Properties of Groups at Infinity

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Properties of Groups at Infinity
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english
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Gentimis,Athanasios
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University of Florida
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Dranishnikov, Alexander N
Committee Members:
Rudyak, Yuli B
Boyland, Philip L
Keesling, James E
Maslov, Dmitrii

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Subjects / Keywords:
aperiodic -- asymptotic -- compactification -- dimension -- groups -- hyperbolic -- limit -- spaces
Mathematics -- Dissertations, Academic -- UF
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In this dissertation we present 3 topics of geometric group theory. The first, asymptotic dimension, was a very important topic around 2006 when all relative theorems where highly sought after. Our corresponding paper has 8 citations already. At least three more researchers have given relative theorems/proofs on the subject. After a thorough introduction to the idea of asymptotic dimension we give a nice geometric/topological proof of an algebraic property. This is a rare moment where topology is used to assist algebra in an algebraic topology setting. It is not common today a topic of mathematics to extend to more than one specialized fields. Our second topic, limit aperiodic groups, does exactly that. It is a nice interplay between geometric group theory, dynamics and logic. Hopefully, mathematicians from those fields will use the common language we established and work on new, or existing ideas, using ?translated? theorems from other fields using our connections. The results were quoted twice in the 2008 Spring Topology and Dynamical Systems Conference, in different fields. Limit aperiodic spaces and limit aperiodic subgroups extend significantly the idea of limit aperiodic groups and may lead to future work. The counter example for limit spaces is a very interesting construction on its own and may be used as a counter example to other questions. The last part of the dissertation, talks about a certain compactification of hyperbolic spaces and its cohomological properties. It answers, partially, a 20 year old question related to the coarse Baum-Connes conjecture. We think that the tools used to answer our modified version of the problem will lead to the eventual solution of the original question. Furthermore the technique we have developed with the use of the l-infinity cohomology might prove more useful than the theorem itself. At least two other researchers are waiting our result to be officially published so they can use it in their own research. Furthermore our work on the l-infinity obstruction theory might yield another way to view cohomology for certain spaces and it seems like an idea worth exploring.
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In the series University of Florida Digital Collections.
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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 Athanasios Gentimis.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Dranishnikov, Alexander N.

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PROPERTIESOFGROUPSATINFINITYByATHANASIOSGENTIMISADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011AthanasiosGentimis 2

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IdedicatethistoMaria 3

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ACKNOWLEDGMENTS IwouldliketothankmyadvisorDr.Dranishnikovforallhishelpandsupport.Thankyou,fortheideasyousharedwithmeandyourguidancethroughthetoughyearsofaPhD.IamgratefultomycommitteemembersfortheireffortsinshapingmeintoanappropriatePhDcandidate;ToDr.KeeslingforourgreatmathdiscussionsandthetimespentinexplainingDynamicalSystemsandChaostome.ToDr.Boylandforhispromptanswerstomyquestionsoneldsunrelatedtohisresearch(includingteaching);ToDr.RudyakforteachingmeAlgebraicTopologyandinsistingthatIlearnAlgebra,andtoDr.Maslovwhotookafewhoursfromhisprecioustimetolistenandcommentonapurelytheoreticalmathresearchpaper.Iamgratefultomyparentsandmybrotherfortheirloveandsupport.Completingthisthesiswouldn'tbepossiblewithoutyou.Also,Iwouldliketothankmyfellowgraduatestudentsandfriends,withoutwhichlifeinGainesville,andthusthecompletionofaPhD,wouldbeveryhard.Finally,Iwanttothankfromthebottomofmyheartmyance,Maria,forallherloveandsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 2ASYMPTOTICDIMENSION ............................. 10 2.1BasicDenitions ................................ 10 2.2CayleyGraphs ................................. 22 2.3MainResults .................................. 26 3LIMITAPERIODICGROUPS ............................ 35 3.1Motivation .................................... 35 3.2LimitAperiodicGroups ............................. 36 3.3G-SetsAndLimitAperiodicity ......................... 40 3.4LimitAperiodicSubgroups ........................... 50 4COMPACTIFICATIONSONHYPERBOLICSPACES ............... 52 4.1ShortHistoryOverview ............................ 52 4.2TheCoHigsonCompactication ....................... 53 4.3`1-cohomology ................................ 56 4.4MainResult ................................... 58 REFERENCES ....................................... 61 BIOGRAPHICALSKETCH ................................ 64 5

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LISTOFFIGURES Figure page 2-1asdimR11 ..................................... 12 2-2asdimR22 ..................................... 18 2-3Z6 ........................................... 23 2-4Z= ....................................... 24 2-5Z= .................................... 24 2-6AuniformlyoneendedspacewithasdimX=1 ................... 32 4-1AnelementofCo(R) ................................. 55 6

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AbstractofdissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyPROPERTIESOFGROUPSATINFINITYByAthanasiosGentimisAugust2011Chair:AlexanderDranishnikovMajor:MathematicsInthisthesis,weexaminethreedifferenttopicsofGeometricGrouptheory.Asymptoticdimensionanditsrelationtogroups,limitaperiodicgroupsandthecohomologicalpropertiesofcertaincompacticationsofcoarsestructuresonhyperbolicspaces.Wegiveallappropriatedenitionsandproperties,presentbasicexamplesandprovesomemaintheorems. 7

