.Ifshewereabletohearme,IwouldtellherIloveandmissyou,always.Lastly,butmostimportantly,IwholeheartedlythankmyanceeMarie.Shehasturnedmylifearoundwithherunendingloveandsupport.IdonotthinkthatIwouldhavehadthestrengthtogobacktonishmyPhDwithouther. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................... 4 LISTOFFIGURES ........................................ 8 ABSTRACT ............................................ 10 CHAPTER 1INTRODUCTION ...................................... 12 1.1NotationandTerms ................................. 14 1.1.1Sets ...................................... 14 1.1.2Topology ................................... 14 1.1.3Partitions ................................... 14 1.1.4InverseLimitofFiniteGroupsandp-adicNotation .......... 16 2HISTORY .......................................... 18 3COMPACTIFICATIONS .................................. 30 3.1ActionsontheSpaceofIrrationalNumbers .................. 30 3.2EquivariantCompactications ........................... 35 3.3ExtendingGroupActions .............................. 40 3.4Examples ....................................... 43 3.5RingsofContinuousFunctions .......................... 46 3.5.1GroupActionsonCXandaMetriconCX ........... 48 3.5.2DimensioninCX ............................ 48 3.5.3Hilbert-SmithinCX ........................... 49 4PEANOSPACES ..................................... 51 4.1EquivariantPartitionsofPeanoContinua .................... 51 4.2LiftingArcsandHomotopies ............................ 54 5EXAMPLESANDINVARIANTSETS .......................... 59 5.1SimpleEffectiveANRAction ........................... 59 5.2MengerCurveAction ................................ 62 5.2.1PasnynkovPartialProductDescriptionofn .............. 62 5.2.2Dranishnikov'sActiononn ........................ 62 5.3InvariantSets ..................................... 64 6HILBERTSPACE ..................................... 76 6.1SpaceofMeasurableFunctions ......................... 76 6.2ViewingSimpleFunctions ............................. 77 6

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6.3SomeSpecialSimplicesinM)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,G`2 ................. 78 6.4InvariantSets ..................................... 79 REFERENCES .......................................... 81 BIOGRAPHICALSKETCH ................................... 85 7

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LISTOFFIGURES Figure page 2-1DavidHilbert ........................................ 18 2-2BrouwerandKerekjarto .................................. 19 2-3AlfredHaar ......................................... 19 2-4J.VonNeumann,L.S.Pontrajagin,C.Chevalley,andH.Fredudenthal ...... 20 2-5A.GleasonandK.Iwasawa ............................... 20 2-6S.Bochner,D.Montgomery,L.Zippin,andH.Yamabe ............... 21 2-7OneofonlytwophotosofP.A.Smithknowntoexist ................ 22 2-8M.H.A.Newman ...................................... 22 2-9A.N.Kolmogoroff ..................................... 23 2-10S.EilenbergandN.E.Steenrod ............................. 23 2-11R.D.Anderson,L.V.Keldys,andD.C.Wilson ..................... 24 2-12C.T.Yang,F.Raymond,andR.F.Williams ....................... 25 2-13A.N.DranishnikovandZ.Yang ............................. 27 2-14Pictured:B.A.Pasynkov(sitting),M.Bestvina,andR.D.Anderson ........ 27 2-15Pictured:E.V.Scepin,D.Repovs,J.F.Nash,andJ.Pardon ............ 29 3-1Pre-compactquotientspace ............................... 39 3-2CompacticationofNZ2 ................................ 43 3-3L.Gillman,M.Jerison,andM.Henriksen ....................... 46 3-4M.H.Stone,E.Hewitt,A.N.Kolmogoroff,andI.M.Gelfand ............ 47 3-5M.Katetov .......................................... 49 5-1CantorTree ......................................... 59 5-2DeformationretractionofatriangletotheCantorTree ............... 60 5-3Constructingasimpleclosedcurvewithdisjointorbits ............... 68 5-4Invariantp-adicsolenoid ................................. 69 5-5pkinvariantp-adicsolenoids ............................... 69 8

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5-6Comparingquotientspacesforfreeactionson1 .................. 74 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEXTENSIONSOFGROUPACTIONSANDTHEHILBERT-SMITHCONJECTUREByJamesR.MaissenMay2013Chair:JamesKeeslingMajor:MathematicsTheHilbert-SmithConjectureproposesthateveryeffectivecompactgroupactiononacompactmanifoldisaLiegroup.TheconjectureisthegeneralizationofHilbert'sfthproblem,anditisstillopenfordimensions4andhigher.Itiswellknownthattheconjectureisequivalenttopostulatingthatthereisnoeffectiveactionofapadicgrouponacompactmanifold.Weexploreseveralwell-knownexamplesoffreeandeffectivep-adicgroupactionsonspacesthatarenotmanifolds.Weprovideconstructionsforotheractionsonsimilarspacesbythegroupofp-adicnumbersandotherpro-nitegroups.WeprovethatforanyPeanocontinuumadmittinganeffectivep-adicaction,thecontinuumcanbeequivariantlypartitioned.Whilethequotientmapoftheactionisnotgenerallyacoveringmap,weextendmanystandardresultsoncoveringmapstoit.WealsopresentadifferentapproachtotheHilbert-Smithconjecturebylookinginternallyatthespaceundertheactionofthegroup.Weshowthatfreep-adicactionsonthespaceofirrationalsareuniqueuptoconjugation.Wealsoshowthatshouldacounter-exampletotheHilbert-Smithconjectureexist,thecounter-examplewouldbeanextensionofthisuniquefreep-adicgroupactiononthespaceofirrationals.Thismotivatestheinvestigationofcompactextensionsofcompactmetricgroupactionsonseparablemetricspaces.Whileweshowthatthereisalwaysanextensionofap-adicactiontosomemetriccompactication,afreeactiondoesnotnecessarily 10

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extendtoafreeaction.Wegivesufcientconditionstoguaranteeanextensionofagroupactiontoagivencompactication.Wepresentexamplesofgroupactionsfailingtoextendwithoutthoseconditions.Theconjectureistranslatedintotermsoftheringofcontinuousfunctionsonthespaceofirrationals.Wegiveanequivalentversionoftheconjectureintheseterms,andexplorehowthenewsettingfacilitatesourinvestigationofextendingactionsandansweringtheconjecture.Wegivesufcientconditionsforensuringthatbeingabletoextendeachhomeomorphisminagroupactiontoacompacticationwillhavethegroupactionextend. 11

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CHAPTER1INTRODUCTIONTheHilbert-Smithconjecturehascapturedthemindsofmathematiciansforgenerations.In1940,PaulAlthausSmithgeneralizedHilbert'sFifthProblem,whichhadaskedwhetherevery(nite-dimensional)locallyEuclideantopologicalgroupisnecessarilyaLiegroup.SmithproposedthateveryeffectivecompactgroupactiononacompactmanifoldisbyaLiegroup.Thisiswellknowntobeequivalenttoaskingwhetherornotap-adicgroupcanacteffectivelyonacompactmanifold.Thisdissertationisfocusedbythedesiretoanswertheconjecture.Wegiveamulti-prongedattackontheproblem.Westudyrelated,known,p-adicactionsonnon-manifoldspacesandconstructsimilar,newones.Weexplorethequotientspacesofsuchactionstodrawconclusionsastothenatureofanypossiblecounter-exampletotheconjecture.Andwealsoapproachtheprobleminadifferentwaybylookinginternallyataspaceundertheactionofthep-adicgroup.Thislastapproachleadsustosomeequivalentwaysofseeingtheconjecture.Instudyingap-adicactiononaspace,thequotientspaceoftheactionisquiteusefultounderstandinwholeorinpart.Manybeforeushaveextensivelystudiedwhatpropertiessuchquotientspacesmusthave.Whilesuchquotientmapsarenot,exceptinthesimplestofcases,coveringmaps,theydosharesimilarpropertiestocoveringmaps.Weprovesomesimilarresultstoclassicalcoveringspacetheoryforthesequotientmaps.Theseresultshelpustoprovethateveryp-adicactiononaPeanocontinuumadmitsequivariantpartitions;which,inturn,helpstofurtherourunderstandingofthenatureofp-adicactionsonsuchcontinua.Itiswellknownthatthereareeffective,andevenfree,p-adicactionsonnon-manifoldspaces.OneexamplewasknownevenpriortoSmithmakingtheconjectureitself,thoughperhapsnotrealizedassuchatthetime.Infact,freep-adicactionsonMengermanifoldsofalldimensionshavebeenconstructed.Wediscusssomeofthoseclassic 12

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partialtopologicalproductbasedconstructionstopresentamoregeometricviewofthem.Werecaphowtoconstructfreep-adicactionsonHilbertspacefromeasytoconstructeffectivep-adicactionsontheHilbertCube.Inattemptstosolvetheconjecture,moreexoticPeanocontinuaofalldimensionshavebeenconstructedinsuchawaytoadmitp-adicactionswithdesiredproperties.Oneofourgoalsistomakemanyofthesesomewhatarcaneconstructionsmoreaccessible.Wedemonstrateeasierconstructionsofsome,providenewp-adicactionsontheMengercurve,andshowinterestingpropertiesofsuchinregardstoanypotentialcounter-exampletotheHilbert-Smithconjecture.Weprovethatanypotentialcounter-exampletotheHilbert-Smithconjecturewouldhaveadensesubsethomeomorphictothespaceofirrationalnumbersuponwhichthegivenp-adicactionwouldactfreelyasasub-action.Infactforalargeclassofexamplesofeffectivep-adicactionsonnon-manifolds(includingthoseonMengercontinua)thisisindeedthecase.Weprovethatthereis,uptoconjugation,onlyonesuchfreep-adicactiononthespaceofirrationalnumbers.Thus,anypotentialcounter-exampletoHilbert-Smith,aswellasthesenumerousothereffectivep-adicactionsonnon-manifolds,canbeseenasanextensionoftheunique,free,p-adicactiononthespaceofirrationals.Inexploringthenatureofextendinggroupactionstocompactications,potentialobstructionswerefound.Wegiveaclearexampleoftheseobstructionsbymeansofasimplespacewithsomefairlysimple0-dimensionalgroupsactinguponit.Wethengivesufcientconditionsforagroupactiontoextend,asagroupaction,toagivencompactication.Likewise,givena0-dimensionalcompactgroupactingonanon-compactspacewedemonstratetheexistenceofacompacticationtowhichtheactionextends(thoughnotnecessarilyfreely).WhileourfocusisonHilbert-Smith,andthusonp-adicgroups,wetreatthisinmoregeneralityasitcanhaveotherapplications. 13

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Whenstudyingcompactications,itisnaturaltoattempttotranslatetheproblemintothesettingofringsofcontinuousfunctions.WereformulatetheHilbert-Smithconjecturehereaswellastranslateotherrelatedproperties.Itisourhopethatthisnewsettingwherecompacticationscanbesoeasilydescribedwillprovemoretractable. 1.1NotationandTermsForourtermsandnotationsandunlessotherwisespecied,wewillassumethatwearedealingwithametricspaceX,dx 1.1.1SetsWeletRdenotetherealnumbers,Qtherationalnumbers,andI)]TJ /F11 11.955 Tf 11.25 0 Td[(RQtheirrationalnumbers.LettheunitintervalbegivenbyI)]TJ /F4 11.955 Tf 9.28 0 Td[()]TJ /F8 11.955 Tf 4.55 0 Td[(0,1.WewilluseZktomeantheabeliangroupofintegerswithadditionmodulok,thatisZk)]TJ /F14 7.97 Tf 9.28 2.77 Td[(Z~kZ. 1.1.2TopologyForY`X,wewilldenoteitsinteriorbyY,it'sboundary(orfrontier)by@Y,andit'sclosureinXbyY.DenotetheopenballaboutasetA`XtobetheopensetBrA)]TJ /F4 11.955 Tf 11.69 0 Td[(x>XSdxx,A@r`XanddenotethediameterofasubsetAbydiamA)]TJ /F8 11.955 Tf 9.28 0 Td[(supdxa,bSa,b>A. Denition1.1. AmapfX)]TJ /F2 11.955 Tf 12.64 0 Td[(Yissaidtobe 1. Anopen(orinterior)mapif,foreveryopensubsetU`X,theimagefU`Yisopen. 2. Aclosedmapif,foreveryclosedsubsetV`X,theimagefV`Yisclosed. 3. Aperfectmapping,ifitisacontinuous,closed,surjectivemapsuchthatforeachy>Ythesetf1yiscompact. 4. Alightmapif,foreveryy>fX,thepre-imagef1yistotallydisconnected. 1.1.3Partitions Denition1.2. [ 8 ]AsubsetK`XhaspropertySifforeachnumberA0,thereisanNA0suchthatK)]TJ /F17 11.955 Tf 9.28 -.94 Td[(Nn)]TJ /F10 7.97 Tf 4.64 0 Td[(1Kn,whereeachKnisconnectedanddiamKn@. 14

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Denition1.3. [ 8 ]AsubsetK`Xcanbepartitionedif,foreachnumberA0,thereisanNA0suchthatthereisacollectionofdisjointsubsetsUnNn)]TJ /F10 7.97 Tf 4.63 0 Td[(1ofK,whereUn`KisconnectedandopenforeachnBN,diamUn@foreachnBN,Uj9Uk)]TJ /F12 11.955 Tf 9.47 0 Td[(gforalljxk,andNn)]TJ /F10 7.97 Tf 4.63 0 Td[(1UnisdenseinK.WeshallsaythatU)]TJ /F4 11.955 Tf 9.28 0 Td[(UnNn)]TJ /F10 7.97 Tf 4.63 0 Td[(1isan-partitioningofK. Denition1.4. [ 9 ]ApartitioningUofK`Xisabrickpartitioningif: (a) EachdomaincontainingapointofKwhichisalimitpointofeachoftwoelementsofUalsocontainsapointofKwhichisalimitpointofeachofthesesametwoelementsofUbutofnootherelementofU. (b) EachelementofUisuniformlylocallyconnectedundertheconnecteddistancemetricwithinK. (c) EachboundarypointinKofanelementofUisaboundarypointofanotherelementofU(i.e.regular)ForapartitionUofasetK`XwedenethemeshofthepartitiontobemeshU)]TJ /F8 11.955 Tf -440.18 -23.91 Td[(supdiamUSU>U.WealsodenethestarofasubsetA`X(orapointa>X)withrespecttoapartitionUtobeStA,U)]TJ /F4 11.955 Tf 9.28 0 Td[(x>XSx>U,U>U,U9Axg.WesaythatapartitionVrenesapartitionUif,foreachV>V,thereisanelementUV>UwithV`UV.IfmeshVB,thenVisan-renementofU.IfbothpartitionsUandVarebrickpartitions,thenwesaythatVbrickrenesU.WedenoteVreningUbyVhU.Wewillusethesamenotationwhendealingwithopencovers.AndifarenementVofanopencoverUissuchthatforeachV>VthereisanelementUV>UwithStV,V`UV,thenwesaythatVstar-renesUanddenotethatbyVhU. Denition1.5. [ 10 ]ThebrickpartitioningVofasetK`XisacorerenementofthebrickpartitioningUofK`Xif: (a) VisarenementofU(i.e.VhU). (b) foreachpairofadjacentelementsU,UofUthereisapairofadjacentelementsV,VofVinUandUrespectivelysuchthatV8VisasubsetoftheinteriorofU8U,and 15

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(c) foreachelementUofU,theelementsofVinUmaybeorderedV0,V1,...,VnsuchthatV0intersectseachVi,whileViintersectstheboundaryofUifandonlyifiA0.WecallV0acoreelementandVi,iA0borderelements. 1.1.4InverseLimitofFiniteGroupsandp-adicNotationWhenwewriteG)]TJ /F8 11.955 Tf 9.28 0 Td[(limGi,i1iwewillhavethefollowingunderstandings: 1. EachGiisanitegroup 2. G0)]TJ /F4 11.955 Tf 9.28 0 Td[(e 3. GiBGi1withequalityifandonlyifGi)]TJ /F2 11.955 Tf 9.28 0 Td[(G 4. ForeachiC0,i1iGi1)]TJ /F2 11.955 Tf 12.65 0 Td[(GiisanontohomomorphismInthecasewherethegroupGisnite,forsimplicity,wewilltakeGi)]TJ /F2 11.955 Tf 9.35 0 Td[(Gandi1iG)]TJ /F2 11.955 Tf 12.72 0 Td[(GtobetheidentitymapforalliC1.Bythep-adicgroup,p,wemeanlimZpn, i1iwherethebondingmaps i1iZpi1)]TJ /F11 11.955 Tf 13 0 Td[(Zpiarethesurjectionsobtainedbytakingthemodulusbypi.Intheliteraturethiszero-dimensionalcompactgroupissometimesalsodenotedbyApor^Zp.Whiletopologicallyoneelementofpgeneratesthegroup(thatisthatoneelementalgebraicallygeneratesadensesetofp),thiselementisbynomeansunique.Whenwehaveneedtopickoneofthesegenerators,wewilldenoteitby.Proper,nontrivialsubgroupsofparethenoftheformpkpforeachchoicek>N.Wewillusethenotationk)]TJ /F5 11.955 Tf 9.27 0 Td[(pkptoconservespaceandeyestrain.WhenpactsfreelyonaspaceXthentherenaturallyarisesasystemofmapsonthequotients: Xp5)]TJ /F6 7.97 Tf 22.27 2.77 Td[(X~4p4)]TJ /F6 7.97 Tf 22.26 2.77 Td[(X~3p3)]TJ /F6 7.97 Tf 22.27 2.77 Td[(X~2p2)]TJ /F6 7.97 Tf 22.26 2.77 Td[(X~1p1)]TJ /F6 7.97 Tf 22.27 2.77 Td[(X~p(1)Wherethep-to-1coveringmapspnX~n)]TJ /F6 7.97 Tf 13.03 2.76 Td[(X~n1wereinducedbythemultiplicativemapnn1)]TJ /F8 11.955 Tf 14.21 0 Td[(n(wherenissimplycompositionby).Whenpactseffectively 16

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ratherthanfreely,thenthesamesystemcanbeobtained;however,themapspnarebranchedcoveringsratherthanfullcoveringmaps.Foreachn>N,letthepn-to-1(branched)coveringmapPnX~n)]TJ /F6 7.97 Tf 13.07 2.77 Td[(X~pbegivenbyPn)]TJ /F2 11.955 Tf 10.13 0 Td[(p1Xp2X)-145()-145()-222(Xpn1Xpn.Wewilltake0X)]TJ /F6 7.97 Tf 13.49 2.77 Td[(X~ptobethequotientmapofthepactionandforeachn>NwewillletthemapnX)]TJ /F6 7.97 Tf 13.27 2.77 Td[(X~nbethequotientmapinducedbythesubgroupaction,meaningthecompositionPnXn)]TJ /F5 11.955 Tf 9.28 0 Td[(0. 17

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CHAPTER2HISTORYAtthestartofthepriorcenturyinAugustof1900,DavidHilbertspokeattheParisconferenceoftheInternationCongressofMathematicswherehechallengedtheworldwithasetof,atthetime,tenunsolvedproblemsinvariousdisciplinesofmathematicssayingWehearwithinustheperpetualcall:Thereistheproblem.Seekitssolution[ 27 ].Hislistofproblemsexpandedtonumbertwenty-threebythetimehepublishedhisspeechlaterthatyearinaGermanjournal[ 26 ].In1902,Dr.MaryWinstonNewsontranslatedthisarticleintoEnglishfortheBulletinoftheAmericanMathematicsSociety[ 27 ]. Figure2-1. DavidHilbertasheappearedin1912forpostcardsofthefacultymembersattheUniversityofGottingen,whichweresoldtostudentsatthetime. DavidHilbert'schallengethatjustaseveryhumanundertakingpursuescertainobjects,soalsomathematicalresearchrequiresitsproblems.Itisbythesolutionofproblemsthattheinvestigatorteststhetemperofhissteel[ 27 ]inspiredmathematiciansforgenerationstocome.Notallofthelistedproblemswereclearlydenedhowever,sothereisambiguityonwhetheragivensolutiontoaproblemcompletelysolvesthatproblem.ThuswhenoneconsidersaHilbertProblem,onemustrstworktounderstandtheproblemandthecontextsinwhichitcanbeviewed.ThefthproblemofHilbert's23wastranslatedintoEnglishasHowfarLie'sconceptofcontinuousgroupsoftransformationsisapproachableinourinvestigationswithouttheassumptionofthedifferentiabilityofthefunctions[ 27 ].Overtheyearssincethenmanypartialsolutionsandinterpretationsofthisproblemhavebeengivenbyacornucopiaoftheworld'sbrightestmathematicians.Thenamesandpicturesthatappearbelowshouldndresonancewithanymathematicianforthesheerbreadthoftheirworks 18

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isaweinspiring.Thateachtouchedandweretouchedinreturnbythisproblemiswhatmakesitsuchatreasure. ABrouwer BKerekjartoFigure2-2. LEJBrouwerandBelaKerekjartoindependentlysolvedthecasefordimension2byexhaustion Earlyadvancesontheproblemweremadein1919byLuitzenEgbertusJanBrouwerinthespecialcasewhendealingwithamanifoldofdimension2.AftertheFirstWorldWar,whenheturnedbacktoTopology,heshowedthatagroupactingeffectivelyona2-manifoldisaLiegroup[ 13 ].SimilarresultstoBrouwer'swereobtainedindependentlybyBelaKerekjarto[ 33 ],whominterestinglyenoughBrouwerhadoncedescribedderogatorilyasrash[ 51 ].Itis,also,perhapsinterestingtonotethatbothpapersappearedoneaftertheotherinthesameissueoftheMathAnnalenjournal. Figure2-3. AlfredHaarwasastudentofDavidHilbertattheUniversityofGottingen In1933,AlfredHaarintroducedtheworldtoHaarMeasure[ 23 ].Thisallowsustodenequiteanaturalmetriconaspaceuponwhichacompactgroupacts.SupposeX,disametricspaceandthatacompactgroupGactsonit,thenwemaydeneametricusingtheHaarmeasure: Gx,y)]TJ /F17 11.955 Tf 9.28 -2.53 Td[(SGdgx,gydg(2) 19