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CHAPTER1INTRODUCTIONTheresultspresentedinthisdissertation,containelementsofgeometricgrouptheory.Theyarerelatedtodiscretegroupsandtheirpropertiesatinnity.Thisthesisconsistsofthreechapters.InChapter2,wediscussthenotionofasymptoticdimensionofdiscretegroupswhichwasintroducedbyGromovin( 23 ).Itisalargescaleanalogoftopologicaldimensionanditisinvariantunderquasi-isometries.ThisnotionhasprovedtobeusefultothefamousNovikov'shighersignatureconjectureanditwasstudiedfurtherbymanyotherresearchers,see( 37 ),( 7 ),( 10 ),( 11 ),( 34 ),( 4 ).IntherstchapterwegivealltherelativedenitionsandbasicpropositionsaboutasymptoticdimensionandprovethatIfGisaone-endednitelypresentedgroupthenGhasasymptoticdimensiongreaterorequalto2.Thisleadstothemoregeneraltheorem:IfGisanitelypresentedgroupwithasdimG=1thenGisvirtuallyfree.Thisisanimportanttheoremandatleasttwomorepeoplehavepresenteddifferentproofsforit(see( 26 ),( 17 )).Itisaniceexampleofusingtopologicalandgeometricalideastoproveapurelyalgebraicproperty.Weconcludechapter1withanexampleofanitelygeneratedgroupwithasdimG=1whichisnotvirtuallyfree.ThethirdChaptertalksaboutlimitaperiodicgroups.Thisideaiscloselyrelatedtoaperiodictilingsofspaces.WeinvestigategroupswiththelimitaperiodicpropertyandextendittosetswhereagroupGactson(G-sets).Wegivetwodenitionsoflimitaperiodicgroupsandshowthattheyareequivalent.WethenprovethatifXisalimitaperiodicG-setandthereexistsanx2XsuchthatX=OrbG(x)withStabG(x)isalimitaperiodicgroupthenGisalimitaperiodicgroup.Wethenusethistheoremtoprovethatdirectproducts,HNNextensions,andamalgamatedproductsoflimitaperiodicgroupsarelimitaperiodic.Thenweproduceanexampleofanonlimit 8

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aperiodicG-set.WenishthischapterwiththenotionoflimitaperiodicsubgroupswhichhelpsinvestigatewhichsetsarelimitaperiodicandwhicharenotundertheactionofG.Chapter4isaboutaspecialcompacticationofspaces,theCocompactication.Westartthischapterwithabriefoverviewofthehistorybehindthequestionandtherelativeliterature.AfterpresentingthebasicdenitionsaboutcoarsestructuresandtheirpropertieswequicklyreviewthedenitionofHyperbolicspacesandobstructiontheory.Weusetheideaofl1-cohomologytoprovetheHigsonconjecturefortheHyberbolicspaceHn:TheCechcohomologyoftheC0-compacticationofHnistrivialindimensionsgreaterthan1.Thisthesiswasbasedonthreepaperswrittenthroughtheacademicyears2006-2011whichareundervariousstages(published,underreview,preprint)( 21 ),( 20 )( 19 ). 9

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CHAPTER2ASYMPTOTICDIMENSION 2.1BasicDenitionsInthissectionwediscussthebasicdenitionsofasymptoticdimension.Wegivevariousequivalentdenitionsandproverelativetheorems.Althoughmostofthetheoremswereproveninotherpaperstheproofsinthissectionareoriginalandsometimesdifferentfromthestandardones. Denition2.1.1. AmetricspaceYissaidtobed-disconnectedorthatithasdimen-sion0onthed-scaleifthereexistsafamilyfBigi2Isuchthat: 1. Y=[i2IBi 2. supfdiam(Bi),i2Ig6D<1 3. dist(Bi,Bj)d8i6=jwheredist(Bi,Bj)=inffdist(a,b)a2Bi,b2Bjg Denition2.1.2. (ASYMPTOTICDIMENSION1)WesaythataspaceXhasasymptoticdimensionn,ifnistheminimalnumbersuchthatforeveryd>0wehaveX=n[k=0XkwhereXkared-disconnected8k.WethenwriteasdimX=n. Denition2.1.3. WesaythatacoveringfBighasd-multiplicity,kifandonlyifeveryd-ballinXmeetsnomorethanksetsBiofthecovering.AlsoacoveringfBigi2IisD-boundedifdiam(Bi)6D8i2I. Denition2.1.4. (ASYMPTOTICDIMENSION2)WesaythataspaceXhasasdimX=n,ifnistheminimalnumbersuchthat8d>0thereexistsacoveringofXofuniformlyD-boundedsetsBisuchthatd-multiplicityn+1.Thelastdenitionofasymptoticdimensioncanbereformulatedintermsofcolors. 10