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ThisproducesaG-invariantmetriconthespacethatistopologicallyequivalenttotheoriginalmetric.ThismakesthemetricinducedbyHaarmeasuretheperhapsmostnaturalmetriconwhichweshouldinvestigateacompactgroupaction.Notsurprisingly,someniceusesofthismetrichaveoccurredinthehistoryoftheproblem[ 40 46 52 ]. AVonNeumann BProntrjagin CChevalley DFreudenthalFigure2-4. J.VonNeumann,L.S.Pontrajagin,C.Chevalley,andH.FredudenthaleachgavesolutionsforvariousspecialcasesofHilbert'sFifthProblem Infact,inthesamejournalinthearticleimmediatelyfollowingthatofHaar,JohnVonNeumannusedHaarmeasuretoproveHilbert'sfthprobleminthecaseofcompactgroups[ 52 ].Inthefollowingyear,LevSemenovichPontrjaginextendedtheresultforthecaseofabeliangroups[ 43 ].In1941,ClaudeChevalleypublishedasolutionforsolvablegroups[ 14 ].Andafewyearspriortothat,HansFreudenthalintroducedtheideaofalocallycompactgroupbeingapproximatedbyLiegroups[ 18 ],namely: Denition2.1. [ 30 ]AlocallycompactgroupGcanbeapproximatedbyLiegroups,ifGcontainsasystemofnormalsubgroupsNsuchthatG~NareLiegroupsandthattheintersectionofallNcoincideswiththeidentitye. AGleason BIwasawaFigure2-5. A.GleasonandK.IwasawaindependentlypavedthewaytothesolutionofHilbert'sFifthProblem ThisideawouldinspireAndrewM.Gleasonwithhisconceptof`'GroupswithoutSmallSubgroups.And,aftertheSecondWorldWar,progressontheproblem 20

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expandedundertheseparateworksofKenkichiIwasawaandAndrewGleason,whobothconjecturedthatthefollowingformofHilbert'sfthproblemheldtrue: Conjecture1(Gleason-Iwasawa[ 21 30 ]). EverylocallycompactgroupisageneralizedLiegroupAsanaside,itisinterestingtoseetheabilityoftheworldwarstohamperresearchinthisproblemthroughtherestrictionoftravel,inrestrictingcommunicationsbetweenmathematicians,aswellassimplydrawingthemtootherthings.InthecaseoftheFirstWorldWar,itwasrestrictingthetraveloftheDutchBrouwerintoGermany,andinthecaseoftheSecondWorldWar,itwasbothbylimitingtheavailabilityofjournalpaperstotheJapanesseIwasawa,andbytakingtheAmericanGleasonfromhisresearchintobreakingJapanesecodesfortheAllies.Germany'spoliticsofthe1930sbroughtSalomonBochnertoPrincetonand,afterthewar,heandDeaneMontgomeryprovedHilbert'sfthproblemforgroupsofdiffeomorphisms[ 11 ]. ABochner BMontgomery CZippin DYamabeFigure2-6. S.Bochner,D.Montgomery,L.Zippin,andH.YamabegavesolutionsfordifferentversionsofHilbert'sFifthProblem Thesesuccessesculminatedin1952,whenGleasonprovedthatalocallycompactnitedimensionalgroupwithoutsmallsubgroupswasaLiegroup[ 22 ].UsingthisDeaneMontgomeryandLeoZippinreceivecreditforsolvingHilbert'sfthprobleminthefollowingform: Theorem2.1(Montgomery-Zippen[ 38 ]). Alocallycompact,nitedimensional,locally-compactgroupisaLiegroupInthefollowingyear,HidehikoYamabeextendedtheirresultforinnitedimensionalgroups,generalizingtheresultsofMontgomeryandZippin.Dependingonhowyouview 21

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Hilbert'sfthproblemdeterminestowhomyouwillcreditforthenalanswer,althoughitismosttypicallyviewedashavingbeensolvedbyMontgomeryandZippin.However,thisisnottheonlywaythatonemayviewHilbert'sfthproblem.Intheperiodofthelate1930samoregeneralizedversionofHilbert'sfthproblemwasconsideredandputforthbyPaulAlthausSmith. Figure2-7. OneofonlytwophotosofP.A.Smithknowntoexist.PhotocourtesyofUniversityArchives,ColumbiaUniversityintheCityofNewYork(for$20) Conjecture2(Smith[ 48 ]). IfGisacompactgroupwhichactseffectivelyonamanifold,thenGisaLiegroup.Heshowedthatthisconjectureisequivalenttothefollowing,whichisknownastheHilbert-SmithConjecturethatinspiresresearchtowardsananswertothisday: Conjecture3(Hilbert-Smith). Ap-adicgroupcannotacteffectivelyonamanifold.ItistowardsthisproblemthatP.A.SmithdevelopedhisSmithTheoryinstudyingtransformationsbygroupsofniteperiods.HealsoprovidedhisownproofofNewman'stheorem[ 48 ]differentfromMHANewman'soriginal.ThetheoremasoriginallystatedbyMaxwellHermanAlexanderNewmanis: Theorem2.2(Newman[ 39 ]). LetMbeaconnectedmanifoldwithmetricd,thenthereexistanA0suchthat,foreveryactionofanitegroupGonM,thereexistsanorbitofdiameterlargerthan. Figure2-8. M.H.A.Newman,themathematicalfatherofColossus(theworld'srstdigitalprogramablecomputer) 22

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Onekeyelementinattemptingtodeterminewhetherornotap-adicgroupcanacteffectively(orfreely)onamanifoldistoconsiderthequotientspacethatarisesfromsuchapotentialp-adicaction.Beingtheimageofanopenmappingofamanifold,thequotientspaceoftheactionisacompact,connected,locallyconnected,metricspace.However,thishypotheticalspacedoesnothavemanymoreofthenicequalitiesthatoneassociateswithmanifolds. Figure2-9. A.N.Kolmorgorofffailedtoprovethatopenmapscouldnotraisedimension In1941,Smithprovedthatifap-adicgroupApactedfreelyonamanifoldX,thenthedimensionofthequotientspacedimX~ApxdimX[ 47 ].Onenaturalquestioniswhetherornotaquotientmap,orinfactanyopenmapcanraisedimension.Justafewyearsprior,in1937,AndreyNikolaevichKolmogoroffshowedthistobepossiblebyconstructinganopenmapfroma1-dimensionalspacetoa2-dimensionalspace[ 34 ].Thisexample,thoughmotivatedsimplytoshowthatanopenmapcouldraisedimension,hasadirectimpactonthestudyoftheHilbert-Smithconjecture,eventhoughneitherofthespacesinvolvedisamanifold. AEilenberg BSteenrodFigure2-10. S.EilenbergandN.E.SteenrodwereinvolvedincarryingonthespiritofHilbert'sProblems In1949andinthespiritofHilbert'sproblems,SamuelEilenbergpublishedalistofopenproblemsinTopology[ 16 ]arisingfromamathematicsconferenceheldaspartof 23

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theBicentennialCelebrationofPrincetonUniversity.ThelistincludedtheHilbert-Smithconjectureandsomeproblemsmotivatedbyit.ThoserelatedquestionsposedbyNormanEarlSteenrodwere: Conjecture4(Steenrod[ 16 ]). Doesthereexistaninteriormapofamanifoldonaspaceofhigherdimension? Conjecture5(Steenrod[ 16 ]). DoesthereexistalightinteriormapofamanifoldsuchthattheinverseimageofsomepointisaCantorset?Inbothofthesethereadershouldbeawarethatby`interior'themeaningis`open'astheformertermhasfallenoutofpracticeinfavorofthelater.Itshould,also,benotedthatanegativeanswertoeitherwouldalsosufcetoanswertheHilbert-Smithconjectureinthenegative. AAnderson BKeldys CWIlsonFigure2-11. R.D.Anderson,L.V.Keldys,andD.C.WilsonansweredSteenrod'sconjectures Therstconjecturewasshowntoholdtrue,in1956byRichardDavisAnderson,whoannouncedfarmorethanwhatwasaskedbySteenrod[ 2 ]. Theorem2.3(Anderson[ 2 ]). IfforanynC3andmC2,Misann-cellorn-sphereandYisanm-cellorm-sphere,thenthereexistsamonotoneopenmappingofMontoY.(WhileIsay`shown'IcannotndaproofofitpublishedbyR.D.Anderson,thoughLyudmillaVsevoldovnaKeldyspublishedasuchpoofin1957inRussian[ 32 ]).Andthesecondconjecturewassolved(alsointheafrmative)in1973byDavidC.Wilson[ 57 ]andhissolutionalsoexceededthedemandsoftheconjecturemotivatingit: 24

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Theorem2.4(Wilson[ 57 ]). IfM3isacompacttriangulated3-manifoldandmC3,thenthereexistsalightopenmappingofM3ontoImsuchthateachpoint-inversesetishomeomorphictothestandardCantorsetIn1960,Chung-TaoYangextendedtheworkofP.A.Smith,andinparticularmadeuseof`amodiedspecialhomologytheoryofSmithinwhichrealsmodulo1areusedascofcients'[ 58 ].Heprovedifap-adicgroupactseffectivelyonann-dimensionalmanifoldX,thentheorbitspaceX~Gisofhomologydimensionn2.Moreover,ifXismerelyalocallycompactHausdorffspace,thenthehomologydimensionoftheorbitspaceisnomorethann3.Thisresultwasalsoprovenindependently,andinadifferentmanner,byGlenEugeneBredon,FrankRaymond,andRobertFonesWilliamsinajointpaperusingspectralsequencesthoughtheygiveprimacytoCTYangfortheresult[ 12 ]. AYang BRaymond CWilliamsFigure2-12. C.T.Yang,F.Raymond,andR.F.Williamsdeterminedthedimensionofthequotientspaceforanypossiblecounter-exampletotheconjecture Ifacounter-exampletotheHilbert-Smithconjectureweretoexist,thenthequotientbysuchanactionwouldperforcebeadimensionraisingopenmap.Towardsunderstandingwhatsuchactionsmightbelike,RaymondandWilliamsconstructedexamplesofp-adicactionsonspaceswherethequotientmapraiseddimension[ 45 ].Oneexample,donebyWilliams[ 55 ],wasamodicationoftherstdimensionraisingopenmap:theexampleKolmogoroffgavein1933[ 34 ].Kolomogoroff'sexampleis,inreality,thequotientmapofaneffectivedyadicactionontheMengercurvewherethequotient(ororbit)spaceisthePontrjagin2-surface[ 42 ].Williamsmodiedtheconstructioninorderthattheactionwouldbefree(aswellasforanarbitraryprimep)andformalizedafunctor[ 55 ]tomorereadilydescribenotonly 25

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thedimensionallydecientPontrjaginsurfaces,buttheexampleofPontrjagin'sstudentVladimirGrigorevichBoltyanskii.Inalaterpaper[ 56 ],WilliamsdetailsoutaconstructiontocreateforanynC1,ann-dimensionalspaceX,acteduponfreelybyap-adicgroupp,insuchawaythattheorbitspaceX~phasdimensionn1.Moreover,theseexampleshaveasecondproperty,namelythatHn1cU)]TJ /F11 11.955 Tf 10.6 0 Td[(ZpforanyopenU`X~p.ThisissomethingthatmustoccurwheneverXisaPeanocontinuum,andsomethingwhichIwilldirectlyevidencelaterinthispaper,butwhichfollowsfromthespectralsequence.ConstructingaspaceXthathasthep-adicgroup,pactuponitfreelysuchthatdimX~p)]TJ /F8 11.955 Tf 10.02 0 Td[(2dimXisfarmoredifcult.FrankRaymondwasabletoconstructsuchanexample[ 45 ].Now,inRaymond'sexamplethesecondpropertythatWilliamswasabletoobtaininhisexamplesthatjustraisedthedimensionoftheorbitspaceby1didnothold.Infact,neitherwasabletoconstructanexamplewithbothfeatures,andwentsofarastoconjecturethatitwasnotpossiblewhendimX~p@.They,indeed,provedthattheirmethodsofconstructioncouldcertainlynotproducesuchanexampleinanycase,andthisservestohighlightthestringentrequirementsthatsuchap-adicactionwouldhavetohaveinordertoacteffectivelyonamanifold.WilliamsandRaymondobtainedtheirdimensionalboundsbystudyingtheclassifyingspaceforthep-adicgroup,andithasbeenasourceofinterestformanyothersinattackingtheproblem.In1992,SergeiMikhailovichAgeevprovedthatnisn-universalforfreeactionsofp[ 1 ].HeshowedthatforgivenfreepactionsonnandanANRX,suchthatdimX~pBnandforanyp-invariantclosedsubsetA`Xthatanyp-equivariantmappingfA)]TJ /F5 11.955 Tf 12.7 0 Td[(nadmitsanequivariantextensionfX)]TJ /F5 11.955 Tf 12.7 0 Td[(n.Andinthesamearticleconjectured Conjecture6(Ageev). Forany0-dimensionalcompactgroupG,ifnmandnarefreeG-spaces,thenthereisnoequivariantmapmn)]TJ /F5 11.955 Tf 13.15 0 Td[(nimplyingthattheorbitspaceR~p 26

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hasinnitedimension,whereRisanycompactANR-spacewithfreeactionofthegrouppofp-adicintegers.Thiswouldhandleoneofthetwopossiblecasesforthedimensionoftheorbitspaceofamanifoldshoulditadmitafreep-adicaction. ADranishnikov BYangFigure2-13. A.N.DranishnikovandZ.Yangconstructedandexplainednon-dimensionraisingfreep-adicactionsonMengercontinuua Thestudyofp-adicactionsonMengeruniversalspacesnisthenveryreasonableandconnectedtotheproblem.InadditiontoKolmogoroff'sexampleandWilliamsmodicationofittoafreeaction,Alexander(Sasha)DranishnikovproducedafreeactionongeneralMengercompacta[ 15 ]inthelate1980s.NowDranishnikov'sactiononthemengercurvewasdifferentfromthatofKolmogoroff'sinthatthequotientspacewasnotahigherdimensionalspace. APasynkov BBestvina CAndersonFigure2-14. Pictured:B.A.Pasynkov(sitting),M.Bestvina,andR.D.Anderson.Theyfounddescriptionsandcharacterizationsforn Dranishnikov'sactionmadeuseofBorisPasynkov'sconceptofpartialtopologicalproducts,specicallytheiruseindescribingtheuniversalcompactan.ZhiquingYang'spaper[ 59 ]recreatesthisactioninamorereadilyaccessibleversion.ThePasynkovpartialproductdescriptionofncanbeseenfromrstprinciplesinthelaterpaper, 27

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withoutanyknowledgeofpartialtopologicalproducts.Further,YangmakesuseMladenBestivina'scharacterizationforn[ 7 ]ratherthanR.D.Anderson's[ 4 ]whichwasallthatPasynkovhadwithwhichtoworkatthetime.FromthePasynkovpartialproductdescriptionofMengercontinua,itisplainwhysuchcontinuaareanaturalsettingforp-adicactions.Inthisdissertation,Iwilldemonstrateasimilarconstructionof1toobtainafreepactiononthemengercurve 5.4 .Theconstruction,thatIwilluse,willformthemengercurveoutofp-adicsolenoids.ItwillbedifferentfromtheactiongivenbyDranishnikov[ 15 ],buttheconstructioncaneasilybemodiedinordertoproducethatactionifitisdesired.RecenttimeshaveshowninterestingpartialsolutionstotheHilbert-Smithconjecture.Asthehistoryoftheproblemismarkedbysuchgroupingsofadvancements,itisreasonabletoexpectmore,andsimilarpartialsolutionstooccurinthenearfuture.In1997,oneofDranishnikov'sadvisors,EugineVitalievichScepintogetherwithDusanRepovswereabletoputtogetheramostelegantproofthattheHilbert-SmithconjectureholdstrueforactionsbyLipchitzmaps[ 46 ].Theirproofisadimensiontheoreticargumentthatresolvesdowntoastringofhalfadozeninequalitiesthatcanberepresentedinasinglechain.TheproofutilizesthemetricformedusingtheHaarmeasurementionedearlier( 2 )fromthe1930s,JohnForbesNashJr'sembeddingtheoremfromthe1950s,someclassicaldimensiontheoryresults[ 28 ],andC.T.Yang'sboundontheIntegralCohomologicaldimensionofthequotientspace,aswellasevenaBaireCategoryargument.FollowinginthewakeofthisproofarederivativeresultsforHolderactionsandquasi-conformalmaps[ 36 37 ].Justthispastyear,ayoungmathematiciannamedJohnPardonannouncedviaAriv,asolutiontoHilbert-Smithinthecaseofmanifoldsofdimension3[ 40 ].Interestinglyenough,itdoesnotapproachtheproblembyattackingthroughthequotient 28

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AScepin BRepovs CNash DPardonFigure2-15. Pictured:E.V.Scepin,D.Repovs,J.F.Nash,andJ.Pardon.RecentadvancesontheHilbert-SmithConjecture spaceofapotentialpaction,butrathertakesthespiritofGleason'sconceptofgroupswithoutsmallsub-groupsandcombinesitwithknownfactsof3-manifolds. 29

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CHAPTER3COMPACTIFICATIONSWewillshowthatforagivenzero-dimensionalcompactgroupG,thereisonlyonefreeG-action,uptoconjugationbyahomeomorphism,onthespaceofirrationals.Moreover,thisG-actionwillbeasub-actionofanyeffectiveG-actiononacompletemetricspacewheretheperiodicpointscontainnoopensetandtheactionhasnoisolatedorbits.Theseresultsareofinterestsinceanypotentialcounter-exampletotheHilbert-Smithconjecturewouldcontainthisuniquefreeactiononthespaceofirrationalsasasub-action.Weturntoinvestigatehowgroupactionsonnon-compactspacescanbeextended,asgroupactions,tocompacticationsofthosespaces.First,weshowthateveryzero-dimensionalcompactgroupactiononaseparablemetricspacehasatleastonecompacticationtowhichtheactionextends.Second,wegiveconditionstoensurethatsuchagroupactionwillextendforagivencompactication.Requiringthateachgroupelementextendstoahomeomorphismonthecompacticationisgenerallynotsufcienttoensurethatthegroupactionextendstothecompactication.Wehaveexamplesandtheoremsthatillustrateexactlywhatcanhappen.Weprovideastraightforwardexampleofaxed-pointfreeeffectiveactiononaseparablemetricspacethatcannotbeextendednon-triviallytoanactiononanycompactication.WethenconcludethechaptergivinganequivalentformulationoftheHilbert-Smithconjectureinlightoftheresultsfoundherein. 3.1ActionsontheSpaceofIrrationalNumbersThemaingoalofthissectionistoproveTheorem 3.3 .Webeginbyshowingthatthequotientspaceofanycompactzero-dimensionalgroupactiononIishomeomorphictoI.ThisresultwillfollowfromTheorem 3.1 ,whereweshowthattheimageofanyperfectopenmaponIishomeomorphictoI. Lemma3.1. IfX,disacompletemetricspaceandthemappingpX)]TJ /F2 11.955 Tf 12.79 0 Td[(Yisaperfectopensurjection,thenthespaceYistopologicallycomplete. 30

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Proof. Sincepisaperfectmapping,letbethemetricforYinducedbytheHausdorffdistancebetweenpre-imagesofpointsunderpusingthecompletemetricd.Inotherwordsfor,>Y,letA)]TJ /F2 11.955 Tf 9.28 0 Td[(p1,B)]TJ /F2 11.955 Tf 9.28 0 Td[(p1anddene,)]TJ /F2 11.955 Tf 9.28 0 Td[(maxsupa>Ainfb>Bda,b,supb>Binfa>Ada,b.WeshallshowthatY,iscomplete.Letnn)]TJ /F10 7.97 Tf 4.63 0 Td[(1`YbeaCauchysequencewithrespecttothemetric.For)]TJ /F10 7.97 Tf 10.42 2.76 Td[(1~21,thereisanN1A0suchthatn,mCN1impliesthatn,m@1~21.Pickb1>p1N1andobservethattheopenballB1)]TJ /F2 11.955 Tf 9.35 0 Td[(B1 21b1intersectsp1mforeverymAN1.Inductivelypickbn1suchthatNn1ANncorrespondstoachoiceof)]TJ /F10 7.97 Tf 11.68 2.77 Td[(1~2n1andbn1>p1Nn19BnthenobservethattheopenballBn1)]TJ /F2 11.955 Tf 11.32 0 Td[(B1 2n1bn1intersectsp1mforeverymANn1.LetN0)]TJ /F8 11.955 Tf 11.32 0 Td[(0andB0)]TJ /F7 11.955 Tf 11.32 0 Td[(I.Chooseasequenceann)]TJ /F10 7.97 Tf 4.64 0 Td[(1bypickingai)]TJ /F2 11.955 Tf 10.01 0 Td[(bkifi)]TJ /F2 11.955 Tf 10.01 0 Td[(NkforsomekC1andbychoosingai>p1i9BkwhenNk@i@Nk1forsomekC0.Byconstructionthesequenceann)]TJ /F10 7.97 Tf 4.63 0 Td[(1isCauchywithrespecttod.Sincedisacompletemetricthislatersequencehasalimita)]TJ /F8 11.955 Tf 9.89 0 Td[(liman.Let)]TJ /F2 11.955 Tf 9.9 0 Td[(paandobservethat)]TJ /F8 11.955 Tf 9.9 0 Td[(limnandthusthemetricisacompletemetricforY.Sincethereexistsacompletemetriconit,thespaceYistopologicallycomplete. Theorem3.1. IfthemappingpI)]TJ /F2 11.955 Tf 14.18 0 Td[(Yisaperfectopensurjection,thenthespaceYI,thespaceofirrationals. Proof. Hausdorffcharacterizedtheirrationalsasthe0-dimensional,nowherelocallycompact,separablemetricspacethatistopologicallycomplete[ 24 ].Sincepisbothanopenandclosedmapping,anysubsetO`IthatisbothopenandclosedwillhavethepropertythatpOisbothopenandclosed.Sincepisasurjectiontheopen-closedbasisofIismappedtoanopen-closedbasisofY,henceYisa0-dimensionalseparablemetricspace. 31

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Further,sincepisaperfectandlightsurjectionandthespaceIisnowherelocallycompact,theimagespacepI)]TJ /F2 11.955 Tf 9.28 0 Td[(Yisaswell.SincethespaceIistopologicallycomplete(sayviewedasNN)thereisametricdforwhichIiscomplete.ByLemma 3.1 thespaceYistopologicallycomplete.HencebyHausdorff'scharacterizationoftheirrationalsthespaceYI)]TJ /F11 11.955 Tf 9.71 0 Td[(RQasstated. Nowweapplythisresultto0dimensionalcompactmetricgroupactionsonthespaceofirrationals.Pontrjaginshowedthatevery0-dimensionalcompactgroupistheinverselimitofnitegroups[ 44 ].Itiseasytoseethatforagiven0-dimensionalcompactmetricgroupG,eitherGisniteorGhasthetopologyofaCantorset.IfGisnotnite,wecallGaCantorGroup.Ineithercase,wecanwriteG)]TJ /F8 11.955 Tf 10.15 0 Td[(limGi,i1iwiththeconventionsestablishedinsection 1.1.4 .Thezero-dimensionalcompactgroupofp-adicnumbersisaCantorgroupoftheformlimZpi,i1i,hencetheresultsbelowwillholdforthemaswell. Corollary3.1.1. LetG)]TJ /F8 11.955 Tf 9.39 0 Td[(limGi,i1ibeazero-dimensionalcompactgroup.IfthegroupGactsfreelyonthespaceofirrationalsI,thenthequotientspaceI~GI. Proof. LetthemappI)]TJ /F26 7.97 Tf 13.18 2.76 Td[(I~Gdenotethequotientmapinducedbythefreeaction.Themappingpisaperfectopensurjection,sobyTheorem 3.1 thequotientspaceI~GI. Wewillusethisresulttoshowthatforanygivenzero-dimensionalcompactgroupG,thereisonlyonefreeactionbyGonthespaceofirrationals.InthecasewhereGisnite,thisisdoneinonestepandthemaps1initiallyobtainedwillbethemapsthatiseventuallydesiredtodenethehomeomorphisminthestatementofthetheorembelow.OtherwisewhenGisinnite(andhenceaCantorgroup),themapsisobtainedbyasequenceofmapssii)]TJ /F10 7.97 Tf 4.63 0 Td[(1. 32

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Theorem3.2. LetG)]TJ /F8 11.955 Tf 11.85 0 Td[(limGi,i1ibeazero-dimensionalcompactgroup.IfthemappingAGI)]TJ /F7 11.955 Tf 12.68 0 Td[(Iisafreetopologicalgroupaction,thenthereisahomeomorphismhI)]TJ /F7 11.955 Tf 12.64 0 Td[(IGsuchthatAg,w)]TJ /F2 11.955 Tf 9.28 0 Td[(h11hw,g2hw,where1IG)]TJ /F7 11.955 Tf 12.64 0 Td[(Iand2IG)]TJ /F2 11.955 Tf 12.64 0 Td[(Garetheprojectionmaps. Proof. LetG)]TJ /F8 11.955 Tf 9.85 0 Td[(limGi,i1i.ForiA0letKi)]TJ /F1 11.955 Tf 9.86 0 Td[(kerii1VGianddenotebyeitheidentityelementforthegroupKi.ThuswecanwriteG)]TJ /F4 11.955 Tf 11.25 0 Td[(gii)]TJ /F10 7.97 Tf 4.63 0 Td[(1Sgi>Ki.ForeverynA0,letGn)]TJ /F4 11.955 Tf 10.88 0 Td[(gii)]TJ /F10 7.97 Tf 4.63 0 Td[(1>GSgi)]TJ /F2 11.955 Tf 10.89 0 Td[(ei>Kiforalli@nVG.LetB)]TJ /F4 11.955 Tf 10.88 0 Td[(Bnn>Nbeacountableopen-closedbasisofI. Claim1. ForagiveniC1andw>IthereisasetBnw,i>Bsuchthatw>Bnw,iandGiBnw,i9GGiBnw,i)]TJ /F12 11.955 Tf 9.28 0 Td[(g Proof. SinceGactsfreelyonI,thesetGiw9GG1w)]TJ /F12 11.955 Tf 11.04 0 Td[(g.SinceGandGiarebothcompactandBisabasisofIthereisanopen-closedsetB>Bsuchthatw>BbutGGiw9B)]TJ /F12 11.955 Tf 10.09 0 Td[(g.FurtherthereisaBnw,i>Bsuchthatw>Bnw,ibBandGGiB9Bnw,i)]TJ /F12 11.955 Tf 9.96 0 Td[(gthusasGisagroupwehaveGiBnw,i9GGiBnw,i)]TJ /F12 11.955 Tf 9.96 0 Td[(gasclaimed. Sincethepointw>Iwasarbitrary,foragiveniC1thereisamaptiI)]TJ /F11 11.955 Tf 13.93 0 Td[(Nbytiw)]TJ /F2 11.955 Tf 10.01 0 Td[(nw,i>NfromClaim 1 above.Fori)]TJ /F8 11.955 Tf 10.02 0 Td[(1,letN1)]TJ /F4 11.955 Tf 10.01 0 Td[(nj!j)]TJ /F10 7.97 Tf 4.63 0 Td[(1bethestrictlyincreasing(possiblyinnite)sequenceofnaturalnumberssuchthatwehavem>njj)]TJ /F10 7.97 Tf 4.63 0 Td[(1ifandonlyifm>t1I.Nowusingtheorderonthissubsetofthenaturalsandtheopen-closednatureofthesetsBnk,wewilldeneacontinuousmaps1I)]TJ /F2 11.955 Tf 14.92 0 Td[(K1.ForkC1andforallw>GBnkGm@kBnm,lets1w)]TJ /F5 11.955 Tf 11.59 0 Td[(>K1ifandonlyifw>gBnkforsomeg)]TJ /F4 11.955 Tf 10.63 0 Td[(g,ii)]TJ /F10 7.97 Tf 4.63 0 Td[(0>G1)]TJ /F2 11.955 Tf 10.64 0 Td[(Gsuchthatg,1)]TJ /F5 11.955 Tf 10.63 0 Td[(.SinceallofthesetsBnmarebothopenandclosed,soaresetsoftheformGBnkGm@kBnmandthemappings1iscontinuous. 33

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Proceedwiththespaces11e1,formasequenceN2,anduseittodeneamappings2I)]TJ /F2 11.955 Tf 13.02 0 Td[(K2suchthatitisconsistentwithourmappings1andthenproceedinductivelytodenesnI)]TJ /F2 11.955 Tf 12.64 0 Td[(Kn.DenethemapsI)]TJ /F2 11.955 Tf 12.77 0 Td[(Gbysg)]TJ /F2 11.955 Tf 9.4 0 Td[(sgii)]TJ /F10 7.97 Tf 4.63 0 Td[(0)]TJ /F4 11.955 Tf 9.4 0 Td[(sigii)]TJ /F10 7.97 Tf 4.63 0 Td[(0>G,letthemapfI~G)]TJ /F7 11.955 Tf 12.77 0 Td[(Ibethehomeomorphismguaranteedbycorollary 3.1.1 ,andletthemappI)]TJ /F26 7.97 Tf 12.75 2.77 Td[(I~GbetheperfectlightopenquotientmapinducedbytheGaction.DenethemaphI)]TJ /F7 11.955 Tf 12.65 0 Td[(IGbyhw)]TJ /F4 11.955 Tf 9.28 0 Td[(fXpw,sw. Wenowshowthatforagivenzero-dimensionalcompactgroupG,thereisonlyonefreeG-actiononthespaceofirrationals,asallothersareconjugatetotheactionbyhomeomorphisms. Corollary3.2.1. IfAGI)]TJ /F7 11.955 Tf 13.21 0 Td[(Iisafreezero-dimensionalcompactgroupactiononI,thenitisconjugatebyhomeomorphismwiththefreezero-dimensionalcompactgroupactionBGIG)]TJ /F7 11.955 Tf 12.64 0 Td[(IG,whereBisdenedbyBj,w,g)]TJ /F4 11.955 Tf 9.28 0 Td[(w,jXg. Proof. LetG)]TJ /F8 11.955 Tf 9.28 0 Td[(limGi,i1ibeazero-dimensionalcompactgroupthenIIG.DeneBGIG)]TJ /F7 11.955 Tf 13.07 0 Td[(IGbyBj,w,g)]TJ /F4 11.955 Tf 9.69 0 Td[(w,jXg.IfAGI)]TJ /F7 11.955 Tf 13.06 0 Td[(Iisafreezero-dimensionalcompactgroupactiononI,thenitisconjugatebyhomeomorphismwithB. LetGbeazero-dimensionalcompactgroup.NoteveryeffectiveG-actiononaspaceXwillcontain,asasub-action,thefreeG-actiononthespaceofirrationals.AtrivialexamplewouldbewhenXisnite.SlightlylesstrivialwouldbelettingGbeaCantorgroupandthespaceX)]TJ /F2 11.955 Tf 10.42 0 Td[(GwithGactingonXbythegroupoperation.InthissecondexampleI`X,butasX~Gisasingleton,obviouslyIX~G.AnothereasyexamplewillbediscussedlaterwhenitappearsinFigure 5.1 .Toavoidthistypeof 34

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problem,werestrictourattentiontoCantorgroups,andimposerestrictionsontheactionandthespace. Theorem3.3. LetG)]TJ /F8 11.955 Tf 9.99 0 Td[(limGi,i1ibeaCantorgroup.IfXisacompletemetricspaceuponwhichGactseffectivelysuchthatthesetofperiodicpointscontainsnoopensetandtheactionhasnoisolatedorbits,thenthereisasubspaceY`X,withYI,suchthatGactsfreelyonY. Proof. LetX)]TJ /F6 7.97 Tf 12.64 2.77 Td[(X~GbethequotientmapinducedbytheeffectiveactiononXbyG.SinceXisacompletemetricspace,thequotientspaceX~Gisaswell.DenotebyPk`XthecollectionofperiodicpointsofXofperiodkundertheGaction.LetP)]TJ /F17 11.955 Tf 9.38 -.94 Td[(Pk.SinceXhasnoisolatedorbits,thespaceX~GPhasnoisolatedpoints.ThereisasetZ`X~GsuchthatZisadenseGsetandZI.Since,foreachk,thesetPkisclosedandcontainsnoopenset,thesubsetP`X~GisanFsetandX~GPisadenseGset.ThusthesetZP)]TJ /F2 11.955 Tf 9.52 0 Td[(Z9X~GPisadenseGsetandZPI.Sinceisanopenmapping,thesetY)]TJ /F5 11.955 Tf 10.31 0 Td[(1ZPisadenseGsetofthecompletemetricspaceX.SinceYhasnoisolatedpoints,YI.FinallythegroupGactsfreelyuponY,sinceY`XPandY)]TJ /F5 11.955 Tf 9.64 0 Td[(1ZPwasthefullpre-imageofasetinX~G. Ifthereisacounter-exampletotheHilbert-Smithconjecture,thenitwillcontainthisuniquefreep-adicactionontheirrationals.Moreoverthiscounter-examplewillbetheextensionofthatp-adicgroupactiontoamanifoldcompacticationoftheirrationals.Thisobservationiswhyweturntoexamineequivariantlyextendinggroupactionsandunderstandthepotentialobstaclesthatmightbefoundtherein. 3.2EquivariantCompacticationsInthissection,wewillusethefactthataCantorgroupisaninverselimitofnitegroupstoprovideconditionswhenagroupactioncanbeextendedfromapre-compactmetricspacetoacompactmetricspace.Inthecaseofap-adicgroupp,if>pisa 35

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generator,thenthesequenceofsub-groupspipi)]TJ /F10 7.97 Tf 4.63 0 Td[(0isaninnitedescendingnormalsequenceandtheorderofpi1p~pipisp.InthecaseoftheCantorgroupi)]TJ /F10 7.97 Tf 4.63 0 Td[(1Z2,whichwewilluseasanexampleinalatersection,considerthesequenceHii)]TJ /F10 7.97 Tf 4.63 0 Td[(1whereHij)]TJ /F6 7.97 Tf 4.63 0 Td[(iZ2Vi)]TJ /F10 7.97 Tf 4.63 0 Td[(1Z2isthesubgroupwherethersti1coordinatesare0>Z2.ThissequenceisaninnitedescendingnormalsequencewheretheorderofHi1~Hiis2.Thetheorembelowdemonstratesaconstructionforanextensionofagroupactiononaseparablemetricspacetoaneffectivegroupactiononitscompactication.Allthatisrequiredisthechoiceofapre-compactmetricandchoiceofasequenceofniteopencoversofthespace. Theorem3.4. LetG)]TJ /F8 11.955 Tf 9.63 0 Td[(limGi,i1ibeaCantorgroupandXbeaseparablespacewithapre-compactmetricdistX.IfGactseffectivelyonX,thenthereisacompacticationCofXsuchthatGextendstoaneffectiveactiononC. Proof. LetGbethemetricdenedinequation 2 inducedfromthetotallyboundedmetricdistX.LetUii)]TJ /F10 7.97 Tf 4.63 0 Td[(1beasequenceofniteopencoversofX,GsuchthatmeshUn)]TJ /F8 11.955 Tf 12.65 0 Td[(0andUnhUn1.SinceGisaCantorgroup,Ghasaninnitedescendingnormalsequence[ 5 ].Towhit,thereisasequenceGkVGk)]TJ /F10 7.97 Tf 4.63 0 Td[(1suchthatGk)]TJ /F4 11.955 Tf 11.24 0 Td[(eandforeachk>NwehavethatexGkVGk1withindexSGk~Gk1S)]TJ /F12 11.955 Tf 6.39 0 Td[(nk@.SinceGiscompact,takeGk)]TJ /F4 11.955 Tf 9.28 0 Td[(gGkg>GtobethenitecoverofGbyopen-closedcosets.Foreachm>N,deneanitecoverVm)]TJ /F4 11.955 Tf 10.57 0 Td[(HUH>Gm,U>Umandnotethat,obviously,wehaveVmhVm1.PleasealsonotethatwhilebothmeshUn)]TJ /F8 11.955 Tf 14.29 0 Td[(0andmeshGn)]TJ /F8 11.955 Tf 13.1 0 Td[(0,wedonotknowthatmeshVn)]TJ /F8 11.955 Tf 13.09 0 Td[(0and,indeed,itmightnotbethecaseforthegivenpre-compactmetric.LetdenotethecollectionVmm)]TJ /F10 7.97 Tf 4.63 0 Td[(1. Claim1. ThecollectionstronglyseparatespointsinX(foranypairofdistinctpointsx,y>Xonlynitelymanyelementsofcontainboth). 36

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Pickx,y>Xdistinct.Case1:Ifx>Gybutxxy,thenthereisaKA0suchthatxGKy.Letd)]TJ /F8 11.955 Tf 10.11 0 Td[(minGGKy,gGKyg>GGKA0.SincemeshUm)]TJ /F8 11.955 Tf 13.48 0 Td[(0,thereisann>NsuchthatnAKandmeshUn@d~4.Ifx>HU>Vn,thenthereisah>GsuchthatH)]TJ /F2 11.955 Tf 11.24 0 Td[(hGn.NowtheorbitGyispartitionedbyGKythatseparatesthepointsxandy.SinceU>UnanddiamU@d,thenthesetJ>GKJy9UxgisasingletonconsistingofthesolerepresentativememberofGKforthepartitionoftheorbitGKyintersectingwithU.SinceH)]TJ /F2 11.955 Tf 10.47 0 Td[(hGn`hGK>GKmerelypermutesthecosetsofGKwehaveonlyonememberofthepartitionoftheorbitGKyintersectingthesetHU)]TJ /F2 11.955 Tf 10.17 0 Td[(hGnU`hGKUthusx>HUimpliesyHUastheyoccurindifferentmembersofthepartitionoftheirorbitGKy.Case2:IfxGy,thenletd)]TJ /F5 11.955 Tf 9.68 0 Td[(GGx,GyA0.SincemeshUm)]TJ /F8 11.955 Tf 13.04 0 Td[(0,thereisann>NsuchthatmeshUn@d~4.Ifx>HU>Vn,thenGx9Uxg.SinceU>Un,wehavediamU@d~4.ThusGy9U)]TJ /F12 11.955 Tf 9.27 0 Td[(gandlikewiseyHU.ThereforethecollectionstronglyseparatespointsinXasclaimed. Claim2. ThecollectionisabasisforuniformityonX.Foreachn>N,wehave,foreveryU>Un,anopensetV>Vn>suchthatU`V.SinceeachUnisauniformcovering,wehaveeachVnisauniformcovering.SinceVmhVm1forallm>N,wehaveisanormalfamily(infact,formsanormalsequence)and,thusisasub-basisforapre-uniformity[ 29 ].SincepairwiseintersectionsofmembersofanyVmcanberepresentedasunionsoflatermembersof,wehavethatis,infact,abasisratherthanjustasub-basis.Wehaveisabasisforuniformityratherthanjustapre-uniformity,sincegivenanytwopointsx,y>X,thereisacoveringVm,suchthatforeveryV>Vm,ifx>V,thenyV. Claim3. ThegroupGisequi-uniformlycontinuouswithrespectto. 37

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LetbeoneofthegeneratorsofG.Sinceforeachk>NthecoverofGbyGkconsistsofthesubgroupGkofniteindexanditscosets,thecollectionnGkn)]TJ /F10 7.97 Tf 4.63 0 Td[(0isnite.Thusitistriviallytruethatforeachm>NthenumberH-n1i)]TJ /F10 7.97 Tf 4.63 0 Td[(0iVmisboundedforalln>N.SinceisabasisforuniformityonX,thefamilynisequiuniformlycontinuouswithrespectto.LetCbethecompacticationdeterminedbythebasisofuniformity.SinceeachVmisanitecoverandthenormalsequenceofthosecoversseparatespoints,thecompacticationCismetric. Claim4. ThegroupGactseffectivelyonthecompacticationCdeterminedbythebasisofuniformity.Again,letbeoneofthegeneratorsforG.Sincecoverelementspermuteunder,thehomeomorphismextendstoahomeomorphism^C)]TJ /F2 11.955 Tf 14.24 0 Td[(C.Since^formsaequiuniformlycontinuousfamilyandwasarbitrary,theentiregroupGextendstoCandfurthermoreextendsasagroupactiononC. EvenifthegroupactiongiveninTheorem 3.4 happenedtobeafreeaction,theextendedgroupactiononthecompacticationmightnotbefree.Thisdepends,inpart,onthepre-compactmetricgivenforthespace.Wewilldemonstratethispossibilitywiththefollowingexample.Westartbyconstructinganon-compactquotientspaceofaCantorgroupaction.LetthespaceYbethespaceconsistingofthehalf-openinterval0,1togetherwithaninnitesequenceoftangentcircles.ImposeuponYthepre-compactmetricinorderthatthecompacticationofYisthe1pointcompactication.Thusthesequenceoftangentcircleswilllimitdowntothecompacticationpoint0.LetG)]TJ /F8 11.955 Tf 11.32 0 Td[(limGi,i1ibeaCantorgroupactingfreelyonanon-compactmetricspaceXsuchthatY)]TJ /F6 7.97 Tf 9.79 2.76 Td[(X~G.ThereareinnitelymanypossiblespacesXthatwillsatisfy 38

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( acg Figure3-1. Pre-compactquotientspaceYwithagivenpre-compactmetric thatrequirement.However,ifCisacompacticationofXconstructedbyTheorem 3.4 whereC~Gisthe1-pointcompacticationofY,andiftheextendedGactiononCisafreeaction,thenXYG.ToseewhathappenswithsomeoftheotherpossibilitiesforthespaceX,xanontrivialgroupelementg>Ge.Toavoidconfusion,denotethisnewpossibleXbyX.LetX0,1G*Z~,whereZ)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1NG.Allthatremainsistodescribehoweachcopyof)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1isattachedto0,1G.PicturethattraversingoncearoundanycircledownstairsinYliftstoapathinXfromonecopyof0,1toit'simageunderthexedgroupelementg.ApplyingourconstructionfromTheorem 3.4 ,wewillobtainthe1-pointcompacticationofX,andassuchtheextendedgroupactiononitwillbe,perforce,onlymerelyeffectiveratherthanfree.ThisisnottosaythatXdoesnothavecompacticationswhereGextendsfreely,butrathertosaythatwiththispre-compactmetricitfailstoextendfreely.ForexampleifZ)]TJ /F4 11.955 Tf 9.28 0 Td[()]TJ /F8 11.955 Tf 10.32 0 Td[(0,1*S1N8~isacompacticationofY,thenthereisacompactication,CofX,andafreeextensionofGsothatC~GZ.Thisleavesopen,thefollowingquestion,forwhichtherelikelyisapositiveanswer: Question3.1(Maissen). LetG)]TJ /F8 11.955 Tf 9.78 0 Td[(limGi,i1ibeaCantorgroupandXbeaseparablespace.IfGactsfreelyonX,thenisthereacompacticationCofXsuchthatGcanbeextendedtoafreeactiononC? 39