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Denition2.1.5. Wesaythatacoveringism-coloredifitistheunionofm1disjoinedfamiliesU=m[i=1UisuchthateveryfamilyUiconsistofuniformlyD-boundedsets.Sowehavethisequivalentdenitionofasymptoticdimension. Denition2.1.6. (ASYMPTOTICDIMENSION3)AmetricspaceXhasasymptoticdimensionn,ifitadmitsann-colored,d-disjoinedcover. Example2.1.7. EverycompactspaceXhasasdimX=0. Proof. SinceXiscompactitisbounded.ThismeansthatthereisaD<1suchthatXB(x0,D).Soforeveryd>0weusethecoveringfB(x0,D)g.Itisobviousthatd-multiplicity=1.AccordingtotheseconddenitionofasymptoticdimensionwehaveasdimX=0. Thisexampleisone,butnottheonly,wheretopologicaldimensionandasymptoticdimensiondiffer.Theexamplesareinnitesincewecanconstructcompactspaceswithtopologicaldimensionnforalln2Nwhichwillhaveasymptoticdimension0.Alsowiththisexamplewestatethatworkingwithasymptoticdimensionproducesresultsonlyininniteandnon-boundedcases.Namely: Proposition2.1.8. Everyboundedmetricspacehasasymptoticdimension0andnonon-compactgeodesicmetricspacehasasymptoticdimension0. Example2.1.9. asdimR=1 Proof. Letd2N.Dene:A0=[0,d+1),B0=[d+1,2(d+1)),Ai=[(i+1)(d+1),(i+2)(d+1))ifiisoddand[)]TJ /F3 11.955 Tf 9.3 0 Td[(i(d+1),)]TJ /F9 11.955 Tf 9.3 0 Td[((i)]TJ /F9 11.955 Tf 11.96 0 Td[(1)(d+1))ifiiseven,alsoBi=[i(d+1),(i+1)(d+1)ifiisoddand[)]TJ /F9 11.955 Tf 9.3 0 Td[((i)]TJ /F9 11.955 Tf 12 0 Td[(1)(d+1),)]TJ /F9 11.955 Tf 9.3 0 Td[((i)]TJ /F9 11.955 Tf 12 0 Td[(2)(d+1))ifiiseven.ThenwehaveA=n[i=0AiandB=n[i=0Biwith: 11

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1. diam(Ai)(d+1)anddiam(Bi)(d+1).ThatmeansthatbothAandBareuniformlyD-boundedcoveringswhereD=d+1. 2. dist(Ai,Aj)(d+1)d,8i6=janddist(Bi,Bj)(d+1)d,8i6=j.ThatmeansthatbothAandBared-disconnected. 3. X=ASB.Thatmeans(accordingtotherstdenitionofasymptoticdimension)thatasdimR1.SinceRisageodesicmetricspaceanditisnotcompactwehavethatasdimR6=0.SoasdimR=1. Theproofaboveisbetterillustratedbythefollowingpicture: Figure2-1. asdimR11 Thedenitionsofasymptoticdimensionareequivalentforgeodesicmetricspaces.Fortheproofwewillneedsomemoredenitions. Denition2.1.10. LetXbeametricspaceanda,b2X.Apathfromatobisacontinuousfunctionp:[0,k]!Xwithp(0)=aandp(k)=b.Mostofthetimesapathpisdenedtobeequaltoitsimagep([0,k]). Denition2.1.11. Wedenethelengthofapathtobelength(p)=supfnXi=0d(p(ti),p(ti+1))g 12

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wheret0,t1,...,tnisadecompositionof[0,1]. Denition2.1.12. AmetricspaceXissaidtobeageodesic metricspaceifforalla,b2Xthereexistsapathp:[0,1]!Xwithp(0)=a,p(b)=bandlength(p)=d(a,b). Proposition2.1.13. Thersttwodenitionsofasymptoticdimension(aswegavetheminthisessay)areequivalentforgeodesicspaces. Proof. LetXbeametricspacewithasdimXnaccordingtotherstdenition.Letd>0.ThenwecanconstructX1,X2,...,Xn+1familiesofsetssuchthat: 1. Xi=[j2JiBij8i=1,2,..,n 2. Xiare2d)]TJ /F3 11.955 Tf 11.96 0 Td[(disconnected 3. X=n[i=1XiWecollectalltheBijandwenamethemBssowehave:X=[s2SBs.ThismeansthatfBsgisacoveringofX.LetxbeapointofXandBd(x)=fy2Xsuchthatdist(x,y)6dgthed)]TJ /F3 11.955 Tf 12.26 0 Td[(ballofx.IfBd(x)intersectsn+2setsofthecoveringfBsgthenatleasttwoofthissetsmustbeinthesameXk.LetthembeB1andB22Xk.SinceXkis2d-disconnectedwehavethatdist(B1,B2)>2d.WealsohavethatBd(x)TB16=;)9x12B1TBd(x)andBd(x)TB26=;)9x22B2TBd(x).Sincex1,x22Bd(x))d(x1,x2)6d.Sincex12B1,x22B2)dist(B1,B2)6d(x1,x2)6d.Thatisacontradiction. 13

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Tosumup8d>09acoveringofX,X=[s2SBssuchthatd-multiplicityn+1.ThatmeansthatasdimXnaccordingtotheseconddenition.NowletXbeametricspacewithasdimXnaccordingtotheseconddenition.a)Ifn=0thenasdimX=0.ThatmeansthatwecanconstructacoveringofX,fBigsuchthateveryballB(x,2d)intersectswithatmostoneBi.WenamethenX1=[i2IBiTheneveryBiisD-bounded.AlsoXiisd-disconnectedsinceifd(Bi,Bj)dthenthereexistx12Biandx22Bjsuchthatd(x1,x2)d.SinceXisageodesicspacewedeneaxinXsuchthatxisonthegeodesic[x1,x2]andd(x1,x)=d(x2,x)(themiddle-point).ThenobviouslytheballB(x,2d)containsbothx1andx2,whichmeansthattheballB(x,2d)intersectswithbothBiandBj.Thatisacontradiction.ThusX1isd-disconnectedandwehavethatasdimX=0accordingtotherstdenition.b)Ifn6=0thenforeveryd>0wecanconstructacoveringfBigi2Isuchthat: 1. diam(Bi)D8i2I 2. Every10n+1d)]TJ /F3 11.955 Tf 12.3 0 Td[(ballofeverypointxintersectswithn+1atmostsetsBiofthecovering.And 3. X=[i2IBiLetdk=4kdfork=1,2,...,n+1.Thenwename:Xk=fx2X:B(x,dk)andB(x,dk+1)intersectwithexactlyksetsofthecoveringBig.Sinceadk-balliscontainedina10n+1d-ballwhichcanintersectwithatmostn+1setsofthecoveringwehavethat 14