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Forsakeofcompleteness,wementionthattheremainingpossiblechoicesforourspaceXinthisexamplecanbeobtainedbyindividualchoicesofgroupelementsforeachcircleinthequotientspaceY.HopefullythiscanstarttogivesomeinsightinthewayinwhichonecanconstructCantorgroupactions,andthechoicesinvolved.Therearesomeminorobstaclesindoingthisforgroupactionsingeneral,whichweaddressinthefollowingsections. 3.3ExtendingGroupActionsThetheorembelowwillshowsufcientconditionstoguaranteetheextensionofthegroupactionasagroupaction.Later,wewilldiscusspossibleconsequencesfromnotimposingtheseconditions. Theorem3.5. LetXbeametricspace,Gbeacompactmetricgroup,andAGX)]TJ /F2 11.955 Tf 12.64 0 Td[(Xbeatopologicalgroupaction.IfC,dcisametriccompacticationofXsuchthatforallg>GthemapAg,)]TJ /F4 11.955 Tf 2.59 0 Td[()]TJ /F2 11.955 Tf 9.47 0 Td[(gxX)]TJ /F2 11.955 Tf 12.83 0 Td[(Xextendscontinuouslyto^Ag,)]TJ /F4 11.955 Tf 2.59 0 Td[()]TJ /F8 11.955 Tf 9.71 0 Td[(^gxC)]TJ /F2 11.955 Tf 12.83 0 Td[(C,then^AGC)]TJ /F2 11.955 Tf 12.65 0 Td[(Cisacontinuousgroupaction. Proof. Foreaseofnotation,deneforeachg>GthemapgC)]TJ /F2 11.955 Tf 12.64 0 Td[(Cbygx)]TJ /F8 11.955 Tf 10.18 2.66 Td[(^Ag,x.Bywayofacontradiction,supposethat^Aisnotcontinuous.Withthisassumptionwewillshowthatthereisanelementh>Gsuchthatthefunctionh)]TJ /F8 11.955 Tf 10.76 2.65 Td[(^Ah,)]TJ /F4 11.955 Tf 2.58 0 Td[(C)]TJ /F2 11.955 Tf 13.23 0 Td[(Cisnotcontinuous.Since^Aisnotcontinuous,Cismetric,andXisdenseinCthereisasequencegi,xii)]TJ /F10 7.97 Tf 4.63 0 Td[(1`GXsuchthatgi,xi)]TJ /F4 11.955 Tf 12.64 0 Td[(g,x>GCbutgixi)]TJ /F2 11.955 Tf 12.65 0 Td[(gx>C. Claim1. Withoutlossofgeneralitywemayassumeg)]TJ /F2 11.955 Tf 9.28 0 Td[(e>G.Ifthesequenceg1gixi)]TJ /F2 11.955 Tf 13.61 0 Td[(x>C,thengg1gixi)]TJ /F2 11.955 Tf 13.62 0 Td[(gxsinceg>Gimpliesgiscontinuous.Butthengixi)]TJ /F2 11.955 Tf 13.64 0 Td[(gxsincegg1gixi)]TJ /F2 11.955 Tf 10.28 0 Td[(gixiasxi>XandAisagroupactiononX.Thusthesequenceg1gixi)]TJ /F2 11.955 Tf 12.64 0 Td[(x>Candwemayassumethatg)]TJ /F2 11.955 Tf 9.28 0 Td[(e>G. Claim2. Withoutlossofgeneralitywemayassumethatgixiconvergestoyxx. 40

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Sincegixi)]TJ /F2 11.955 Tf 13.11 0 Td[(x>CandCisacompactmetricspacethesequencegixii)]TJ /F10 7.97 Tf 4.63 0 Td[(1hasaconvergentsubsequencesuchthatgikxik)]TJ /F2 11.955 Tf 14.1 0 Td[(y>Cx.Sowecanassumethatgixi)]TJ /F2 11.955 Tf 12.64 0 Td[(yxx. Claim3. Thereisasequenceofgroupelementshj>Gj)]TJ /F10 7.97 Tf 4.63 0 Td[(0,setsWjbXj)]TJ /F10 7.97 Tf 4.63 0 Td[(1,andpointsuj>Xj)]TJ /F10 7.97 Tf 4.63 0 Td[(1,vj>Xj)]TJ /F10 7.97 Tf 4.63 0 Td[(1suchthat 1. diamWj)]TJ /F8 11.955 Tf 12.64 0 Td[(0, 2. uj>Wj9Xandvj>Wj9X,and 3. hjui>Uandhjvi>VforalliBj.Let0@r@dcx,yx0,letn0)]TJ /F8 11.955 Tf 9.33 0 Td[(0,letU)]TJ /F2 11.955 Tf 9.33 0 Td[(Br 2xandletV)]TJ /F2 11.955 Tf 9.33 0 Td[(Br 2y,soU9V)]TJ /F12 11.955 Tf 9.34 0 Td[(g.Forsakeofinduction,wedeneh0)]TJ /F2 11.955 Tf 9.28 0 Td[(e>GandW1)]TJ /F2 11.955 Tf 9.28 0 Td[(Br 2h10x)]TJ /F2 11.955 Tf 9.28 0 Td[(U.Sincexi)]TJ /F2 11.955 Tf 13.23 0 Td[(xandgixi)]TJ /F2 11.955 Tf 13.23 0 Td[(yinC,thereexistsanumberM0A0suchthatwhenevernAM0bothxn>Uandgnxn>V.Letm1)]TJ /F2 11.955 Tf 9.28 0 Td[(M01andletu1)]TJ /F2 11.955 Tf 9.28 0 Td[(h10xm1)]TJ /F2 11.955 Tf 9.27 0 Td[(xm1>U)]TJ /F2 11.955 Tf 9.27 0 Td[(W1.Sincegi)]TJ /F2 11.955 Tf 12.97 0 Td[(einG,forthecompact(nite)setxm1,thereisanumberN1Am1suchthatwhenevernAN1thepointgnxm1>U.Letn1)]TJ /F2 11.955 Tf 9.98 0 Td[(N11andletv1)]TJ /F2 11.955 Tf 9.99 0 Td[(h10xn1)]TJ /F2 11.955 Tf 9.99 0 Td[(xn1>U)]TJ /F2 11.955 Tf 9.28 0 Td[(W1.Leth1)]TJ /F2 11.955 Tf 10.59 0 Td[(gn1Xh0)]TJ /F2 11.955 Tf 10.59 0 Td[(gn1>G,thenh1u1)]TJ /F2 11.955 Tf 10.59 0 Td[(gn1xm1>Uandh1v1)]TJ /F2 11.955 Tf 10.59 0 Td[(gn1xn1>V.SupposeforanumberkC1andall1BjBkthat 1. hj>G 2. Wj)]TJ /F2 11.955 Tf 9.28 0 Td[(Br jh1j1x 3. uj>Wj9Xandvj>Wj9X 4. njAmjAnj1 5. hj1uj)]TJ /F2 11.955 Tf 9.28 0 Td[(xmjandhj1vj)]TJ /F2 11.955 Tf 9.27 0 Td[(xnj 6. hjui>Uforall1BiBjandhjvi>Vforall1BiBjLetWk1)]TJ /F2 11.955 Tf 9.28 0 Td[(Br k1h1kx.Sincegi)]TJ /F2 11.955 Tf 14.61 0 Td[(einG,forthecompact(nite)sethkui,hkvi1BiBkthereisanumberMkAnksuchthatwhenevernAMkthepointh1kxn>Wk1,thepointsgnhkui>Uforall1BiBkandthepointsgnhkvi>Vforall1BiBk.Letmk1)]TJ /F2 11.955 Tf 9.27 0 Td[(Mk1andletuk1)]TJ /F2 11.955 Tf 9.28 0 Td[(h1kxmk1>Wk1. 41

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Sincegi)]TJ /F2 11.955 Tf 13.19 0 Td[(einG,forthecompact(nite)setxmk1thereisanumberNk1Amk1suchthatwhenevernANk1thepointgnxmk1)]TJ /F2 11.955 Tf 10.09 0 Td[(gnhkuk1>U.Letnk1)]TJ /F2 11.955 Tf 10.09 0 Td[(Nk11andletvk1)]TJ /F2 11.955 Tf 9.28 0 Td[(h1kxnk1.Lethk1)]TJ /F2 11.955 Tf 9.81 0 Td[(gnk1Xhk>Gthenhk1ui>Uandhk1vi>Vfor1BiBk1.Thenasclaimedthereisthefollowingforalln>NandallkBn 1. hn>G 2. Wn)]TJ /F2 11.955 Tf 9.28 0 Td[(Br n1h1n1xbX,thusdiamWn)]TJ /F8 11.955 Tf 12.65 0 Td[(0 3. un>Wn9Xandvn>Wn9X 4. hnuk>Uandhnvk>V Claim4. Thereisanh>Gsuchthathx)]TJ /F8 11.955 Tf 11.44 2.65 Td[(^Ah,)]TJ /F4 11.955 Tf 2.58 0 Td[(C)]TJ /F2 11.955 Tf 13.91 0 Td[(Cisnotcontinuouswhichcontradictsourassumptionthat^Aiscontinuous.SinceCisacompactmetricspace,thereisasubsequenceofukk)]TJ /F10 7.97 Tf 4.63 0 Td[(1whichconvergestoasinglepointinC.Withoutlossofgeneralityignoresubindices.Letz>Csuchthatuk)]TJ /F2 11.955 Tf 12.64 0 Td[(zinC.SincediamWn)]TJ /F8 11.955 Tf 12.65 0 Td[(0,wealsohavevk)]TJ /F2 11.955 Tf 12.64 0 Td[(zinCaswell.SinceGisacompactmetricgroup,thesequencehkk)]TJ /F10 7.97 Tf 4.63 0 Td[(1hasaconvergentsubsequencetoanelementinG.Withoutlossofgeneralityignoresubindices.Leth>Gsuchthathk)]TJ /F2 11.955 Tf 12.64 0 Td[(hinG.Foraxedi>Nandthecorrespondingcompact(nite,twopoint)setui,visincehk)]TJ /F2 11.955 Tf 12.69 0 Td[(hinGitfollowsthathkui)]TJ /F2 11.955 Tf 12.69 0 Td[(huiandhkvi)]TJ /F2 11.955 Tf 12.69 0 Td[(hvi.ForkAithepointshkui>Uandhkvi>Vthushui>Uandhvi>V.Sinceh>GweshouldhavehC)]TJ /F2 11.955 Tf 14.35 0 Td[(Cbeingcontinuous,sohui)]TJ /F2 11.955 Tf 14.35 0 Td[(hz>Uandhvi)]TJ /F2 11.955 Tf 13.63 0 Td[(hz>V.Thuswehavehz>U9V)]TJ /F12 11.955 Tf 10.27 0 Td[(gwhichmeansthathC)]TJ /F2 11.955 Tf 13.62 0 Td[(Cisdiscontinuousasclaimed.Thisisacontradictionandthusitmustbethecasethat^Aiscontinuousasdesired. Whilethepriorsectiondemonstratedaconstructionforchoosingacompacticationtowhichthegroupactionwouldperforceextend,withTheorem 3.5 wehavecriteriato 42

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guaranteeanextensiontoacompacticationofourchoice.Inthenextsectionofthechapterbelow,wewillexploreandjustifythecriteriawerequireinthetheorem. 3.4ExamplesOnemightthinkthatsimplyrequiringeveryelementofagroupactionuponaspacetoextendwouldbesufcienttoforcethegroupactiontoextend.However,weimposebyTheorem 3.5 twoadditionalcriteria,namelythatthegroupbecompactandthatthecompacticationbemetric.Wenowgiveanexampleofasimplespace,provideeffectivegroupactionsonthatspace,anddemonstratethattheywillnotextendwhenoneofthecriteriafromthetheoremisnotmet.ConsiderthespaceX)]TJ /F11 11.955 Tf 11.11 0 Td[(NZ2,whereNdenotesthenaturalnumbersandZ2thegroupoftwoelements.LetCbeacompacticationofNandletY)]TJ /F2 11.955 Tf 10.08 0 Td[(CZ2bethecorrespondingcompacticationofX. Z21YYYY...CN10YYYY...CN01234...NCNFigure3-2. CompacticationofNZ2,whichwillbereferredtoasthespaceY Wewillrstshowthattheconditionthatthegroupbecompactisnotaspuriouschoice.WewilldeneaneffectivegroupactiononXbyanon-compactgroup,andshowthatthisgroupactiondoesnotextendtoanycompactication,Y,denedabove.Deneforeachi>N,thehomeomorphismfiX)]TJ /F2 11.955 Tf 12.65 0 Td[(Xgivenby:fin,z)]TJ /F4 11.955 Tf 9.28 21.11 Td[(n,z`Z21n)]TJ /F2 11.955 Tf 9.27 0 Td[(in,znxi 43

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LeteX)]TJ /F2 11.955 Tf 12.68 0 Td[(Xbetheidentity,andletGwbethegroupgeneratedbyfii)]TJ /F10 7.97 Tf 4.63 0 Td[(1(inotherwordsthecountableweakproductofZ2actions).ForanycompactsetK,letNK)]TJ /F2 11.955 Tf 10.5 0 Td[(maxn>NSn,z>K,thenforiANKwehavefiSK)]TJ /F2 11.955 Tf 10.64 0 Td[(e.Wehavethatfi)]TJ /F2 11.955 Tf 14.01 0 Td[(ewiththetopologygeneratedbyconvergenceoncompactsets.Foreachi>Nthehomeomorphismfiextendstoa^fiY)]TJ /F2 11.955 Tf 12.64 0 Td[(Yby:^fi)]TJ /F4 11.955 Tf 9.28 21.11 Td[(fi>X>YXLet^eY)]TJ /F2 11.955 Tf 13.6 0 Td[(YbetheidentityonY.Theextension,^fi,isalsoahomeomorphismsincethefunctionfiisahomeomorphism,thecomposition^fiX^fi)]TJ /F8 11.955 Tf 9.44 0 Td[(^e,andforanyneighborhoodNyofYXwhereNy`Yn,zSnBi,z>Z2wehave^fiSNy)]TJ /F8 11.955 Tf 11.49 0 Td[(^eand,thus^fiiscontinuous.Sinceeveryg>Gwisthenitecompositionofelementsfromfii)]TJ /F10 7.97 Tf 4.64 0 Td[(1,themapgX)]TJ /F2 11.955 Tf 12.65 0 Td[(Xextendstoahomeomorphism^gY)]TJ /F2 11.955 Tf 12.64 0 Td[(Yaswell.Yet,inthesequence^fii)]TJ /F10 7.97 Tf 4.63 0 Td[(1,wedonothave^fi)]TJ /F8 11.955 Tf 13.37 0 Td[(^e.Toseethis,consider>YXandanyneighborhood,N`Y,of.ForeveryN>N,thereisannNANsuchthatnN,z>Nforsomez>Z2.Now^fnNnN,zx^enN,z)]TJ /F4 11.955 Tf 9.28 0 Td[(nN,z,solim^fix^e.EventhougheveryelementofthegroupGwextendstoahomeomorphismfromYtoitselfandthegroupstructureonGwismaintained,thetopologyofthegroupactionisnot.ItispossibletodeneextensionsofGwtosomecompacticationsofX,butnotthosecompacticationsintheformthatwedenedforY.GivenaYdenedabove,deneZ)]TJ /F6 7.97 Tf 9.28 2.77 Td[(Y~byc,zc,zifandonlyifc)]TJ /F2 11.955 Tf 9.28 0 Td[(c>CN.ThegroupactionGwextendsasagroupactiontoZ,whereforeveryf>GwwehavefSZX1Z,theidentityonthespaceZ.Next,wewilljustifyourdecisiontorequirealsothatthecompacticationbeametriccompactication.Letusconsideralargergroup,acompactgroupthatwasmentionedearlier,namelytheCantorgroupGsi)]TJ /F10 7.97 Tf 4.63 0 Td[(1Z2.Forevery)]TJ /F4 11.955 Tf 11.64 0 Td[(i>i)]TJ /F10 7.97 Tf 4.64 0 Td[(1Z2,denefX)]TJ /F2 11.955 Tf 12.82 0 Td[(Xbyfn,z)]TJ /F4 11.955 Tf 9.45 0 Td[(n,z`Z2n.ThesefunctionsyieldaGsgroupactiononX.If,for 44

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every>Gs,thehomeomorphismfextendstoahomeomorphism^fY)]TJ /F2 11.955 Tf 13.13 0 Td[(Y,thenwewillshowthatC)]TJ /F5 11.955 Tf 9.28 0 Td[(N,theStone-CechcompacticationofN.Foreach)]TJ /F4 11.955 Tf 10.43 0 Td[(i>i)]TJ /F10 7.97 Tf 4.63 0 Td[(1Z2,deneafunctionFN)]TJ /F4 11.955 Tf 13.8 0 Td[(0,1byFn)]TJ /F5 11.955 Tf 10.44 0 Td[(n.SinceC0isopeninYandsincefextendstoahomeomorphism^f,thereisacontinuousextension^FC)]TJ /F4 11.955 Tf 12.65 0 Td[(0,1givenby^F)]TJ /F4 11.955 Tf 9.28 33.06 Td[(F>N0>CN,^f,0>C01>CN,^f,0>C1SinceeveryfunctionFN)]TJ /F4 11.955 Tf 13.83 0 Td[(0,1extendscontinuouslytoafunction^FC)]TJ /F4 11.955 Tf 13.83 0 Td[(0,1bydenitionoftheStone-Cechcompactication,wehavethatthecompacticationC)]TJ /F5 11.955 Tf 9.66 0 Td[(N.SinceY)]TJ /F2 11.955 Tf 9.66 0 Td[(CZ2,wehaveYN.SincetheonlycompactsetsofNthataremetrizablearenitesets,theorbitsofacompactgroupactiononitwouldbeperforcetrivial.Considertheextension^fY)]TJ /F2 11.955 Tf 13.81 0 Td[(Yoffwhere)]TJ /F4 11.955 Tf 10.45 0 Td[(1i)]TJ /F10 7.97 Tf 4.64 0 Td[(1whichwouldperforcehavetoswapCN0withCN1.ThusGsdoesnotactonYasagroupaction.Again,aswasthecasewithGwactingonX,thereareextensionsofthegroupactionGstocompacticationsofX,butwecannotmandatethatthosecompacticationsbeoftheformgiventoY.SinceGsisaCantorgroup,wecanuseTheorem 3.4 toconstructacompactication,Z,ofthespaceXwherethereisanextensionofthegivenGsgroupactiontoaGsgroupactiononZ.ThiscompacticationwillbetheZ)]TJ /F6 7.97 Tf 9.7 2.77 Td[(Y~thatwasdenedbyc,zc,zifandonlyifc)]TJ /F2 11.955 Tf 10.3 0 Td[(c>CNthatwedescribedearlierinthesection.TherewillbeanextensionofthegivenfreeGsactiononXtoaGs-actiononZ.Forevery>Gs,weextendthehomeomorphismfX)]TJ /F2 11.955 Tf 13.23 0 Td[(Xtothehomeomorphism^fZ)]TJ /F2 11.955 Tf 12.65 0 Td[(Zby^f)]TJ /F5 11.955 Tf 9.28 0 Td[(forevery>ZX.MoreovertheuseofZ2asafactorofXinourexamplecouldhavebeenreplacedbyamoreinterestingcompactgroupsuchasthep-adicnumbersorsomeotherCantor 45

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groupwithoutdifculty.Likewise,givenazero-dimensionalcompactgroupactiononaseparablemetricspace,itisnothardtoseeanexampleofthiskindoccurringwithinitasasub-action.Thustheobstaclesthatthissimpleexamplehighlightsare,insomesense,fundamental.Thisexampledemonstratestheimportanceofunderstandinghowgroupactionsonspacescanbeextendedtogroupactionsoncompacticationsofthosespaces.EveryeffectiveCantorgroupactiononacompletemetricspacethatdoesnothaveanopensetofperiodicpointsandthattheactiondoesnothaveisolatedorbits,isanextensionoftheuniquefreeaction,bythatCantorgroup,onthespaceofirrationals.Sincethegroupofp-adicnumbersisjustaCantorgroup,ifacounter-exampletotheHilbert-Smithconjectureweretoexist,wehaveshownthatthiscounter-examplewouldhavetobeanextensionoftheuniquefreep-adicactiononthespaceofirrationals.ThustheHilbert-Smithconjecturecanbeviewedasaquestiononthepossibleextensionsofthisonefreep-adicactionofthespaceofirrationalstomanifoldcompactications. 3.5RingsofContinuousFunctions AGillman BJerison CHenriksenFigure3-3. L.Gillman,M.Jerison,andM.HenriksenranthePurdueseminarthatleadtothefamousbookonringsofcontinuousfunctions Since,intheearliersectionsofthischapter,wehaveestablishedtheHilbert-SmithConjectureasaquestionofextensionsofagroupactionfromametricspacetoamanifoldcompacticationofit,itisnaturaltoseekasettinginwhichcompacticationsandextendingmapstothemarereasonablywell-suited.ForthereaderunfamiliarwithRingsofContinuousFunctions,Irecommendthewell-knownbooksimilarlyentitledbyLeonardGillmanandMeyerJerison[ 20 ].Itisaveryusefulbookthatsprangfrom 46