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Xk=;8kn+2.ObviouslyX=n+1[k=1XkWewillprovethatalltheXkared-disconnectedsobytherstdenitionofasymptoticdimensionwewillhavethatasdimX6n.InthesubsetXkwedenetherelationxyif9x1,x2,...,xtsuchthat: 1. xi2Xk8i=1,2,...,t 2. x=x1andy=xt 3. d(xi,xi+1)dk8t=1,2,...,t)]TJ /F9 11.955 Tf 11.95 0 Td[(1Theaboverelationisanequivalencerelationbecauseifxythenbytakingtheenumerationofxibackwardswehavethatyx.Alsoxxisobvious.Ifnowxyandyzthenthereexist:x=x1,x2,...xm=yandy=x01,x02,...,x0s=zwithsatisfyingalloftheabove.Thenwehavex=x1,x2,...xm=y=x01,x02,...,x0s=z.Weputxm+i=x0isowehavex=x1,x2,...,xm,xm+1,...,xm+s=zsatisfyingxi2Xk8i=1,2,...,m+sandd(xi,xi+1)dk8i=1,2,...,m+s)]TJ /F9 11.955 Tf 11.96 0 Td[(1soxz.WenameS1,S2,...,St,...theequivalenceclasses.ThenSiareclosedsetsandwehave:Xk=[i2ISiAlsodist(Si,Sj)>dkfori6=jbecauseifdist(Si,Sj)dk,thentherewouldbex12Siandx22Sjwithdist(x1,x2)dk.Butthenx1x2.Thatmeansthatx12Sjandx22Si.Alsox12SiwhichmeansthatSiTSj6=;.Thatisacontradictionbecausefori6=jthetwoequivalentclassesaredisjoint.Sodist(Si,Sj)>dk)dist(Si,Sj)>4kd>d.FurthermoreeverySiisL-bounded.IndeedifSiwasnotL-boundedthen:ForeveryL>0therewouldbeani2Isuchthatdiam(Si)>L.WechooseL>(n+1)DwhereDisthediameteroftheBisetsofthecovering.AndletS0betheclassofXkwithdiam(S0)>L.Thensincediam(Si)>LwehavethatS0intersectswithmorethann+1 15

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setsoftheBicovering.Sincekislessorequalton+1wehavethatS0intersectswithmorethanksetsofthecovering.Sotherearex0andx1inS0suchthat: 1. B(x0,d)intersectswithksetsoftheBicovering,namelyB1,B2,...,Bk+1 2. B(x1,d)intersectswithksetsoftheBicovering,namelyB01,B02,...,B0k+1withatleastoneoftheB0jbedifferentthantheBj. 3. dist(x0,x1)dkThenwedenetheballB(x0,dk+1)=B(x0,4dk)whichclearlycontainsbothB(x0,dk),B(x1,dk)balls.ThustheballB(x0,dk+1)intersectswithatleastk+1setsoftheBicovering.ThatisacontradictionbecausetheballB(x0,dk+1)mustintersectwithksetsofBisincex0isinXk.Thusdiam(Si)
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Formoredetailsonthisdenitionandtheproofoftheproducttheoremsee( 1 ),( 3 )and( 34 ).ForthefollowingexampleweneedtheLebesqueCoveringTheoremfoundin( 25 ). Theorem2.1.16. LebesqueCoveringTheorem:LetInbethen-dimensionalcubeandletInbethesumofniteclosedsetsBi,i=1,2,...,r(thefamilyfBigissaidtobeacoveringofIn)noneofwhichcontainspointsfromtwooppositefaces.Thenatleastn+1oftheseclosedsetshaveacommonpoint. Example2.1.17. FortheEuclideanspaceofdimensionnwehaveasdimRn=n Proof. FirstletsprovethatasdimRnn.Letssupposethatthestatementisfalse.ThenasdimRnn)]TJ /F9 11.955 Tf 12.67 0 Td[(1.Letd>0thenwecanconstructacoveringfBigofRnsuchthat82XtheballS(x,d)intersectswithatmostnsetsofthecovering.LetInbeacubeofRnwithfCi,C0ig,i=1,2,...,nbeitsopposingfaces,withdist(Ci,C0i)>100D+100d.WenameKi=[x2BiS(x,e).Where0
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1. Inm[i=1Hi. 2. Hiareclosedsetsforalli=1,2,...,m.SupposenowthatanHiintersectswithtwoopposingfacesofInnamelyC1,C01.Thentherewouldbex1,x22Hisuchthatx12C1andx22C01.Thusdist(x1,x2)>100D+100d.SodiamHi>100D+100d.ThatisnottruesincediamHi=diamKi6diamBi+2e
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Denition2.1.18. LetXandYbetwometricspaces.Andf:X!Yafunctionsuchthatthereisac>0withthefollowingproperties: 1. 1 cd(x,y))]TJ /F3 11.955 Tf 11.96 0 Td[(c6d(f(x),f(y))6cd(x,y)+c 2. 8y2Y,9x2X:d(f(x),y)6cThenwesaythatfisaquasi)]TJ /F3 11.955 Tf 11.95 0 Td[(isometry andthatthetwospacesarequasi-isometric. Remark2.1.19. Therelationofquasi-isometryisanequivalencerelation. Remark2.1.20. TwoisometricspacesX,Yarequasi-isometrici.e.Thereexistsanf:X!Yisometrywhichisaquasi-isometrywithc=1. Proposition2.1.21. Twoquasi-isometricspaceshavethesameasymptoticdimension. Proof. LetXbeametricspacewithasdimXnandf:X!Yaquasi-isometrybetweenXandY.Letd>0thenweconstructacoveringfBigofXsuchthat: 1. X=[i2IBi 2. 8i2I,diam(Bi)DforaD>0 3. EveryballB(x,4c2+4dc)intersectswithatmostn+1setsofthecoveringfBig.WethendenefB0igacoveringofYtobeB0i=fy2Y:9x2Bisuchthatd(f(x),y)
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havethat:y2[i2IB0iThusfB0igisacoveringofY.Furthermorelety1andy2betwoelementsofB0i.Thenthereexistx1andx2bothinBisuchthatd(f(x1),y1)
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Thepropositionabovestatesthatasymptoticdimensionisaquasi-isometryinvariant.Alsoitisveryhelpfulinsomeexampleslikethefollowing. Example2.1.22. AninnitetubeXwithdiameterk,hasasdimX=1Thisisobvioussincetheabovetubeisquasi-isometrictotheEuclideanlineandasdimR=1.Letsstatenowadenitionofasymptoticdimensionbasedonmapsonsimplicialcomplexes. Denition2.1.23. (ASYMPTOTICDIMENSION)LetXbeapropermetricspace.Wesaythatitsasymptoticdimensionisnifforeach>0thereexistsan-LipschitzanduniformlycoboundedmapfromXtoann-dimensionalpolyhedra(whichisthegeometricalrealizationofann-dimensionalsimplicialcomplex)equippedwithitsafnemetric.TheabovedenitionwasintroducedbyGromovin( 23 )andrenedbyDranishnikov.Itisequivalenttotheothertwo.Ashortproofoftheequivalencecanbefoundin( 34 ). Corollary2.1.24. LetXbeametricspaceandYX.ThenasdimYasdimX Proof. Letd>0andletasdimX=nthenweconstructacoveringUofXbyuniformlyD-boundedsetswithd-multiplicityn+1.ThentherestrictionofthiscovertoYgivesusauniformlyS-boundedcoveringwithSDwithd-multiplicityatmostn+1.SoasdimXn)asdimYasdimX ThefollowingveryusefulpropositionisduetoBellandDranishnikovanditisfoundin( 1 ). Proposition2.1.25. LetX=X0SX00beametricspace.Then:asdimX=maxfasdimX0,asdimX00g 21

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2.2CayleyGraphs Denition2.2.1. Agraph )]TJ /F11 11.955 Tf 10.1 0 Td[(ismadeoftwosets: 1. )]TJ /F8 7.97 Tf 6.78 -1.79 Td[(0=V=fverticesof)]TJ /F2 11.955 Tf 6.78 0 Td[(g 2. )]TJ /F8 7.97 Tf 6.78 -1.79 Td[(1=E=fedgesof)]TJ /F2 11.955 Tf 6.77 0 Td[(gandtwofunctions: 1. (o,t):E!VVwithl!(o(l),t(l))whereo(l)iscalledthestartingvertexandt(l)iscalledtheendingvertex. 2. )]TJ /F9 11.955 Tf 12.62 0 Td[(:E!Ewithl! lthereversesideofl.suchthat: i) l=l ii) l6=l iii) o(l)=t(l)A'geometricalside'isthepairfl, lg(anedgeanditsreverseedge). Denition2.2.2. IfGhasnitelymanygeneratorsnamelyfg1,g2,...,gngthenwecallitnitelygenerated.ThatmeansthateveryginGcanbewrittenintheformg=ga1i1ga2i2...gakikwithai2Zforalli=1,2,..,kandgij Denition2.2.3. LetG=bearepresentationofGsuchthatSisanitesetofgeneratorsandSisanitesetofrelations.ThenGissaidtobenitelypresented. Denition2.2.4. (CAYLEYGRAPH)LetGbeanitelypresentedgroupnamelyG=.ThenwedenetheCayleygraphofG()]TJ /F4 7.97 Tf 6.77 -1.79 Td[(S(G))asfollows.Wename: 1. V()]TJ /F11 11.955 Tf 6.77 0 Td[()=fgjg2GgalltheelementsofBtobethesetofvertices 2. E()]TJ /F11 11.955 Tf 6.77 0 Td[()=f(g,gs)and( g,gs),g2G,s2Sgthesetofalledges. 3. o(g,gs)=g 4. t(g,gs)=gsEveryedge(g,gs)is'labeled'withs. 22