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aseminaratPurduein1954-1955thatwasorganizedbyMelvinHenriksen.TherootsofthesubjectlieintheworksofMarshallHarveyStone[ 50 ],EdwinHewitt[ 25 ],andKolmogorov'spaperwithhisstudentIzrailMoisevichGelfand[ 19 ]. AStone BHewitt CKolmogoroff DGelfandFigure3-4. M.H.Stone,E.Hewitt,A.N.Kolmogoroff,andI.M.Gelfandlaidthefoundationsforstudyingringsofcontinuousfunctions Denition3.1. LetXbeacompletelyregulartopologicalspace.DeneCX)]TJ /F4 11.955 Tf 10.54 0 Td[(fX)]TJ /F11 11.955 Tf 13.6 0 Td[(RSfiscontinuousandCX`CXbethesubsetofboundedfunctions.FromtheringstructureonR,thecollectionsCXandCXbothinheritaringstructure.WecallCXtheringofcontinuousfunctionsonthespaceX,andCXtheringofboundedcontinuousfunctionsonX.TheringCXofboundedcontinuousfunctionsencodesallofthetopologicalinformationofthecompletelyregulartopologicalspaceX.Thefollowingtheoremshelptoillustratethis: Theorem3.6(Gelfand-Kolmogorov[ 19 ]). ThemaximalidealsinCXareinonetoonecorrespondencewiththepointsp>X,theStone-CechcompacticationofX. Theorem3.7([ 20 ]). IfaclosedsubringF`CXseparatespointsinX,thenthereisacompacticationCFofXsuchthatCCFF.Moreover,ifthesubringFisseparable,thenthecompacticationCFismetric.Conversely,ifthereisacompacticationCofXthenthereisaclosedsubringF`CXthatgeneratesthetopologywhereCCF.WewillusethenotationCFtodenotethecompacticationassociatedwiththeclosedsubringF(andnotethatCCXX).Fromthetwotheorems,itbecomesapparentthattheringofboundedcontinuousfunctionsonthespaceofirrationalsCIisanaturalsettingforourproblem. 47

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3.5.1GroupActionsonCXandaMetriconCXEverycontinuousfunctionhX)]TJ /F2 11.955 Tf 14.74 0 Td[(XnaturallyinducesaringhomomorphismhCXCX.IfGX)]TJ /F2 11.955 Tf 13.89 0 Td[(Xisatopologicalgroupaction,thenforeachg>GthereisacorrespondinggCXCX.Onemightthinkthatitwouldbenaturaltoimposethesup-normmetriconCX,howeverthiswindsupbeingproblematic.Foronething,theGactiononXdoesnotinduceatopologicalgroupactiononCXequippedwiththesup-normtopologyevenifitperforceretainsitsalgebraicstructure.WecaninsteaddeneanothermetriconCX,basedonanormtakenovermaximalideals: Denition3.2. Foreveryf>CX,denedf,0)]TJ /F4 11.955 Tf 11.39 0 Td[(YfY)]TJ /F8 11.955 Tf 11.39 0 Td[(supSMfS,wherethesupremumistakenoverallmaximalidealsMinCX.Underthismetric,thegroupactionGinducesagroupactionGonCX.Moreover,iftheGactiononXiseffective(free),thentheinducedGactioniseffective(free)onCX.Givenaseparable,closedsubringF`CIthatgeneratesthetopologyitsufces,inlightofTheorem 3.5 ,merelythatFispinvariant.Inthiscase,eachofthefunctionsgI)]TJ /F7 11.955 Tf 14.55 0 Td[(IinthecompactgrouppwouldextendtothemetriccompacticationCF.Thus,byTheorem 3.5 ,thegroupactionpI)]TJ /F7 11.955 Tf 13.67 0 Td[(IwouldextendasagroupactiontopCF)]TJ /F2 11.955 Tf 13.63 0 Td[(CF.WeneednotconsiderametricinwhichpactsuponF,butratherjustneedtocheckforeachg>pthatFg`F. 3.5.2DimensioninCXTheringofboundedcontinuousfunctionsCXencodesthetopologyofthespaceX,andrecognizingthedimensionofXfromit'sboundedringofcontinuousfunctionsissomethingthatisofinteresttous.Inordertodothis,wemustrstintroduceafewdenitionsthatwillbeusedindeningwhatisknownastheanalyticdimensionofasubringofCX. Denition3.3(Analyticsubring[ 20 ]). AsubringA`CXisananalyticsubringif 48

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1. allconstantfunctionsbelongtoA;and 2. f2>Aimpliesf>A. Denition3.4. LetBbeanysubfamilyofCX.LetAbetheintersectionofallanalyticsubringsofCXcontainingB.Sincetheintersectionofanalyticsubringsareanalytic,Aisananalyticsubring.ThefamilyBiscalledtheanalyticbaseforA.Inthissetting,thenotionofanalyticdimensionwasdenedbyMiroslavKatetov. Figure3-5. M.Katetovwasnotonlyafamousmathematician,butwasalsoaninternationalchessmasterandthePraguechampionin1942and1946. Denition3.5(AnalyticDimension[ 31 ]). LetXbeacompletelyregularspace.Theanalyticdimension(orKatetovdimension),adCX,istheleastcardinalmsuchthateverycountablefamilyinCXiscontainedinananalyticsubringhavingabaseofpowerBm.Fromthefollowingtheorem,weseethatanalyticdimensionofCXwillcoincidewiththeLebesguecoveringdimensionofX. Theorem3.8(Katetov[ 20 ]). ThefollowingareequivalentforanycompletelyregularspaceX. 1. dimXBn, 2. adCXBn, 3. EverynitesubfamilyofCXiscontainedinananalyticsubringhavingabaseofcardinalBn. 3.5.3Hilbert-SmithinCXWithinthisframework,wecanrestatetheHilbert-SmithConjectureasfollows: Conjecture7(Hilbert-Smith). LetpI)]TJ /F7 11.955 Tf 14.18 0 Td[(Ibethefreep-adicgroupactiononthespaceofirrationals.IfF`CIisaclosed,separablesubringthatgeneratesthe 49

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topology,hasanalyticdimensionn>N,andisgeneratedbynfunctionsfini)]TJ /F10 7.97 Tf 4.63 0 Td[(1,thenFisnotpinvariant.ShouldsuchaclosedsubringFexist,thenthecorrespondingcompacticationCFwouldimbedinInusingthesetfini)]TJ /F10 7.97 Tf 4.63 0 Td[(1.SinceadF)]TJ /F2 11.955 Tf 10.64 0 Td[(nimpliesdimCF)]TJ /F2 11.955 Tf 10.64 0 Td[(n,thesetCF`Inxg.RestrictingtheextendedgroupactiontopCF)]TJ /F8 11.955 Tf 14.12 2.66 Td[(CFandthenchoosingasufcientlysmallnon-trivialsubgroupofpwouldproducethecounter-exampletoHilbert-Smith.Conversely,shouldtherebeacounter-exampletoHilbert-Smithforsomedimensionn,thentherewouldbeaclosed,separablesubringF`CIwiththeaboveproperties.Ibelievethatinthissettingtheproblemwillprovetobemoretractableandallowforacomplete,ratherthanmerelypartial,solution.TranslatingproblemsintoRingsofContinuousFunctionshasprovedinthepasttobeanicemediumforeasysolutionstoharderproblems[ 20 ],anditisourhopethatitwillhappenagain.Afterall,onecantranslatethequotientspacesX~ktosubringsofCXthatdonotseparatekorbits,whichseemsaneasierpackagethantheoriginalwhichwouldbeacollectionofdimensionallydecientPeanocontinua.Further,itwouldbewonderfultohighlighttheusefulnesstotopologyforwhichtheseringsclearlyholdthepotentialtodeliver. 50

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CHAPTER4PEANOSPACES 4.1EquivariantPartitionsofPeanoContinuaAPeanocontinuum,X,isdenedasacompact,connected,locallyconnectedmetricspace.SincesatisfyingthesecriteriaisequivalenttosayingthatthespaceXisthecontinuousimageofanarc,Peanocontinuaarealsoreferredtoascontinuouscurves.In1949,R.H.Bingprovedthateverycontinuouscurveispartitionable[ 8 ].Ifthereisaneffectivep-adicgroupactiononaPeanocontinuumX,thenanaturalquestionistoask,whetherornottherearepartitionsofXthatrespectthegroupaction.Thissectiondemonstratesthat,foreveryA0,thespaceXcanbepartitionedbypartitionablesetsofdiameterlessthansuchthatthegroupactionmerelynitelypermutesthesesets.Webeginwiththeproofofanelementarytheoremthatisbynomeansanythingnew,andcanbefoundinelementarytextbooks(e.g.Theorem3.7.2inEngelking[ 17 ]). Theorem4.1. IffX)]TJ /F2 11.955 Tf 12.79 0 Td[(Yisaperfectmapping,thesetA`Xisclosed,thesetB`Yiscompact,andfA)]TJ /F2 11.955 Tf 9.28 0 Td[(B,thenthesetAiscompact. Proof. LetC)]TJ /F4 11.955 Tf 9.28 0 Td[(xiN`Abeaninnitesubset,andletD)]TJ /F2 11.955 Tf 9.28 0 Td[(fC.Case1:ThesetDisnite.Thereisay>Bsuchthatfxi)]TJ /F2 11.955 Tf 10.25 0 Td[(yforinnitelymanyvaluesofi.Sincefxisaperfectmapping,thesetf1yiscompact.Byassumption,thesetAisclosed,sotheintersectionA9f1yiscompact,whichimpliesthatthesetCadmitsalimitpointinA.Case2:ThesetDisinnite.SinceBiscompact,thesetDadmitsalimitpoint,y>B.Suppose,bywayofcontradiction,CadmitsnolimitpointsinA.SinceAisaclosedset,thesetCisalsoclosedinA,andthusinX.LetE)]TJ /F2 11.955 Tf 10.76 0 Td[(f1y9A,whichisnon-emptyandcompactsincepxisaperfectmapping,fA)]TJ /F2 11.955 Tf 9.4 0 Td[(B,andAisclosed.ThesetC9Emustbenite,soCEisinniteandclosedinX.Sincepxisaperfectmap, 51

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pxisaclosedmap;andthesetpCE)]TJ /F2 11.955 Tf 10.14 0 Td[(Dyisclosed.ThisisacontradictionwiththepointybeingalimitpointofD.IneithercasethesetCadmitsalimitpointinA,sothesetAiscompact. Corollary4.1.1. IffX)]TJ /F2 11.955 Tf 12.95 0 Td[(YisaperfectmappingandB`Yiscompact,thenf1Biscompact. Proof. LetA)]TJ /F2 11.955 Tf 9.93 0 Td[(f1B`X.Sincefxiscontinuous,thesetA`Xisclosed,andthusthesetAiscompactbyTheorem 4.1 ,. Lemma4.1. LetXandYbeconnected,locallyconnectedmetricspaces.IffX)]TJ /F2 11.955 Tf 13.29 0 Td[(Yisaperfect,lightopenmapandU`YisopenwiththepropertythatUiscompactandlocallyconnected,thenV)]TJ /F2 11.955 Tf 9.28 0 Td[(f1Uhasnitelymanycomponents. Proof. Pickapointy>U.Sincefxiscontinuous,thesetV`Xisopen.SinceXislocallyconnected,foreachpointx>W)]TJ /F2 11.955 Tf 10.6 0 Td[(f1y`V,thereisaconnectedopensetOx`V.Sincefxisperfect,thesetWiscompact.ThecollectionOxisanopencoverofW,sothereareanitesub-collectionofopensetsOxiwhichcoverW.SinceeachOxiisconnected,therearenitelymanycomponentsofVcontainingpointsofW,whichimpliesthateachofthosecomponentsisbothopenandclosedinV.Sincefxisanopenmap,theimageofeachcomponentisopen.Likewise,sincefxisaperfectmapping,itisaclosedmap,whichimpliestheimageofeachcomponentisalsoclosed.SinceUisconnected,theimageofeachcomponentisallofU.Thus,thepre-imageVhasonlynitelymanycomponents. Theorem4.2. LetXandYbeconnected,locallyconnectedmetricspaces.IffX)]TJ /F2 11.955 Tf 12.88 0 Td[(Yisaperfect,lightopenmapandU`Yisopen,withthepropertythatUiscompactandlocallyconnected,thenf1Uisalsocompactandlocallyconnected. 52

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Proof. FromTheorem 4.1 ,thepre-imageV)]TJ /F2 11.955 Tf 11.28 0 Td[(f1Uiscompact.Thus,itsufcestoshowthatVhaspropertyS.ByBing'spartitioningtheorems[ 8 ],thisconditionisequivalenttoVbeingpartitionable.SinceUiscompactandlocallyconnected,UhaspropertyS,andisthuspartitionable.LetG)]TJ /F4 11.955 Tf 9.91 0 Td[(gibeapartitioningofU.FromLemma 4.1 ,thenitecollectionh`VShisacomponentoff1gi,forsomegi>Gformsapartitioning,H,ofV.ItsufcestoshowthatthesizeoftheelementsofpartitionsofVcanbecontrolled.Chooseapointy>U.Sincefxisalightmapping,thesetf1yis0-dimensional,thustotallydisconnected.GivenA0;coverf1ybyanitenumberofdisjointopensetsofdiameterlessthan 3.DenotethiscoverbyJi.Sincefxisanopenmap,considertheopenneighborhoodKy)]TJ /F17 11.955 Tf 10.58 -.94 Td[(fJiofthepointy.Thesetf1Ky`Jiisopen,containsthepointsf1y,andthecomponentsoff1Kyhavediameterlessthan 3.SincethepointywasanarbitrarilychosenpointinU,formanopencoverKyy>UofU.SinceUiscompact,takeanitesub-coverofKy,anddenoteitbyKi.LetA0betheminimumdistancebetweennon-intersectingmembersoftheniteclosedcoverKi.LetGbea-partitionofU,andletHbetheassociatedpartitionofVunderthemappingfx.Sinceanymember,g>G,willlieinthestarofthenitesub-cover,Ki,thecollectionHisatmostanpartitionofV.Sincewasarbitrary,thesetVispartitionablethusVhaspropertyS,whichimpliesthatitislocallyconnectedasdesired. Inthespeciccaseofap-adicgroupactingonaPeanocontinuum,Theorem 4.2 providesameanstoconstructequivariantpartitions.Justasthegroupofp-adicnumberscanbeseenasaninverselimitofnitep-powergroups,ap-adicactiononaPeanocontinuumcanbeapproximatedbynitepermutations.ByCorollary 4.2.1 below, 53

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anypossiblecounter-exampletotheHilbert-Smithconjecturewouldadmitareningsequenceofpartitionseachwithacorrespondingniteactiononitsmembers. Corollary4.2.1(EquivariantPartitions). Ifthegroupofp-adicnumbers,pactsonaPeanocontinuumX,thenforeachA0,thereisan-partitionofX,wherethegrouppactsonthemembersofthepartitionbynitepermutation. Proof. Sincethequotientmap0X)]TJ /F6 7.97 Tf 13.11 2.77 Td[(X~piscontinuousandXisaPeanocontinuum,thequotientspaceY)]TJ /F6 7.97 Tf 11.26 2.77 Td[(X~pisalsoaPeanocontinuum.Moreover,themap0isaperfect,lightopenmap,andbothXandYareconnected,locallyconnected,compactmetricspaces,sothecriteriaforTheorem 4.2 andLemma 4.1 aresatised.SinceYisaPeanocontinuum,byBing'spartitioningtheorems[ 8 ],thespaceYcanbepartitioned.LetG)]TJ /F4 11.955 Tf 10.49 0 Td[(gibeapartitionofY.FromLemma 4.1 ,thenitecollectionH)]TJ /F4 11.955 Tf 9.77 0 Td[(h`XShisacomponentof10gi,forsomegi>GformsapartitioningofX.Foragivengi>G,thefullpre-imagehi)]TJ /F5 11.955 Tf 10.58 0 Td[(10gi`Xisxedbythegroupp.Since,byLemma 4.1 ,eachhihasonlynitelymanycomponents,thegrouppnitelypermutesthecomponentsofhi,andthusthemembersofthepartitionH.JustasintheproofofTheorem 4.2 ,thesizeofelementsoftheseinducedpartitionsofXcanbecontrolled,andtheresultfollows. Itiseasytogeneralizethisresult(andothers)tomorearbitraryzero-dimensionalcompactgroups.Recallthatwecanwriteanyzero-dimensionalcompactgroupasC)]TJ /F8 11.955 Tf 10.83 0 Td[(limCi, i1iandthatthegroupofp-adicnumbersisaCantorgroupoftheformp)]TJ /F8 11.955 Tf 9.28 0 Td[(limZpn,n1n. Corollary4.2.2. IfC)]TJ /F8 11.955 Tf 10.45 0 Td[(limCi, i1iisazero-dimensionalcompactgroupactingonaPeanocontinuumX,thenforeachA0,thereisan-partitionofX,wherethegroupCactsonthemembersofthepartitionbynitepermutation. 4.2LiftingArcsandHomotopiesWhilesomepreliminaryworkinthissectionisdoneinmoreabstraction,themotivationisstillthesettingwhereap-adicgroup,p,actsonaspaceX.Whilethe 54

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resultswillalsoholdforanyzero-dimensionalcompactgroup,thefocuswillbewhenthegroupisap-adicgroup.AlthoughallofthemapspnX~n)]TJ /F6 7.97 Tf 13.2 2.77 Td[(X~n1arecoveringorbranchedcoveringmaps,theprojection0X)]TJ /F6 7.97 Tf 14.46 2.77 Td[(X~pisnot,ingeneral,acoveringmap.Nevertheless,somestandardresultsoncoveringspacescanstillbeobtained.ItisalwayspossibletoliftanarcfromthequotientspaceX~ptoanarcinX.Thereisanisomorphismbetweenthehigherhomotopygroups,thusnXX~pforallnC2. Theorem4.3. IfpX)]TJ /F2 11.955 Tf 13.48 0 Td[(Yisalightopenperfectmap,andA`Yisanarcwith>A,thenforanyg>p1thereisanarcA`Xsuchthatg>A,pA)]TJ /F2 11.955 Tf 10.42 0 Td[(A,andpSAisanembedding. Proof. LetA`Ybeanarcwithendpointsand.Picka>p1.SinceAisanarc,Aischainable.LetTnn>N)]TJ /F4 11.955 Tf 11.17 0 Td[(Dk,n`YS1BkBknn>NbeanestedsequenceofchainsofopenconnectedsetscoveringAsuchthatDm,n1`Dk,nDk1,nimpliesthatDm,n19Dkj,n)]TJ /F12 11.955 Tf 9.61 0 Td[(gforalljA0,8Tn1`8Tn,and8Tn)]TJ /F2 11.955 Tf 9.61 0 Td[(A.Foreachn>NandkBkn,letUk,n`XbesuchthatUk,nisconnected,pUk,n)]TJ /F2 11.955 Tf 10.27 0 Td[(Dk,n,andthecollectionSn)]TJ /F4 11.955 Tf 10.07 0 Td[(Uk,nS1BkBknisachainofconnectedsetscontainingthepointareningbutnotcrookedinthechainSn1.LetA)]TJ /F17 11.955 Tf 9.74 -.94 Td[(8Sn.Sincea>8Snforalln>N,a>Axg.SinceSnn>Nisasequenceofnon-crookedchainswithmeshgoingtozero,Aisanarc.AndsincepUk,n)]TJ /F2 11.955 Tf 9.51 0 Td[(Dk,nforallkBkn,itfollowsthatp8Sn)]TJ /F12 11.955 Tf 9.28 0 Td[(8Tn,thuspA)]TJ /F2 11.955 Tf 9.28 0 Td[(p8Sn)]TJ /F17 11.955 Tf 9.27 -.94 Td[(p8Sn)]TJ /F17 11.955 Tf 9.28 -.94 Td[(8Tn)]TJ /F2 11.955 Tf 9.28 0 Td[(A.SincepSAisonetooneandpisperfect,therestrictionpSAisanembedding. ThisisessentiallyarestatementofaTheorembyWhyburn,withadifferentproofandmoremodernterminology. Theorem4.4(WhyburnX.2.1[ 54 ]). LetTA)]TJ /F2 11.955 Tf 10.96 0 Td[(Bbealightinteriortransformation,whereAiscompact.ThenifpqisanysimplearcinBandp0isanypointinT1p,thereexistsasimplearcp0q0inAsuchthatTp0q0)]TJ /F2 11.955 Tf 9.28 0 Td[(pqandTistopologicalonp0q0. 55

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Thisshowsusthatarcscanbeliftedfromthequotientspace. Corollary4.4.1. Ifap-adicgroup,p,actsonaspaceX,thesetA`X~pisanarc,andthemap0X)]TJ /F6 7.97 Tf 13.25 2.77 Td[(X~pisthequotientmapinducedbythegroupaction,thenthereisanarcA`XsuchthatpA)]TJ /F2 11.955 Tf 9.28 0 Td[(Aand0SAisanembedding.Inthecaseofafreeaction,astrongerstatementispossible. Theorem4.5. IfpactsfreelyonXandif0X)]TJ /F6 7.97 Tf 12.8 2.76 Td[(X~pisthequotientmap,thenforanyarcA`X~p,thepre-image10Ap)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1. Proof. Leth)]TJ /F8 11.955 Tf 4.55 0 Td[(0,10X~pbeaparameterizationofthearcA)]TJ /F2 11.955 Tf 9.28 0 Td[(h)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1`X~p.Foreachpointa>h10,thereisanarcAa`X,byCorollary 4.4.1 .Denethemapha)]TJ /F8 11.955 Tf 4.55 0 Td[(0,10Xsothathat>10ht9Aa.Since0SAaisanembedding,themaphaisawell-denedembedding.Fixthepointa>h10,andlet~A)]TJ /F12 11.955 Tf 10.36 0 Td[(8g>pgAa.SinceAa`~A,theset0~A)]TJ /F2 11.955 Tf 10.36 0 Td[(A.Ifapointx>10A,then0x>Athusthereisaparametertx>)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1suchthat0x)]TJ /F2 11.955 Tf 10.28 0 Td[(htx.Lety)]TJ /F2 11.955 Tf 10.28 0 Td[(hatx>Aa.Since0y)]TJ /F5 11.955 Tf 10.28 0 Td[(0hatx)]TJ /F2 11.955 Tf 10.28 0 Td[(htx)]TJ /F5 11.955 Tf 10.29 0 Td[(0x,thereisagx>psuchthatgxy)]TJ /F2 11.955 Tf 9.28 0 Td[(x.SincegxAa`~A,thepointx>~Aandthus~A)]TJ /F5 11.955 Tf 9.28 0 Td[(10A.Suppose,forsakeofcontradiction,thereisanontrivialelementg>psuchthatthereisapointx>Aa9gAa.Letta>)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1suchthathata)]TJ /F2 11.955 Tf 11.34 0 Td[(x,andhencesuchthathta)]TJ /F5 11.955 Tf 11.41 0 Td[(0x.Sincegisahomeomorphism,thereisaparameterizationhg)]TJ /F8 11.955 Tf 4.56 0 Td[(0,1)]TJ /F2 11.955 Tf 14.13 0 Td[(gAasuchthatghat)]TJ /F2 11.955 Tf 10.76 0 Td[(hgt.Since0gz)]TJ /F2 11.955 Tf 10.76 0 Td[(pzforallz>X,itfollowsthat0hat)]TJ /F5 11.955 Tf 9.99 0 Td[(0ghat)]TJ /F5 11.955 Tf 9.98 0 Td[(0hgtforallt>)]TJ /F8 11.955 Tf 4.56 0 Td[(0,1.Lettb>)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1suchthatx)]TJ /F2 11.955 Tf 10.42 0 Td[(hgtb.Sincehisanembeddingandhta)]TJ /F5 11.955 Tf 10.42 0 Td[(0x)]TJ /F5 11.955 Tf 10.42 0 Td[(0hgtb)]TJ /F5 11.955 Tf 10.42 0 Td[(0ghatb)]TJ /F5 11.955 Tf -443.05 -23.91 Td[(0hatb)]TJ /F2 11.955 Tf 9.43 0 Td[(htb,thentheparameterta)]TJ /F2 11.955 Tf 9.43 0 Td[(tb.Sincex)]TJ /F2 11.955 Tf 9.43 0 Td[(hgtb)]TJ /F2 11.955 Tf 9.43 0 Td[(hgta)]TJ /F2 11.955 Tf 9.43 0 Td[(ghata)]TJ /F2 11.955 Tf 9.43 0 Td[(gxandgisnon-trivial,thisisacontradictionwithpactingfreely.SincetheintersectionAa9gAa)]TJ /F12 11.955 Tf 9.28 0 Td[(gforeveryg>pe,thepre-image10Ap)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1. Whilethequotientmap0X)]TJ /F6 7.97 Tf 13.53 2.77 Td[(X~pisnotacoveringmap(ingeneral),themapspnX~n)]TJ /F6 7.97 Tf 13.17 2.76 Td[(X~n1arep-foldbranchedcovers.Givenamappinghintothequotientspace 56

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X~p,asequenceofliftsofhtothequotientspacesX~ncanbeconstructedinmanycasesinordertoobtainaliftingofhtoX. Theorem4.6. Ifap-adicgroup,p,actsonaspaceX,themap0X)]TJ /F6 7.97 Tf 14.04 2.77 Td[(X~pisthequotientmapinducedbythegroupaction,andthemaphK)]TJ /F6 7.97 Tf 12.74 2.77 Td[(X~pisacontinuousmapfromasimplyconnectedcompactspaceK,thenthereisaliftofthismap^hK)]TJ /F2 11.955 Tf 12.66 0 Td[(Xsuchthat0X^h)]TJ /F2 11.955 Tf 9.28 0 Td[(h. Proof. Sincethemaphiscontinuous,thespaceKiscompact,andthemap0isperfect,thesetY)]TJ /F5 11.955 Tf 10.37 0 Td[(10hKiscompact.Letbeageneratorofthegroupp.TherestrictionSYisuniformlycontinuous,andpnSYn)]TJ /F3 7.97 Tf 7.05 0 Td[()]TJ /F8 11.955 Tf 28.36 0 Td[(1Yuniformly.Pickabasepointk0>Kandthenxabasepointx!>10hk0`Y`X.Fromtheinversesequenceofp-foldcoveringmapsonthequotientspacesX~n(labeled( 1 )onpage 16 )thereis,forourbasepointx!>X,acorrespondingsequencexnn)]TJ /F10 7.97 Tf 4.63 0 Td[(0suchthatx0)]TJ /F2 11.955 Tf 10.42 0 Td[(hk0,nx!)]TJ /F2 11.955 Tf 10.43 0 Td[(xn,andpnxn)]TJ /F2 11.955 Tf 10.42 0 Td[(xn1foralln>N.Likewisewiththischoicexed,sinceKissimplyconnectedthereisasequenceofmapsHnK)]TJ /F6 7.97 Tf 13.46 2.76 Td[(X~nn)]TJ /F10 7.97 Tf 4.63 0 Td[(0suchthatHnk0)]TJ /F2 11.955 Tf 9.28 0 Td[(xn>X~nandPnXHn)]TJ /F2 11.955 Tf 9.28 0 Td[(h.Dene^hK)]TJ /F2 11.955 Tf 12.69 0 Td[(Y`Xby^hk>1nHnk,foranyk>K.Themap^hiswell-dened,since limn)]TJ /F3 7.97 Tf 7.06 0 Td[(diam1nHnk)]TJ /F8 11.955 Tf 9.28 0 Td[(0, theset1n1Hn1k`1nHnk,foreachn>N,and theset1nHnkiscompact,foreachn>N.Themap^hiscontinuousbecausethemapHNiscontinuousandthatforanyA0,thereisanNA0suchthatYpNSYY@~3. Corollary4.6.1. Ifap-adicgroup,p,actsonaspaceX,themap0X)]TJ /F6 7.97 Tf 14.44 2.77 Td[(X~pisthequotientmapinducedbythegroupaction,themaph)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F6 7.97 Tf 13.46 2.77 Td[(X~pisapathinthequotientspace,andthemapH)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F6 7.97 Tf 12.8 2.77 Td[(X~pisahomotopyinthequotientspacewhereht)]TJ /F2 11.955 Tf 10.81 0 Td[(Ht,0foreacht>)]TJ /F8 11.955 Tf 4.56 0 Td[(0,1,thenthereisapath^h)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F2 11.955 Tf 14.18 0 Td[(Xsuchthat 57

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0X^h)]TJ /F2 11.955 Tf 9.34 0 Td[(handahomotopy^H)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F2 11.955 Tf 12.71 0 Td[(Xsuchthat^ht)]TJ /F8 11.955 Tf 10.9 2.66 Td[(^Ht,0foreacht>)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1and0X^H)]TJ /F2 11.955 Tf 9.28 0 Td[(H. Proof. Boththespaces)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1and)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1arecompactsimplyconnectedspaces. Corollary4.6.2. Ifap-adicgroup,p,actsonaspaceX,thenforanynC2thereisanisomorphismofthehigherhomotopygroupsnXnX~p. Proof. ForanynC2,thespaceSniscompactandsimplyconnected. 58

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CHAPTER5EXAMPLESANDINVARIANTSETS 5.1SimpleEffectiveANRAction Example5.1. Aneffectivep-adicactionona1-dimensionalANR.InthestandardconstructionoftheCantorseton)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1,theCantorsetisformedbytheremovalofopenintervals,whichwillbereferredtohereastheCantorintervalsthatareorderedbylength.FormtheCantorTreebytakinglinesegmentsofslopes1and1fromthepoints0,0,1,0,andtheendpointsoftheCantorintervalson)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1inorderasshownherewithhighestvertexbeingthepoint1 2,1 2: )]TJ 9.96 9.96 Td[()]TJ 9.97 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.97 Td[()]TJ 9.96 9.96 Td[()]TJ 9.97 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.97 Td[()]TJ 9.96 9.96 Td[()]TJ 9.97 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.97 Td[()]TJ 9.96 9.96 Td[()]TJ 9.97 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.97 Td[()]TJ 9.97 9.96 Td[(@@@@@@@@@@@@@@@@@@@@@@@@)]TJ -9.97 -9.96 Td[()]TJ -9.96 -9.97 Td[()]TJ -9.96 -9.96 Td[()]TJ -9.96 -9.96 Td[()]TJ -9.97 -9.96 Td[()]TJ -219.17 9.96 Td[(@@)]TJ -9.96 -9.96 Td[()]TJ 179.33 9.96 Td[(@@)]TJ -9.96 -9.96 Td[()]TJ /F8 11.955 Tf -232.28 115.63 Td[(0000100000101001111011100101110111Figure5-1. CantorTree(withrstthreestagesofconstructionlabeled) ContinueinthismannerforeachendpointoftheCantorintervalsontheinterval)]TJ /F8 11.955 Tf 4.55 .01 Td[(0,1inorderandthentaketheclosuretoobtaintheCantorTree,T.TheintersectionoftheCantorTreewiththex-axiswillbethestandardCantorsetandTwillliewithinthetrianglewithvertices0,0,1 2,1 2,1,0. 59

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Claim5.0.1. TheCantorTreeTisanANR Proof. ItsufcestoshowthatTisaretractofthetrianglewithvertices0,0,1 2,1 2,1,0.BegintheretractbypushingtheCantorintervalsintotheinteriorofthetrianglesoastomisstheCantorTreeasshownbelow: )]TJ 9.96 9.97 Td[()]TJ 9.97 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.97 Td[()]TJ 9.97 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.97 Td[()]TJ 9.97 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.97 Td[()]TJ 9.96 9.96 Td[()]TJ 9.97 9.96 Td[()]TJ 9.96 9.96 Td[()]TJ 9.96 9.97 Td[()]TJ 9.97 9.96 Td[(@@@@@@@@@@@@@@@@@@@@@@@@)]TJ -9.97 -9.97 Td[()]TJ -9.96 -9.96 Td[()]TJ -9.96 -9.96 Td[()]TJ -9.96 -9.96 Td[()]TJ -9.97 -9.97 Td[()]TJ -219.17 9.97 Td[(@@)]TJ -9.96 -9.97 Td[()]TJ 179.33 9.97 Td[(@@)]TJ -9.96 -9.97 Td[()]TJ /F28 9.963 Tf -199.46 49.62 Td[(' $ 6@@I)]TJ 9.96 9.96 Td[()1000( 6@I)1000( 6@I)999(Figure5-2. DeformationretractionofatriangletotheCantorTree Theremainderoftheretractisthenextendingradiallyfromthesecurvestothetreeitselfasthearrowsintheabovediagramindicate.DothisforeachCantorintervalandthetriangleisretractedontoT,thustheCantorTreeisanANR. Theorem5.1. ThereisaneffectivedyadicactionontheCantorTreeTintheplane. Proof. Asillustratedbygure 5.1 beginbylabelingthehalfopenarcsinTbetweenadjacentvertices,notincludingthevertexwiththegreaterycoordinatewhichwillbereferredtoasthe'higher'vertex.Letthehalfopenarcfrom1 2,1 2downto1 6,1 6belabeledas0andthehalfopenarcperpendiculartoitat1 2,1 2goingdownto5 6,5 6belabeledas1.Ingenerallabelahalfopenarcbetweenadjacentverticeswiththeconcatenationofthelabelofthehighervertexinfrontofeithera0ora1.Thisrightmostdigitisdependentonwhethertheslopeofthehalfopenarcbeinglabeledisparallelor 60

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perpendiculartothehalfopenarccontainingthehighervertex.Thuslabelthehalfopenarcsfrom1 6,1 6downto1 18,1 18and5 18,5 18with00and01respectivelyandlabelthehalfopenarcsfrom5 6,5 6to13 18,13 18and17 18,17 18with11and10respectivelyandsoon.ContinueinthismannertolabelallofthearcsinTandallowtheinnitesequencestolabeltheintersectionofTandthex-axis(i.e.thestandardCantorseton)]TJ /F8 11.955 Tf 4.56 0 Td[(0,1)asthelimitofthearcswithcommonstem.Deneasuccessorfunctiononthelabelsthatisconsistentwithbinaryaddingmachineswithrightcarry-overthatpreservesthelengthofthelabel.Denethenitestringofn0sastheimmediatesuccessortothestringofn1s.Thesuccessorfunctiononinnitesequencesisthenormaladdingmachinewithrightcarry-overthathastheinnitestringof0sastheimmediatesuccessortotheinnitestringof1s.Denotethesuccessorofalabelxibyxi1.DeneasurjectivehomeomorphismT)]TJ /F2 11.955 Tf 12.65 0 Td[(Tbythefollowing:1)Ift)]TJ /F4 11.955 Tf 9.28 0 Td[(x,0>Twithlabelxi1thent)]TJ /F4 11.955 Tf 9.28 0 Td[(xi1.2)Ift)]TJ /F4 11.955 Tf 9.67 0 Td[(x,y>TwithyA0andt>AwhereAisahalfopenarcwithlabelxin1forsomenA0thentisthepointonthehalfopenarcBwithlabelxin11withthesameycoordinate.3)Thepoint1 2,1 2isxedby.ThishomeomorphismgeneratesadyadicactiononTwhichisfreeonthecantorsetC)]TJ /F2 11.955 Tf 9.28 0 Td[(T9)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1andthusisaneffectiveactiononT. Corollary5.1.1. Therearep-adicactionson1-dimensionalANRs Proof. TheCantorTreeis1-dimensional,isanANRfromClaim 5.0.1 ,andadmitsaneffectivedyadicactionfromTheorem 5.1 61

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5.2MengerCurveActionAsmentionedearlier,anaturalsettingforp-adicactionsistheMengercurve,andthehigherdimensionalanalogsn.TherstdemonstrationofadimensionraisingopenmapwhichwasgivenbyKolmogoroffwasthequotientmapofaneffective2-adicactionon1[ 34 ].ThereisacorrespondingfreeactionthatWilliamsconstructedbaseduponKomogoroff'sexample[ 55 ],but1(andningeneral)admitsanotherp-adicactionwherethequotientmapdoesnotraisedimension[ 15 ]. 5.2.1PasnynkovPartialProductDescriptionofnWhilemostpeoplearefamiliarwiththeconstructionbySolomonLefschetz[ 35 ]oftheuniversalk-dimensionalspacen,theconstructionbyBorisA.Pasnykov[ 41 ]isdifferentandmuchmoreusefulinthissetting.Thislaterdescriptionisbasedupontheconcept,duetoPasnykov,ofaPartitalTopologicalProduct: Denition5.2. (PasynkovPartialProduct[ 7 15 41 49 ])LetXbeacompactmetricspaceandletU)]TJ /F4 11.955 Tf 11.48 0 Td[(Ui`Xi>NbeacountablebasisforXwithdiamUi)]TJ /F8 11.955 Tf 14.84 0 Td[(0.LetF)]TJ /F4 11.955 Tf 10.72 0 Td[(Fii>Nbeacountablefamilyofnitesetsofatleasttwoelementseach.DeneP0X,U,F)]TJ /F2 11.955 Tf 9.28 0 Td[(Xandforeachr>NdenePrX,U,F)]TJ /F4 11.955 Tf 9.28 0 Td[(XrMi)]TJ /F10 7.97 Tf 4.63 0 Td[(1Fi~~wheretherelationshipx,f1,f2,...,fi,...,fr~x,f1,f2,...,fi,...,frholdsfor1BiBrandallfi,fi>FiifxUi.Theprojectionr1Pr1X,U,F)]TJ /F7 11.955 Tf 13.2 0 Td[(PrX,U,Fisinducedbytheprojectionr1Lr1i)]TJ /F10 7.97 Tf 4.63 0 Td[(1Fi)]TJ /F17 11.955 Tf 13.18 -.95 Td[(Lri)]TJ /F10 7.97 Tf 4.63 0 Td[(1Fi.ThePasynkovpartialproductofthecollectionFoverXwithrespecttoUisthendenedtobetheinverselimitPX,U,F)]TJ /F8 11.955 Tf 9.28 0 Td[(limPrX,U,F,r 5.2.2Dranishnikov'sActiononnIn[ 15 ]A.N.DranishnikovusedPasynkov'spartialproductdescriptionofntoprove: 62

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Theorem5.2. (Dranishnikov[ 15 ])OneverycompactMengermanifoldthereexistsafreeactionofthep-adicgrouppand,whatismore,ofanyzero-dimensionalcompactgroup.ZhiqingYangpresentedthisconstructioninaslightlymoreaccessiblemanner[ 59 ]whichwillbeenumeratedhere.Toconstructtheuniversalspacen,Yangdoesthefollowingsteps: 1. Startwithn1disjointcollectionsofcountabledensesubsetsCi`I. 2. ForanynitesetF`IdeneLF)]TJ /F17 11.955 Tf 9.28 -.94 Td[(nk)]TJ /F10 7.97 Tf 4.63 0 Td[(1x1,x2,...,xnSxk>F`In. 3. Takeapositivegeometricallydecreasingsequencekk)]TJ /F10 7.97 Tf 4.63 0 Td[(1 4. Foreach1BiBn1deneFki`Ci`Itobeank~ n-densenitesubsetwithFkiFk1i. 5. DeneLki)]TJ /F2 11.955 Tf 9.28 0 Td[(LFki`In.FinallyhethentakesthespaceX)]TJ /F2 11.955 Tf 11.31 0 Td[(Inpn1anddenesarelationRgivenbyx,r1,r2,...,rn1Rx,r1,r2,...,rn1ifforsomekC1andsome1BjBn1wehavex>Lkj,r1>krj`prj,andri)]TJ /F2 11.955 Tf 9.38 0 Td[(riforallixj.YangthenshowsthatthequotientspaceX~RnusingBestvina'scharacterizationofn[ 7 ].Thebeautyofthisconstructionisthatthepactionisnaturaltoseeasthediagonalactiononthepfactors.ThisisequivalenttotheactionrstdescribedbyDranishnikov[ 15 ]wherewearesimplytakingadvantageofthefactthatlimZpk)]TJ /F8 11.955 Tf 11.02 0 Td[(p.ThismakesiteasiertovisualizewhatisoccurringwiththePasynkovpartialproduct.Itisquitereasonabletotrytovisualizeiteitherway;infactZhiqingYanginshowingthatthequotientspacesarenbythecharacterization,demarcatesacollectionofsubspacesbyWm`nwhichareinfactthePmspacesofthePasynkovpartialproductconstruction.Bestvina,inhisdissertation[ 7 ],alsonotesthisidenticationasahelpfulobservationwherehenotesthatthebondingmapWm1)]TJ /F2 11.955 Tf 12.64 0 Td[(Wmissimplyaretraction.Thisconstructioninthecasewhenn)]TJ /F8 11.955 Tf 10.84 0 Td[(1isofparticularinterestinthatasimilarconstructionwillbemade,thistimewithanambientpactioninthefollowingsection.Likewisefamiliaritywithitwillproveusefulinshowingthenegativetowhatwould 63

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otherwisebeareasonablequestioninregardstohigherdimensionaluniversalcurvesnthatwillalsofollowinthischapter. 5.3InvariantSetsWhiletheresultsofthissectionextendtomorearbitraryCantorgroups,theattentionisrestrictedtop-adicgroups.Shouldacounter-exampleexisttotheconjecture,thenitwouldperforcehavecertainproperties.Theaimhereistoaddtothoseknownproperties.Inthissection,theactionisassumedtobeafreeaction.Thisassumptioncouldbelessenedtoanactionthatisfreeexceptonasetthatnowherelocallyseparatesthespacewithoutsignicantlylesseningtheresults.Whiletherequirementinthetheoremsbelowthatanitenumberofarcsnowherelocallyseparatethequotientspacemightseemlikeanimposingrestriction,inthecontextoftryingtondacounter-examplefortheHilbert-Smithconjecture,itisatrivialrequirement.Itisalreadyknownthatsuchacounter-examplewouldhavetobeamanifoldofdimensionfourorhigher,andassuchcannotbelocallyseparatedbyanyonedimensionalsubset.Thepairoflemmasbelowserveastheworkhorseforthissection.Theyshowthat,betweenanytwopoints,xandy,onecanconstructanarcJthatcanbemadetonotonlyavoidtheremainderoftheorbitsofitsendpoints,butalsotoavoiditsownorbit(excludingthepossibilitythatxisintheorbitofy).Theremainderofthesectiondemonstrateswhatcanbemadefromsuchconstructionsbylookingatthefullorbitoftheseconstructedarcs.Thesectionconcludeswithaconstructionofap-adicinvariantMengercurvefromthesearcs. Lemma5.1. Ifap-adicgroup,pactsfreelyonaspaceX,thenforanyx,y>XsuchthatthereisaconnectedpropertySsub-spaceY`Xwithx,y>YwhosequotientY~pcannotbelocallyseparatedbyanitenumberofarcsthereis,foreverysubgrouphBp,anarcJ0Xfromxtosomez>hywithJ9px8py)]TJ /F4 11.955 Tf 11.42 0 Td[(x,zand 64