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Denition2.2.5. Apath pin)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(S(G)isasequenceofedgesp=(l1,l2,...,ln)suchthatt(li)=o(li+1)fori=1,2,...,n)]TJ /F9 11.955 Tf 11.87 0 Td[(1.Wedeneo(p)=o(l1)andt(p)=t(ln)thebeginningaddtheendofthepathp. Denition2.2.6. LetGbeanitelypresentedgroupG=and)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(S(G)theCayleygraphofG.Then)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(S(G)becomesametricspaceasfollows: i) Everyedgeisregardedequalto[0,1] ii) Foreveryx,y2VWedened(x,y)=minflength(p),papathwitho(p)=xande(p)=yg Remark2.2.7. Theabovefunctiondiseasilyprovedtobeametric.Ifweconnedontheverticesof)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(S(G)thenwedenedswhichiscalledthewordmetric. Example2.2.8. F2=thefreegroupwithtwogeneratorshasaCayleygraphwhichisatreesuchthatfromeveryverticefouredgesstart. Example2.2.9. IfG=Z6=theCayleygraphisahexagon(seepreviouspicture). Figure2-3. Z6 Remark2.2.10. IfGisnitelypresentedwehavethat)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(S(G)isageodesicmetricspace. 23

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Denition2.2.11. Let)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(S(G)beaCayleygraphofGandginG.Wedenejgj=dS(g,e)whereeistheidentityofthegroup. Denition2.2.12. LetGbeanitelygeneratedgroupwithnitesymmetricgeneratingsetS.WedeneanormonthegroupGcorrespondingtoSbysettingkwkSequaltotheminimalnumberofS-lettersnecessarytopresentawordequaltow.Thisnormyieldsaleft-invariantwordmetricon)]TJ /F4 7.97 Tf 6.78 -1.8 Td[(S(G)bydS(g,h)=kg)]TJ /F8 7.97 Tf 6.59 0 Td[(1hkS. Remark2.2.13. ACayleygraph)]TJ /F4 7.97 Tf 6.77 -1.8 Td[(S(G)changes'shape'ifwechangethegeneratorsofthegroup. Example2.2.14. Z=withS1=xandZ=withS2=fx2,x3gwhichgivesthefollowingtwoCayleygraphs)]TJ /F4 7.97 Tf 6.77 -1.79 Td[(S1(G)and)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(S2(G) Figure2-4. Z= Figure2-5. Z= Lemma2.2.15. If)]TJ /F4 7.97 Tf 6.78 -1.8 Td[(S(G)isthecayleygraphofagroupwithdSitsmetricandginGwehavethatd(h,hg)6jgj. 24

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Proof. Letjgj=nsog=s1s2...snwheresiaregeneratorsofG.Thenhg=hs1s2...snwhichgivesthatd(h,hg)6d(h,s1h)+d(s1h,s2s1h)+...+d(s1...sn)]TJ /F8 7.97 Tf 6.59 0 Td[(1h,s1...snh)=1+1+...+1=n=jgjandwehavethelemma. Proposition2.2.16. (MILNORSVARETHEOREM)LetGbeanitelygeneratedgroupswithG=andG=whereS1andS2aretwodifferentnitesetsofgenerators.Then)]TJ /F4 7.97 Tf 6.78 -1.8 Td[(S1(G)isquasi-equalto)]TJ /F4 7.97 Tf 6.78 -1.8 Td[(S2(G). Proof. LetS1=fa1,a2,...angandS2=fb1,b2,...bkg.Weputl=maxfjaijS2:ai2S1gandm=maxfjbijS1:bi2S2g.Wewillprovethatid:)]TJ /F4 7.97 Tf 13.77 -1.79 Td[(S1(G)!)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(S2(G)isl+m-bilipsitzmeaningthat1 l+mdS1(g1,g2)6dS2(g1,g2)6(l+m)dS1(g1,g2).Thesecondpropertyiseasilycheckedforeveryc>0becauseforeveryg2)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(S2(G)thereisthesamegin)]TJ /F4 7.97 Tf 6.78 -1.79 Td[(S1(G)suchthatd(f(g),g)=d(g,g)=0
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Remark2.2.18. Weremarkthatd(x,y)>d(f(x),f(y))foreveryx,yinthevanKampenDiagramwhered(x,y)isthedistancebetweentwopointsintheVK-diagramandd(f(x),f(y))isthedistanceoftheimageofthosepointsontheCayleyGraph.ThesediagramswerediscoveredbyR.vanKampenin1993,butvanKampendidnotmakemuchuseofthem.Theholeideawasneglectedforthirtyyears.Theywererediscoveredin1966whentheywereuseinsmallcancelationtheorybyR.C.LyndonandC.M.Weinbaum.ThroughCayleyGraphsonecanextendthenotionofasymptoticdimensiontogroups.Namely: Denition2.2.19. LetGbeanitelygeneratedgroup.ConsidertheCayleygraphofG,withthewordmetric)]TJ /F11 11.955 Tf 10.09 0 Td[(whichisageodesicmetricspace.ThenwedeneasdimG=asdim)]TJ /F11 11.955 Tf 6.77 0 Td[(.BellandDranishnikovin( 2 )haveprovedmanyusefulpropertiesofasymptoticdimensioningroupsincludingthefollowingformula: Proposition2.2.20. (Hurewicztypeformula)Considerashortexactsequenceofcountablegroups:0!A!B!C!0then:asdimBasdimA+asdimCThedenitionofAsymptoticdimensionisgeneralizedtoarbitrarydiscretegroupsbyDranishnikovandSmith(?)asfollows: Denition2.2.21. LetGbeadiscretegroup.Dene:asdimG=supfasdimFjFG,Fnitelygeneratedg 2.3MainResultsThedenitionofone-endedspacesfollows.Weneeditinordertobeabletoprovethemaintheoremofthischapter. 26