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gJ9J`x,zforeveryg>p.Moreover,ifL`X~pisaniteunionofarcs,thenthearcJcanbechosensuchthat0J9L`0x,0y. Proof. Givenx,y>Y`XandhBp.LethpY)]TJ /F6 7.97 Tf 12.94 2.77 Td[(Y~hand0pY)]TJ /F6 7.97 Tf 12.94 2.77 Td[(Y~pbethequotientmapsinducedbytheactionofp.SinceY~G)]TJ /F6 7.97 Tf 9.79 2.76 Td[(GY~GforanygroupGactingonX,theymaybeusedinterchangeablyinthisproof.SinceYisconnectedwithpropertyS,thequotientspacesY~pandY~hareaswell.Thereisaph-to-1coveringmapPhY~h)]TJ /F6 7.97 Tf 12.99 2.77 Td[(Y~p,sincepisactingfreelyonX(andhenceonpY`X).Ifthepointx>hy,thenthetrivial,degeneratearcwouldsufce,sowithoutlossofgeneralityassumethatxhy.Denotebyx1)]TJ /F5 11.955 Tf 10.33 0 Td[(hxandx2)]TJ /F5 11.955 Tf 10.33 0 Td[(hy.Sincexhy,immediatelyx1xx2.SincePhisacoveringmapandY~hhaspropertyS,thereisaA0suchthat,foreveryw>Y~h,thedistancedw,mwCforall1Bm@ph.SinceY~phaspropertyS,thereisabrickpartition,ofY~psuchthat,foreachN>,thepullbackP1hNhasexactlyphcomponentseachofwhichhavingdiameterlessthaninY~h.MoreoverthecollectionofthesecomponentsformstheinducedbrickpartitionofY~hwhereeachpartitionmemberM>hasexactlyorderphunderthefreeZphgroupactioninducedbyp.Thiscanbedonesothatx1>IntMforsomeuniqueM>.SincethespaceY~hisconnectedandhaspropertyS,thespaceisarcwiseconnected.LetMini)]TJ /F10 7.97 Tf 4.64 0 Td[(1`beaminimal(withrespectton)chain,inorder,connectingthepointsx1tox2.Thusx1>M1,x1MiforiA1,x2>Mn,x2Mjforj@n,andif@Mk9@Mjxg,thenSkjSB1.SincepactsonasafreeZphaction,forany0@k@phandany1Bi,jBn,ifkMj9Micontainsanon-emptyopenset,thenkMj)]TJ /F2 11.955 Tf 9.28 0 Td[(Mi.LetB0)]TJ /F4 11.955 Tf 9.6 0 Td[(x1.For1BiBn,letBi`@Mi9@Mi1bethelargestsetsuchthatifb>Biandb>@M,theneitherM)]TJ /F2 11.955 Tf 10.1 0 Td[(MiorM)]TJ /F2 11.955 Tf 10.1 0 Td[(Mi1.Sinceisabrickpartition,eachsetBiisnon-emptyandalocalseparator. 65

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LetBi)]TJ /F17 11.955 Tf 9.28 -.94 Td[(ij)]TJ /F10 7.97 Tf 4.63 0 Td[(1Bj.LetK0)]TJ /F4 11.955 Tf 11.3 0 Td[(x2andL0)]TJ /F4 11.955 Tf 11.3 0 Td[(lK0ph1l)]TJ /F10 7.97 Tf 4.64 0 Td[(1.SinceL0isanitecollectionofpoints,itisnotalocalseparatorthusB1L0isnon-empty.PickK1tobeanarcinM1fromthepointk0)]TJ /F2 11.955 Tf 11.21 0 Td[(x1tosomeotherpointk1>B1L0suchthatK19B1)]TJ /F4 11.955 Tf 11.21 0 Td[(k1.DeneL1)]TJ /F2 11.955 Tf 10.06 0 Td[(L08ph1l)]TJ /F10 7.97 Tf 4.64 0 Td[(1lK1.SinceL1L0`Y~hM1,K19L0)]TJ /F12 11.955 Tf 10.06 0 Td[(g,andK19@M1)]TJ /F4 11.955 Tf 10.06 0 Td[(k1,theintersectionL19K1)]TJ /F12 11.955 Tf 9.28 0 Td[(g.Suppose,forsomekwith1Bk@n,thereare,forall1BiBk,arcsKi`Migoingfromki1>Bi1toki>BisuchthatthesetofobstaclesLk)]TJ /F17 11.955 Tf 9.31 -.94 Td[(kj)]TJ /F10 7.97 Tf 4.64 0 Td[(0ph1l)]TJ /F10 7.97 Tf 4.63 0 Td[(1lKjconsistsofanitenumberofdisjointarcsandisolatedpoints,theunionKk)]TJ /F17 11.955 Tf 9.53 -.94 Td[(kj)]TJ /F10 7.97 Tf 4.63 0 Td[(1Kjisanarcfromx0tokk,theintersectionofthetwoKk9Lk)]TJ /F12 11.955 Tf 9.28 0 Td[(g,andKk9Bk)]TJ /F4 11.955 Tf 9.28 0 Td[(kjkj)]TJ /F10 7.97 Tf 4.63 0 Td[(1.SinceLkL09Bn1bph1l)]TJ /F10 7.97 Tf 4.63 0 Td[(1lKk9Bn1)]TJ /F6 7.97 Tf 9.28 12.03 Td[(ph1l)]TJ /F10 7.97 Tf 4.63 0 Td[(1lKk9Bk)]TJ /F6 7.97 Tf 9.28 12.03 Td[(ph1l)]TJ /F10 7.97 Tf 4.63 0 Td[(1kj)]TJ /F10 7.97 Tf 4.63 0 Td[(1lkjisanitesetofpoints,itisnotalocalseparator.TheremainderofthesetBk1Lkisnon-empty.Pickkk1>Bk1Lk.SinceLkisanitenumberofarcsandisolatedpoints,itdoesnotlocallyseparateMk1.DeneKk1tobeanarcfromkk>Bktokk1>Bk1suchthatKk19Lk)]TJ /F12 11.955 Tf 10.35 0 Td[(gandKk19Bk1)]TJ /F4 11.955 Tf 9.28 0 Td[(kk,kk1.DenotebyLk1)]TJ /F2 11.955 Tf 9.28 0 Td[(Lk8ph1l)]TJ /F10 7.97 Tf 4.63 0 Td[(1lKk1.ObservethatLk19Kk1)]TJ /F12 11.955 Tf 9.28 0 Td[(gfollowsfromtheunionLk8Kkbeingthefullp-orbitofKk.Byniteinduction,deneK)]TJ /F2 11.955 Tf 9.28 0 Td[(Kn,wheretheendpointknistakentobex2.LetJbethearc(obtainedfromCorollary 4.4.1 )thatistheliftofKstartingatthepointx>Yandthelemmaisdone.Moreover,givensomenitesetofobstaclearcsL,simplyrealizethatthesetYL80x,0yhaspropertySandhasthesamelocalseparatorconditionssoYL80x,0ycouldhavebeenusedintheproofinsteadofY. 66

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Whilethepriorlemmacouldonlydrawanarcwithinasubgroupofthedesiredtarget,thefollowingonesequentiallyusesthersttoobtainanarcgoingtotheexactdesiredendpoint. Lemma5.2. Ifap-adicgroup,pactsfreelyonaspaceX,thenforanyx,y>XsuchthatthereisaconnectedpropertySsub-spaceY`Xwithx,y>YwhosequotientY~pcannotbelocallyseparatedbyanitenumberofarcs,thereisanarcJ0XfromxtoywithJ9px8py)]TJ /F4 11.955 Tf 9.29 0 Td[(x,yandgJ9J`x,yforeveryg>p.Moreover,ifL`X~pisaniteunionofarcs,thenthearcJcanbechosensuchthat0J9L`0x,0y. Proof. Letx,y>Y`Xasabove.Thestatementistrivialifx)]TJ /F2 11.955 Tf 11.4 0 Td[(y,sowithoutlossofgeneralityassumethatxxyandletd0)]TJ /F2 11.955 Tf 11.25 0 Td[(dx,y.SinceYhaspropertyS,Yispartitionable.Let0bead0~2-partitionofYsuchthaty>Y1forsomeY1>0.Pickasubgrouph0Bpsuchthath0y`Y1.ByLemma 5.1 ,thereisanarcJ0fromxtosomepointy0>h0y`Y1.DeneJ0tobeasub-arcofJ0fromxto@Y1suchthatJ09@Y1)]TJ /F12 11.955 Tf 6.39 0 Td[(x1isasingleton.Letd1)]TJ /F8 11.955 Tf 9.86 0 Td[(mind0 2,dx1,y.SinceY1haspropertyS,thereisad1 2-partition,1ofY1,suchthaty>Y2forsomeY2>1.Pickasubgrouph1Bh0suchthath1y`Y2andagainapplyLemma 5.1 toobtainanarcJ1fromx1tosomepointy1>h1y`Y2suchthatJ09pJ1)]TJ /F4 11.955 Tf 9.28 0 Td[(x1.NowdeneJ1tobethearcJ0togetherwiththesub-arcofJ1fromx1to@Y2suchthatJ19@Y2isasingleton,x2andrepeatthisprocessadinnitumtoobtainasequenceofarcsJkk)]TJ /F10 7.97 Tf 4.63 0 Td[(0,diametersdkk)]TJ /F10 7.97 Tf 4.63 0 Td[(0,andendpointsykk)]TJ /F10 7.97 Tf 4.63 0 Td[(0thereof.Sincethesequencedkissummable,thelimitJ)]TJ /F8 11.955 Tf 10.19 0 Td[(limJnisanarc.Sinceyk)]TJ /F2 11.955 Tf 13.56 0 Td[(y,thearcJrunsfromxtoy.Byconstruction,thearcJhasthedesiredproperties. Applyingthislemmawhentheendpointyisintheorbitofxandthenseeingthefullorbitoftheresultingarc,someinterestingsub-continuacanbeformeduponwhichp,perforce,actsfreely. 67

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Theorem5.3. Ifap-adicgroup,pactsfreelyonaspaceXwiththepropertythatforeverypairofpointsx,y>XthereexistsaPeanocontinuumY`XcontainingbothpointsandthequotientY~pcannotbelocallyseparatedbyanitenumberofarcs,thenforeveryx>X,thereisaspaceZx`pY`Xwithx>Zx,therestrictionpSZxisafreeaction,andcanbeconstructedsothatZxandZx~parehomeomorphicto: 1. theproductpS1andthecircleS1respectively,wherethegrouppsimplyactsontherstfactor, 2. asolenoidandacirclerespectively,or 3. pkdistinctsolenoidsandacirclerespectively. Proof. Let>pbeageneratorforthegroup. 1. Pickzpx.ApplyingLemma 5.2 ,thereisanarcJ1fromxtozsuchthat0J1isanarc.Applyingthelemmaagain,thereisanarcJ2fromztoxsuchthat0J2isanarc,theintersectionJ19J2)]TJ /F4 11.955 Tf 9.72 0 Td[(x,z,and0J18J2isasimpleclosedcurve.TakeZx)]TJ /F8 11.955 Tf 9.3 0 Td[(pJ18J2.ThespaceZxpS1andtheorbitspaceZx~pS1.Seegurebelow. rxrz S1pZx`X ?0 r0xr0zS10Zx`X~pFigure5-3. Constructingasimpleclosecurvewithdisjointorbits 2. UsingLemma 5.2 ,thereisanarcJfromxtoxwhereZx)]TJ /F8 11.955 Tf 10.65 0 Td[(pJpisap-adicsolenoid.Moreover,theorbitspaceJ~p)]TJ /F6 7.97 Tf 10.11 2.77 Td[(Zx~pisasimpleclosedcurve.Seegurebelow. 68

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r0xS10Zx`X~p pZx`X ?0pxpxFigure5-4. Invariantp-adicsolenoid 3. UsingLemma 5.2 ,thereisanarcJfromxtopkx.ThefullimageZx)]TJ /F8 11.955 Tf 9.32 0 Td[(pJZpkpformspkdistinctp-adicsolenoids.AgaintheorbitspaceJ~pS1isasimpleclosedcurve.Seegurebelow. rxS1Zx`X~p ZpkpZx`Xppp ?0pxppkxpxFigure5-5. pkInvariantp-adicsolenoids Amorecomplicatedexampleisconstructedbelowusing,asbuildingblocks,theinvariantspacesformedinparts2and3ofTheorem 5.3 (namelyp-adicsolenoids).Givenafreep-adicactiononaPeanocontinuum,X,whereXcannotbelocallyseparatedbyaonedimensionalsubset,weshowthatateachandeverypoint,xofthecontinuum,thereisagroupinvariantMengercurvex.Lesseningthecondition 69

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thattheactionbefreetoamerelyeffectiveactiononXbutonesuchthatthespaceXcannotbelocallyseparatedbyanyonedimensionalsubsettogetherwiththesetofperiodicpointswouldonlyreducetheresulttoinvariantMengercurvesexistingateverypointx>Xwheretheactionwasfree.Itisinterestingthatthequotientx~pisonedimensionaleveninthecaseshouldthemap0X)]TJ /F6 7.97 Tf 12.64 2.77 Td[(X~praisedimension. Theorem5.4. Ifap-adicgroup,p,actsfreelyonaPeanocontinuumXsuchthatX~pcannotbelocallyseparatedbyanitenumberofarcs,thenforeachpointx>X,thereisaMengercurvecontainingx,wheretherestrictionpSisafreep-adicaction. Proof. Welabelsometypesofcontinua:asubspace,V,ofX~pwillbeoftypeI,ifitishomeomorphictoanarcwithtwodistinctendpointsintheambientspace,oftypeQifitishomeomorphictoaspaceoftypeItogetherwithacirclethatistangentatapointintheinteriorofthatarc,andoftypeXifitishomeomorphictotheunionoftwospacesoftypeIthathaveasinglepointofintersectionandthatsinglepointoccursintheinteriorofbotharcs.ThenamingschemeforthetypesshouldbeindicativeoftheappearanceoftherespectiveV.Letx>X.FromTheorem 5.3 ,thereisasimpleclosedcurve,W0`X~psuchthatx)]TJ /F5 11.955 Tf 10.07 0 Td[(10W0isasolenoiduponwhichpactsfreely.Forsakeofalaterinduction,let1)]TJ /F4 11.955 Tf 9.28 0 Td[(X~pbethetrivialbrickpartitionofX~p,letB1)]TJ /F2 11.955 Tf 9.28 0 Td[(C0)]TJ /F12 11.955 Tf 9.28 0 Td[(g,andletL0)]TJ /F8 11.955 Tf 9.28 0 Td[(0.SinceW0isasimpleclosedcurveinaPeanocontinuumX~p,choose0tobeabrick1-partitionofX~psuchthatthereisasub-collection0withthefollowingproperties: 1. W09!)]TJ /F12 11.955 Tf 9.28 0 Td[(gforany!>00. 2. Foreach!>0,theintersectionW09!isoftypeI,havingboundarypoints!and!.LetB0)]TJ /F4 11.955 Tf 9.27 0 Td[(!S!>08!S!>0.Foreach!>0,letK!`!bethecoreelementinacorerenementof!constructedsuchthatK!9W0xg.Pickc!>K!9W0.Thereisanl!C0suchthat 70

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eachcomponentofthepre-image10K!isinvariantunderl!.PickL1C1suchthatL1Cmaxl!S!>0.FromTheorem 5.3 ,thereisasimpleclosedcurveJ!`K!`!`X~pwithbasepointc!suchthatJ!9W0)]TJ /F4 11.955 Tf 10.78 0 Td[(c!and10J!isexactlypL1solenoidsthatpermuteunderp.LetC1)]TJ /F4 11.955 Tf 9.28 0 Td[(c!S!>0,andletW1)]TJ /F2 11.955 Tf 9.28 0 Td[(W08!>0J!.ForsomekC1,assumethatforall1BnBk: 1. ThereisasetCn`X~pthatisnite,withCn1`CnandCn1xCn. 2. ThereisanaturalnumberLnALn1. 3. ThespaceWnxWn1,withWn1`Wn`X~p,isacontinuumdecomposedintoanitenumberofsimpleclosedcurvesJi,suchthatifJi`Wn1andJixWn1,thenJi9Wn1`CnCn1isasingletonand10Ji`XispLnsolenoids. 4. Thecollectionn1isabrick21n-partitionofX~pthatrenesn2suchthatthesub-collectionn1)]TJ /F4 11.955 Tf 10.32 0 Td[(!>n1S!9Wn1xgsatisesWn19!)]TJ /F12 11.955 Tf 10.32 0 Td[(gforany!>n1n1 5. Moreover,thepartitionn1isconstructedsuchthat,forany!>n1,thesub-spaceWn9!isaspacethatiseitheroftypeQ,oroftypeXhavingboundaryWn9@!beingrespectivelyeithertwoorfourisolatedpoints. 6. ThesetofalltheseboundarypointsBn1,whereBn2`Bn1)]TJ /F12 11.955 Tf 9.47 0 Td[(8!>n1x>WnSx>@!,locallyseparateWn.Letkbeabrick2k-partitionrening0suchthatthesub-collectionk)]TJ /F4 11.955 Tf 10.75 0 Td[(!>kS!9Wkxgsatises: 1. Wk9!)]TJ /F12 11.955 Tf 9.28 0 Td[(g,forany!>kk. 2. If!>k,then!9WkiseitheroftypeIoroftypeX. 3. If!1,!2>ksuchthat!19!29Wkxg,thentheintersectionisasingletonandatmostoneofthesetwosetsisoftypeX.LetBk)]TJ /F2 11.955 Tf 9.28 0 Td[(Bk18!>kWk9@!beanitenumberofpointsthatobviouslylocallyseparatesWk.LetOk)]TJ /F4 11.955 Tf 9.27 0 Td[(!>kS!9WkisoftypeI.Foreach!>Ok,letK!`!bethecoreelementinacorerenementof!suchthatK!9Wkxg.Pickc!>K!9Wk.Chooseanumber 71

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Lk1ALksuchthat,foreveryK!,eachcomponentofthepre-image10K!isinvariantunderLk1.UsingTheorem 5.3 ,thereisasimpleclosedcurve,J!`K!`!`X~pwithbasepointc!suchthatJ!9Wk)]TJ /F4 11.955 Tf 9.28 0 Td[(c!and10J!isexactlypLk1solenoidsthatpermuteunderp.LetWk1)]TJ /F2 11.955 Tf 10.83 0 Td[(Wk8!>kJ!,andletCk1)]TJ /F2 11.955 Tf 10.83 0 Td[(Ck8c!S!>Ok.Thiscompletestheinduction,andsothereisastrictlyincreasingsequenceofnaturalnumbersLnaswellassetsW)]TJ /F17 11.955 Tf 9.28 -.94 Td[(k)]TJ /F10 7.97 Tf 4.63 0 Td[(0Wk,C)]TJ /F17 11.955 Tf 9.28 -.94 Td[(k)]TJ /F10 7.97 Tf 4.63 0 Td[(0Ck,andB)]TJ /F17 11.955 Tf 9.28 -.94 Td[(k)]TJ /F10 7.97 Tf 4.63 0 Td[(0Bk.Let)]TJ /F5 11.955 Tf 11.3 0 Td[(10W.Bestvina'scharacterizationofaMengercurveisthatitistheuniqueconnectedandlocallyconnectedcompactumofdimension1havingthedisjointarcproperty[ 7 ]. Claim1. Thespacehascoveringdimensiondim)]TJ /F8 11.955 Tf 9.28 0 Td[(1.DeneB)]TJ /F4 11.955 Tf 10.41 0 Td[(!>i8b>BiStb,iSi>N.Sincemeshi)]TJ /F8 11.955 Tf 13.77 0 Td[(0asi)]TJ /F12 11.955 Tf 13.77 0 Td[(,thecollectionBformsapropertySbasisforW.ForanyB>B,theboundary@Bisanitenumberofpoints,thusWhascoveringdimensiondimW)]TJ /F8 11.955 Tf 9.28 0 Td[(1.ForeachB>B,thepre-image10BconsistsofanitenumberofopensetsandthecollectionB)]TJ /F4 11.955 Tf 9.98 0 Td[(B`XSBisacomponentof10BforsomeB>Bformsabasisfor.Since,foreachB>B,theboundary@Bisaniteset,foranyB>Btheboundary@BisanitenumberofCantorsetsandisthus0-dimensional.Therefore,wehavedim)]TJ /F8 11.955 Tf 9.28 0 Td[(1. Claim2. Thespaceisconnected.Pickpointsy1,y2>.Thereisann>Nsuchthat0yi>Wnforbothi)]TJ /F8 11.955 Tf 9.72 0 Td[(1,2.Thereisanarcjoining0yitoW0whichliftstoarcsinstartingatyiforbothi)]TJ /F8 11.955 Tf 10.65 0 Td[(1,2,andsincethepre-image10W0)]TJ /F8 11.955 Tf 9.82 0 Td[(xisthesolenoidwhichisconnected,thepointsy1andy2lieinthesamecomponentof.Sincey1,y2wereanarbitrarypairofpointsin,thespaceisconnected. Claim3. Thespaceislocallyconnected. 72

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UsingthebasesBandBdenedinClaim 1 ,itsufcestoshowthateachB>Bhasonlynitelymanycomponents.ThisfurtherreducestoshowingthateachB>Bhasonlynitelymanycomponents,whichinturnreducestoshowingthatforeveryi>N,theintersection!9Whasnitelymanycomponentsforevery!>i.Foranarbitrary!>i,either!9WiisoftypeI(andthusisanarc)oroftypeX(andthusfourarcsmeetinginacommonpoint).Ineithercase,!9Wiisconnected.Letw>!9W,thenw>WnforsomenCi.FromtheconstructionofWn,nitelymanyarcsin!joinwtoWi.Sincewwaschosenarbitrarily,!9Wisconnected.HenceeveryB>Bisconnected,andthuseveryB>Bhasnitelymanycomponentsandislocallyconnected. Claim4. ThespacehastheDisjointArcProperty(DD1P).Letf1,f2)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F5 11.955 Tf 13.37 0 Td[(bearcsin.GivenA0,thereisann>Nthatissolargethatthefollowingholdtrue: 1. Theprojectionofthestartingpoint0fi0>Wnfori)]TJ /F8 11.955 Tf 9.28 0 Td[(1,2. 2. Theloopsize(i.e.meshn1)isatmost2n@ 2. 3. ThesubgroupLnBpissuchthatdx,gx@ 2foreveryx>Xandg>Ln.Considerthearcprojections0fi)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F2 11.955 Tf 13.34 0 Td[(Wfori)]TJ /F8 11.955 Tf 9.97 0 Td[(1,2,thenapproximateeachbyWnobtainingarcsfi)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F2 11.955 Tf 12.77 0 Td[(Wnfori)]TJ /F8 11.955 Tf 9.4 0 Td[(1,2.Foranyg>Lnandi)]TJ /F8 11.955 Tf 9.4 0 Td[(1,2,anylift^fi)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F5 11.955 Tf 12.77 0 Td[(offitakingfi0togfi0issuchthatdf1,^fi@.SincetheimageofeachficanbedecomposedintonitelymanysimplyconnectedpiecesinWnwhichliftuniquelyuptochoiceofbase-pointandthereareuncountablymanychoicesforliftsbychoosingabase-pointbyg>Ln,itfollowsthattherearelifts^f1,^f2suchthatdfi,^fi@and^f19^f2)]TJ /F12 11.955 Tf 9.28 0 Td[(g.ThushastheDisjointArcProperty(DD1P).Sinceisacompact,connected,locallyconnected1-dimensionalmetricspacewiththedisjointarcproperty,usingBestivina'scharacterizationoftheMengercurve[ 7 ],thespaceisaMengercurve. 73