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Denition2.3.1. AgeodesicmetricspaceXissaidtobeone-endedifforeveryboundedK,X)]TJ /F3 11.955 Tf 12.03 0 Td[(Khasexactlyoneunboundedconnectedcomponent.WesaythatXisuniformlyone-endedifforeveryn2R+thereism2R+suchthatforeveryKXsuchthatdiam(K)m. Example2.3.2. R2isuniformlyone-endedsinceforeveryxinR2wedeneK= B(x,d)=fy2R2:d(y,x)n)]TJ /F9 11.955 Tf 12.56 0 Td[(1theballaroundxwithradiusn)]TJ /F9 11.955 Tf 12.56 0 Td[(1.ThenKiscompactandX)]TJ /F3 11.955 Tf 12.02 0 Td[(K=fy2R2:d(x,y)>n+1g.ObviouslyX)]TJ /F3 11.955 Tf 12.03 0 Td[(Khasoneconnectedcomponentofdiameter>mform=n+1anddiam(K)n0.SincetheCayleygraphofGisaconnectedgeodesicmetricspace,thereexistsageodesicpathtconnectingeandg.Sothelengthofthepathismorethann0.Obviouslywecanndasub-patht0oftstartingfromewithlengthn0.Bydenitiont02Cn0.thisisacontradiction.ThusG2 S(e,n0). 27

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ButGisnitelygeneratedandboundedandthusitisalsonitewithjGjn0i=0ki=kn0+1)]TJ /F9 11.955 Tf 11.96 0 Td[(1 k)]TJ /F9 11.955 Tf 11.95 0 Td[(1wherekisthenumberofgeneratorsofthegroup.ThatisacontradictionsinceGisone-endedmeaningithasoneinnitecomponent.b)AllpathsinCnarenaturallyinS(e,n).InsidethespherewehaveatmostjGjni=0ki=kn+1)]TJ /F9 11.955 Tf 11.95 0 Td[(1 k)]TJ /F9 11.955 Tf 11.95 0 Td[(1vertices.Sinceourverticesarenite,soareouredges.ThusthepathswecanconstructwiththisedgesarealsoniteandsothegeodesicpathswecanconstructarealsonitesojCnj<1.c)Consider:Cn!Cn)]TJ /F8 7.97 Tf 6.59 0 Td[(1,whichtakeseverygeodesicpathoflengthnfromCnandcutsoffthelastedgeifthatpath.Clearlyweobtainageodesicpathoflengthn)]TJ /F9 11.955 Tf 12.46 0 Td[(1andsoitiscontainedinCn)]TJ /F8 7.97 Tf 6.59 0 Td[(1.Thusweobtainthefollowinginverselimitsequence:feg C1 C2 ... Cn Cn+1 ...withthearrowsbeingthemapsn.Sinceeveryinverselimitofcompactspacesexistsanditisalsocompactwehavethatthereexists:F=lim fCn,ngLetr2F.Clearlyrisageodesicpathforeveryn2Nandrisinnitestartingfrome.Thusrisaninnitegeodesicraystartingfrome.NowconsiderKn=fg2Gsuchthatd(g,e)=ng 28

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thesetofallverticesofGthataredistancenfrome.AlsodeneL1tobethesetofedgesthatstartfrome.Finallydene:En=fri:riaregeodesicraysofG,ri(0)2Kn,e2rigthesetofgeodesicraysstartingfromKnandpassingthroughe.ClearlyE1canbeseenasasubsetofFL1.NamelyifYisthesubsetofE1suchthatallgeodesicsstartwiththeedgethen:E1=[2L1YLet=(e,se).ThenY2sF.ObviouslyE1isnon-empty.Toseethisletrbethegeodesicraywedenedstartingfrome.Let(e,se)beitsrstsegment.Thendenes)]TJ /F8 7.97 Tf 6.59 0 Td[(1r.Sincemultiplyingbyanelementdoesn'tchangetherespectivedistancess)]TJ /F8 7.97 Tf 6.59 0 Td[(1risalsoageodesicstartingfroms)]TJ /F8 7.97 Tf 6.58 0 Td[(1eandpassingthroughe=s)]TJ /F8 7.97 Tf 6.59 0 Td[(1s.ThusE16=;.WiththesameideawecanprovethatEn6=;.Sonowwecantaketheinverselimit:feg E1 E2 ... En En+1 ...wherethemapsnowdropthestartingedgeoftheray.ThenagainwehavethatE=lim Enexistsanditsnonemptyandcompact.Considerak2E.Thenkisapathpassingthroughe,itisageodesicline,andforeveryn2Nthereexistexactlytwopointsinksuchthatthedistanceofthemis2nandthedistanceofbothofthemfromthepointeisn.Thatkisourbi-innitegeodesicline. Wearenallyreadyforthemaintheoremofthissection. Theorem2.3.5. IfGisanone-endednitelypresentedgroupthenasdimG2 29