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ThusTheorem 5.4 produces,foreachx>X,ap-invariantMengercurve.Ifwerestrictthefreep-actiontooneoftheseMengercurves,weobtainafreepactionon1withaone-dimensionalorbitspace.Thisissimilar,butnotidentical,tothefreepactionon1obtainedbyA.N.Dranishnikov[ 15 ],andlaterdescribedbyZhiqingYang[ 59 ].Toreadilyseethatthetwoactionsaredifferent,simplycomparetheorbitspacesoftherespectiveactions.Belowisarepresentationofeachorbitspaceuptothethirdstageofeachoftheirconstructions.Theorbitspace1~pfromTheorem 5.4 onthelefthascutpoints,whiletheorbitspacefromDranishnikov'sactionontherightonlyhaslocalcutpoints.1~pfromTheorem 5.4 Dranishnikov's1~p[ 15 59 ] rrrr qqqqqqqqqqqqqqqqkkkkkkkkkkkkkkkkrrrr qq qq qq qq Figure5-6. Comparingquotientspacesforfreeactionson1 WithslightmodicationstotheconstructioninTheorem 5.4 ,thefollowingcanbeconstructedwithoutdifculty: Corollary5.4.1. TheMengercurvecanbeconstructedsothattheinheritedfreepactiononitispreciselytheonedescribedbyA.N.Dranishnikov[ 15 59 ]. Corollary5.4.2. TheMengercurve,,canbeconstructedinsuchafashionthatthereisapointx>suchthatpxlocallyseparatesbutnootherorbitdoesso. 74

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Corollary5.4.3. TheMengercurve,,canbeconstructedinsuchafashionthattheorbitspaceofthissub-action,~p1isaMengercurveaswell.ThelastactionontheMengercurvewithorbitspacealsotheMengercurvewasrstshownbyR.D.Andersonin1957[ 3 ],buttheworkheregivesasettingthatisnotonlyfarmoreaccessible,buteasilyandreadilyconstructed.Thishighlightstheusefulnessofthetechniqueinthatitsimpliesandeventrivializestheconstructionofeverynon-dimensionraisingp-adicactionona1-dimensionalPeanocontinuum. 75

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CHAPTER6HILBERTSPACE 6.1SpaceofMeasurableFunctionsInEilenberg'spublishedlistofproblemsinTopologyfromthe1946conferencecelebratingPrincetonUniversity'sBicentennial[ 16 ]beyondquestionsdirectlyrelatingtotheHilbert-SmithconjectureincludedseveralproblemsfromP.A.Smith.Thefollowingisenumerated34inthearticle: Problem6.1(Smith[ 16 ]). DoesHilbertspaceadmitaperiodictransformationofperiod2withoutxedpoints?ThiswasansweredbyJamesWest[ 53 ]aspartofhisPhDthesis.HeshowedthateverycompactgroupactsfreelyonHilbertspace.Themaintheoremwas: Theorem6.1(West[ 53 ]). IfXisaseparable,innite-dimensional,FrechetspaceandifGisatopologicalgroupforwhichthereisaneffectivetransformationgroupactiononX,theneachclosedsubsetofXisthexed-pointsetofasemifreeactionofGonX.Byseparable,innite-dimensional,FrechetspacewemeanHilbertspace,andforourpurposeswetaketheempty-settobetheclosedsubsetofX.AllthatwouldremainwouldbetoexhibitaneffectivetransformationonHilbertspacebyZ2orforourconcernsforp.West'sclevertricktoachievingafreeactionfromamerelyeffectiveoneisinmakinguseofthefollowing: Theorem6.2(West[ 53 ]). IfXisthesubsetofscomposedofallpointswhosesetofcoordinatesisdenseins,thenXishomeomorphictos.Hereheisusings)]TJ /F11 11.955 Tf 9.44 0 Td[(R.Fromthereitisnothardtoseehowtoobtainafreeactionfromaneffectiveone,simplybydeningtheeffectiveactionons,restrictingtothesubsetX`swheretheactionperforcebycontinuitymustbefree,andthenrealizingthatthisisafreeactiononHilbertspace.Toconstructaneasytovisualizeeffectivep-adicactiononHilbertspace(ortheHilbertcube),rstconsiderHilbertspaceastheinniteproductofopendisks(orclosed 76

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disksinthecaseofthecube),thatis`2Li)]TJ /F10 7.97 Tf 4.63 0 Td[(1DiwhereeachDiisanopen2-disk.LetriDi)]TJ /F2 11.955 Tf 14.67 0 Td[(DibetheperiodpirotationaboutthecenterofthediskDianddeneLi)]TJ /F10 7.97 Tf 4.63 0 Td[(1Di)]TJ /F17 11.955 Tf 12.64 -.94 Td[(Li)]TJ /F10 7.97 Tf 4.63 0 Td[(1Dibyxii>N)]TJ /F4 11.955 Tf 9.28 0 Td[(rixii>N.Thisisnotafreeaction.Ifthepoint0i>Diisthecenterofthei-thdisk,thenthepoint0ii>Nisxedbylikewisehasinnitelymanyperiodicpointsoftheformx)]TJ /F4 11.955 Tf 9.5 0 Td[(xii>Nwhereforsufcientlylargevaluesofi,(i.e.thereisanumberNxA0suchthatifiANthen)wehavexi)]TJ /F41 11.955 Tf 9.28 0 Td[(0i.AfewyearsafterWest'sthesis,CzesawBessagaandAleksanderPeczynskiobtainedaslightlystrongerresult Theorem6.3(Bessaga-Peczynski[ 6 ]). EveryseparablemetricgroupadmitsafreeactionintheHilbertspace`2Theydidsobyobtainingamuchmoreusefulresult.TakingthenotationMXtorepresentthespaceofmeasurablefunctionsfromtheinterval)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1toX,undertheequivalencerelationf~gifandonlyiff)]TJ /F2 11.955 Tf 9.28 0 Td[(galmosteverywhere, Theorem6.4(Bessaga-Peczynski[ 6 ]). LetXbeametricspace.ThenMX`2ifandonlyifXiscomplete-metrizable,separable,andXhasmorethanonepoint.WhenonetakesXtobethep-adicgroupporthep-adicsolenoidgrouppthenthefreeactionbyeitherofthesegroupson`2issimplythatofasubgroupactingonthegroup`2.Thegroupoperationon`2isviewedbyusingthegroupoperationofXpoint-wise.OnecanviewX0MX`2asbeingthesetofequivalenceclassesoftheconstantfunctions. 6.2ViewingSimpleFunctionsWhendealingwithmeasurablefunctionsitsalmostalwaysbenecialtoobservethesimplefunctions,andhereisnoexception.Wewillbreakdownthesimplefunctionsbynumberofdistinctintervalsofvalues.LetM)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,G`2bethespaceofmeasurablefunctionsfromtheunitintervalintotheseparablemetricgroupG.LetS)]TJ /F8 11.955 Tf 9.76 0 Td[(0,1,G`M)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,Gbethesubspaceofsimple 77

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functions.DenotethesetoftheequivalenceclassesoftheconstantfunctionsbyF1G)]TJ /F4 11.955 Tf 9.28 0 Td[()]TJ /F2 11.955 Tf 10.4 0 Td[(f>S)]TJ /F8 11.955 Tf 9.76 0 Td[(0,1,Gthereisag>Gsuchthatf1g)]TJ /F8 11.955 Tf 9.28 0 Td[(1.Forsakeofnotation,foreachg>Gdenotetheconstantfunctionsendingtheentireintervaltogbycg)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F2 11.955 Tf 12.78 0 Td[(Gwherecgt)]TJ /F2 11.955 Tf 9.41 0 Td[(gforallt,thusF1G)]TJ /F4 11.955 Tf 9.41 0 Td[()]TJ /F2 11.955 Tf 10.41 0 Td[(cgg>G.Foranequivalenceclassofasimplefunction)]TJ /F2 11.955 Tf 4.55 0 Td[(f>S)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,Gandelementg>G,letNg)]TJ /F2 11.955 Tf 9.75 0 Td[(f)]TJ /F2 11.955 Tf -458.72 -23.9 Td[(minn>NSf1gisndisjointintervalsforsomef>)]TJ /F2 11.955 Tf 4.55 0 Td[(fwhereNg)]TJ /F2 11.955 Tf 9.75 0 Td[(f)]TJ /F8 11.955 Tf 9.95 0 Td[(0ifandonlyiff1gcontainsnointervalsforeveryf>)]TJ /F2 11.955 Tf 4.56 0 Td[(f.FornA1,letFnG`S)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,Gbesuchthatforevery)]TJ /F2 11.955 Tf 4.55 0 Td[(f>S)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,GwehavePg>GNg)]TJ /F2 11.955 Tf 9.75 0 Td[(f@n. 6.3SomeSpecialSimplicesinM)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,G`2Whenviewing`2asthespaceM)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,G,itisveryeasytoenvisionsimpliceswithverticesinF1G.Givenanytwoelementsg,h>Gwepicka1-simplexfromgtohtobegivenbyaspecicmappingAg,h)]TJ /F8 11.955 Tf 4.55 0 Td[(0,10M)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,G.WewilldeneAg,hbyusingrepresentativefunctionsfromF2G.Fort>)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1,letft)]TJ /F8 11.955 Tf 4.56 0 Td[(0,1)]TJ /F2 11.955 Tf 12.64 0 Td[(Gbegivenbyftx)]TJ /F4 11.955 Tf 9.28 21.1 Td[(gxCthx@t.ThenwedeneAg,ht)]TJ /F4 11.955 Tf 10.42 0 Td[()]TJ /F2 11.955 Tf 4.55 0 Td[(ft>F2G`M)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,Gforeacht>)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1.ObservethatAg,h0)]TJ /F4 11.955 Tf 10.11 0 Td[()]TJ /F2 11.955 Tf 4.55 0 Td[(cg,andAg,h1)]TJ /F4 11.955 Tf 10.12 0 Td[()]TJ /F2 11.955 Tf 4.55 0 Td[(ch.ThemappingAg,hgivesusa1-simplexbetween)]TJ /F2 11.955 Tf 4.55 0 Td[(cgand)]TJ /F2 11.955 Tf 4.56 0 Td[(ch.LetS1G)]TJ /F2 11.955 Tf 9.28 0 Td[(F2Gdenotethecollectionof1-simplicesobtainedbythisprocessrangingoveralldistinctpairingsg,h>G.Pleasenotethatthesesimplicesaredirected,andweareobtainingtwo1-simplicesjoininggwithh(asordermatters).Likewisegivenany3distinctelementsg,h,k>G,itisalsonothardtoconstruct2-simplicescompatiblewithourpriorchoices.Thistimewewilluserepresentative 78

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functionsfromF3G.Foru@t>)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1,letfu,t)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1)]TJ /F2 11.955 Tf 12.64 0 Td[(Gbesuchthatfu,tx)]TJ /F4 11.955 Tf 9.28 33.06 Td[(gxCthuBx@tkx@u.ThenwedeneAg,h,ku,t)]TJ /F4 11.955 Tf 9.28 0 Td[()]TJ /F2 11.955 Tf 4.55 0 Td[(fu,t>M)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,Gforeachu@t>)]TJ /F8 11.955 Tf 4.55 0 Td[(0,1.InthecaseofinnitegroupsG,onecanreadilyseeaneasywaytochoosesimplicesofalldimensionswithin`2.FornC2,letSnGFn1Gbethecollectionofallthen-simplicesobtainedbythisprocess,andletS!G)]TJ /F17 11.955 Tf 10.97 -.94 Td[(n)]TJ /F10 7.97 Tf 4.63 0 Td[(1SnG.Whileitisknownthat`2isagroupinit'sownright,viewedasM)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,G,itiseasytoseeagroupoperationon`2inheritedfromthatofG.Wecantakethiscollectionofsimplicesandshiftthembyanygiven,xedelementinM)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,Gtoobtainanothercollection. 6.4InvariantSetsBytheactionofGonM)]TJ /F8 11.955 Tf 9.76 0 Td[(0,1,G,thefactthatGisagroup,andthatSnGisallpossibledistinctpairings,itisobviousthatSnGisaG-invariantsubspaceofM)]TJ /F8 11.955 Tf 9.75 0 Td[(0,1,Gforeachn>N.PerforceS!GisalsoG-invariant.OnecanpictureFn1Gasthen-foldjoinofGwithidenticationscollapsingwhatwouldotherwisebearcsfromanelementtoitself.Butevenforsmall,nitegroupsGthesesetsbecomelargeandthesubsetsSnGdonotimprovethingssufciently.Byconsideringgroupinvariantrelationsbetweenchoicesofvertices,wecanobtainsmallerinvariantsubsetsof`2.Thesehavealreadybeenevidencedinotherspacesearlierinthisdissertation(e.g.inTheorem 5.3 ).ConsiderthecasewhenG)]TJ /F8 11.955 Tf 10.48 0 Td[(p,thegroupofp-adicintegers.Fixagenerator,,ofpthendeneO1G`S1Gtobethecollectionof1-simplicesjoiningeachelementgtoitssuccessorg.ClearlyO1Gishomeomorphictoap-adicsolenoid.LikewisewecandeneK1G`S1Gtobethecollectionof1-simplicesjoiningeachelementg>ptopkgforsomexednaturalnumberk.ThespaceK1Gishomeomorphictopkdistinctp-adicsolenoids. 79

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Foranypointx>M)]TJ /F8 11.955 Tf 9.76 0 Td[(0,1,p,thesetxO1Gisap-invariantp-adicsolenoidcontainingthepointx.Andthuswecanseethatin`2,usingthestrengthofBessagaandPeczynski'sTheorem,thatwecaneasilyshowthatTheorem 5.3 holdsforthisactionon`2.Moreover,itmotivatesthegoaltotrytoattempttoextendtheresultofTheorem 5.3 forthehigherdimensionalanalogsofS1GandO1G.Commonwisdomisthat`2andtheHilbertCubebehavemorelikeaninnitedimensionalversionoftheMengermanifoldsnthanthatofmanifoldsandRn.Inmymind,theunderlyinggoaloftheHilbert-SmithConjectureistobetterunderstandthedistinctionbetweenMengermanifoldsandEuclideanmanifolds.Assuch,itisnaturaltowishtoexplorep-adicactionson`2,theHilbertCube,andQ-manifolds.Iwouldnothavefeltmydissertationtobecompletewithoutatleastthiscursoryaddressingforthereader. 80

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REFERENCES [1] S.M.Ageev,Classifyingspacesforfreeactionsandthehilbert-smithconjecture,Acad.Sci.Sb.Math.75(1993),137. [2] R.D.Anderson,Openmappingsofcompactcontinua,Proc.Nat.Acad.Sci.U.S.A.42(1956),347. [3] ,Zero-dimensionalcompactgroupsofhomeomorphisms,PacicJ.Math.7(1957),no.1,797. [4] ,Acharacterizationoftheuniversalcurveandaproofofitshomogeneity,Ann.ofMath67(1958),313. [5] M.F.AtiyahandI.G.MacDonald,Introductiontocommutativealgebra,Addisson-Wesley,Reading,Mass,1969. [6] CBessagaandA.Pelczynski,Onspacesofmeasurablefunctions,StudiaMath.44(1972),597. [7] MladenBestvina,Characterizingk-dimensionaluniversalmengercompacta,MemoirsoftheAmericanMathematicalSociety71(1988),no.380,iii. [8] R.H.Bing,Partitioningaset,Bull.Amer.Math.Soc.55(1949),1101. [9] ,Partitioningcontinuouscurves,Bull.Amer.Math.Soc.58(1952),536. [10] R.H.BingandE.E.Floyd,Coveringswithconnectedintersections,Trans.Amer.Math.Soc.69(1950),387. [11] S.BochnerandD.Montgomery,Locallycompactgroupsofdifferentiabletransfor-mation,AnnalsofMathematics47(1946),no.4,639. [12] G.E.Bredon,FrankRaymond,andR.F.Williams,p-adicgroupsoftransfor-mations,TransactionsoftheAmericanMathematicalSociety99(1961),no.3,488. [13] L.E.J.Brouwer,Uberdieperiodischentransformationenderkugel,Math.Annalen80(1919),39. [14] C.Chevalley,Onatheoremofgleason,Proc.Amer.Math.Soc.2(1951),122. [15] A.N.Dranishnikov,Onfreeactionsofzero-dimensionalcompactgroups,Izv.Akad.NaukUSSR32(1988),217. [16] SamuelEilenberg,ontheproblemsoftopology,AnnalsofMathematics50(1949),247. [17] RyszardEngelking,Generaltopology,PolishScienticPublishers,Warsaw,Poland,1977. 81

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[18] VonHansFreudenthal,topologischegruppenmitgenugendvielenfrastperiodis-chenfunktionen,AnnalsofMathematics37(1936),no.1,57. [19] IGelfandandAKolmogorov,Onringsofcontinuousfunctionsontopologicalspaces,Dokl.Akad.Nauk.SSSR(1939),no.22,11. [20] LeonardGillmanandMeyerJerison,Ringsofcontinuousfunctions,D.NostrandCompany,1960. [21] A.Gleason,Thestructureoflocalycompactgroups,DukeMathJ.18(1951),85. [22] ,Groupswithoutsmallsubgroups,AnnalsofMathematics56(1952),no.2,193. [23] A.Haar,Dermassbegriffindertheoriederkontinuierlichengruppen,AnnalsofMathematics2(1933),147. [24] F.Hausdorff,Dieschlichtenstetigenbilderdesnullraums,Fund.Math.29(1937),151. [25] EdwinHewitt,Ringsofreal-valuedcontinuousfunctionsi,Trans.Amer.Math.Soc.64(1948),45. [26] D.Hilbert,Mathematischeprobleme,Nachr.Akad.Wiss.Gottingen(1900),253. [27] ,Mathmaticalproblems,Bull.Amer.Math.Soc.8(1902),437. [28] WitoldHurewiczandHenryWallman,Dimensiontheory,revisededition1948ed.,PrincetonMathematicalSeries,no.4,PrincetonUniversityPress,Princeton,1941. [29] J.R.Isbell,Uniformspaces,AmericanMathematicalSociety,Providence,RhodeIsland,1964. [30] KenkichiIwasawa,Onsometypesoftopologicalgroups,AnnalsofMathematics50(1949),no.3,507. [31] M.Katetov,Onringsofcontinuousfunctionsandthedimensionofcompactspaces,CasopisPest.Mat.Fys.75(1950),1. [32] L.V.Keldys,Transformationofamontoneirriduciblemappingintoamontone-interiormappingandamonotone-interiormappingofthecubeontothecubeofhigherdimension(russian),Dokl.Akad.NaukSSSR(N.S.)114(1957),472. [33] B.Kerekjarto,Uberdieperiodischentransformationenderkreisscheibeundkugelasche,Math.Annalen(1919),36. [34] A.Kolmogoroff,Uberoffeneabbildungen,AnnalsofMathematics38(1937),36. 82

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[52] J.vonNeumann,Dieeinfuhrunganalytischerparameterintopologischengruppen,AnnalsofMathematics2(1933),170. [53] J.West,Fixedpointsetsoftransformationgroupsoninnite-productspaces,Proc.Amer.Math.Soc.21(1969),575. [54] G.T.Whyburn,Analytictopology,AmericanMathematicalSocietyColloquiumPublications,vol.XXVIII,AmericanMathematicalSociety,NewYork,1942. [55] R.F.Williams,Ausefulfunctorandthreefamousexamplesintopology,TransactionsoftheAmericanMathematicalSociety106(1963),319. [56] ,theconstructionofcertain0-dimensionaltransformationgroups,TransactionsoftheAmericanMathematicalSociety129(1967),no.1,140. [57] D.Wilson,Openmappingsonmanifoldsandacounterexampletothewhyburnconjecture,DukeMathematicalJournal40(1973),no.3,705. [58] C.T.Yang,p-adictransformationgroups,MichiganMathematicsJournal7(1960),201. [59] ZhiqingYang,Aconstructionofclassifyingspacesforp-adicgroupactions,TopologyanditsApplications153(2005),161. 84

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BIOGRAPHICALSKETCH JamesR.MaissenwasborninSarasota,Florida.TheonlysonofJoanI.MaissenandGiachenM.Maissen,hegrewupinSarasota,Florida,graduatingfromPineViewschoolforthegifted.HisparentsmetinVenezuelaandlivedtherefor20yearspriortohisbirth.HisfatherisSwiss,whilehismotherisdescendedfromWilliamWhiteontheMayowermakinghimsimultaneously1stand20thgenerationAmerican.HeattendedtheUniversityofColoradoatBoulderfortwoyearsbeforeswitchingmajorsfromcomputerscienceengineeringtomathematicsandtransferringtotheUniversityofFloridawhereheearnedhisB.S.inmathematics.HecontinuedattheUniversityofFloridaseekingaPhDforseveralyearsuntilleavingtocareforhismotherwhosehealthhadturnedfortheworse.Afterherpassing,heworkedinthecomputerscienceeldinOrlandobeforereturningtoGainesvilletocompletehisPhD.UponcompletionofhisPhDprogram,Jameswishestopursuehisloveofteaching.JamesisengagedtoMarieMandel,whomhemetplayingbridgeandwithwhomhefoundmorethanjustabridgepartner. 85