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Proof. SinceGisoneendedwehavethatGisinniteandthuswehaveasdimG>0.SupposeasdimG=1.LetM=maxfjrij,i=1,2,...,n,jriisarelationofGgwherejrj=lengthofthewordr.Fixd>100M+100.SinceasdimG=1thereisacoveringB=fBigwithG=[i2IBianddiamBi2N.Wehavethenlength(w)=length([x,y])+length(p)2N+length(p))length(w)>200D.Soinordertocoverthepathwweneedatleast3setsofthecoveringfBig.WeconsidernowtheVan-KampendiagramDthatcorrespondstothepathwandthefunctionffromD(1)totheCayleygraphofG.Sof(@D)=w.Fornotationalconveniencewelabelverticesandedgesof@DLetBbeasetofthecoveringthatintersect[x,y].Weconsiderf)]TJ /F8 7.97 Tf 6.58 0 Td[(1(B).LetC(B)betheunionofall2-cellsofDwhichhavethepropertythattheirboundaryiscontainedinf)]TJ /F8 7.97 Tf 6.58 0 Td[(1(B).LetUbethecollectionofallsuchsetsBwiththefollowingproperty:Forsomeconnectedcomponent,KofC(B),[x,y]\Kiscontainedinaninterval[a,b]with 30

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a,b2Ksuchthatx02[a,b].Letd(K)=d(a,b)forsickacomponentandletd(B)bethemaximalvalueforalld(K)fortheKcomponentofC(B)suchthatx02[a,b].LetB1beasetinUforwhichd(B1)ismaximal.LetK1betheconnectedcomponentofC(B1)forwhichd(K1)=d(B1).Let'ssaythatK1\[x,y]iscontainedin[a1,b1]witha1,b12K1.Letebetheedgeof[x,y]adjacenttoa1whichdoesnotliein[a,b].Ifristhe2-cellcontainingethereissomeB22BsuchthatC(B2)containsrLetCbethesubsetoftheVanKampendiagramDwhichcontainsall2-cellswithboundarycontainedinf)]TJ /F8 7.97 Tf 6.59 0 Td[(1B1[B2.Sinced(x0,p)N,Cdoesnotintersectp.ThusD)]TJ /F3 11.955 Tf 9.97 0 Td[(C6=;.LetLbetheconnectedcomponentofCwhichcontainsx0.LetP= D)]TJ /F3 11.955 Tf 11.96 0 Td[(L\ L.PisconnectedsinceKisconnected.EachedgeofPiscontainedintwo2-cells.Oneofthese2-cellsliesinf)]TJ /F8 7.97 Tf 6.59 0 Td[(1(B1[B2),andonedoesnotlieinthisset.ItisnotpossiblethatalledgesofParecontainedinone2-celloff)]TJ /F8 7.97 Tf 6.59 0 Td[(1(B2).Indeedinthiscasewewouldhaved(B2)>d(B1)whichisimpossible.SinceriscontainedinC(B2),someedgeofPisnotcontainedina2-celloff)]TJ /F8 7.97 Tf 6.59 0 Td[(1(B1).Itfollowsthattherearetwoadjacentedgese1,e2inPsuchthatoneofthemiscontainedina2-celloff)]TJ /F8 7.97 Tf 6.58 0 Td[(1(B1)andtheotherina2-celloff)]TJ /F8 7.97 Tf 6.59 0 Td[(1(B1).Ifcisthe2-cellthatcontainse1andisnotinf)]TJ /F8 7.97 Tf 6.59 0 Td[(1(B1[B2),thencliesinasetf)]TJ /F8 7.97 Tf 6.59 0 Td[(1(B3)withB32BandB36=B1,B2.Theedgese1,e2andthe2-cellchaveavertexvincommon.Sov2f)]TJ /F8 7.97 Tf 6.59 0 Td[(1(B1\B2\B3).ItfollowsthatB1\B2\B36=;whichisacontradiction. Remark2.3.6. Weremarkthattheproofaboveappliesalsoforuniformlyone-endedsimplyconnectedsimplicialcomplexes.Ofcoursetheresultdoesnotholdforone-endedsimplyconnectedsimplicialcomplexes,ahalf-linegivesacounterexample.Furthermorethereareone-endedgraphs,evenuniformlyone-ended,whichhaveasymptoticdimension1.Thefollowingpictureisanexample:Recallnowthedenitionofvirtuallyfreegroups. 31

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Figure2-6. AuniformlyoneendedspacewithasdimX=1 Denition2.3.7. WesaythatagroupGisvirtuallyfreeifthereexistsaniteindexsubgroupHofGwhichisafreegroup.Wenotethatthefollowingtheoremholds: Theorem2.3.8. (Dunwoody-Stallings( 15 ))IfGisanitelypresentedgroupthenGisthefundamentalgroupofagraphofgroupssuchthatalltheedgegroupsareniteandallthevertexgroupsare0or1ended.Alsoitisknownthat: Proposition2.3.9. Ifallthevertexgroupsare0-ended(i.e.nite)thenGisvirtuallyfree.(see( 35 ),page120,prop.11)Furthermorethefollowinglemmaisobvious: Lemma2.3.10. IfH
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RememberthattheCayleygraphofHisasubgraphofGandthatasymptoticdimensionismonotonous.UsingtheaboveandDunwoody-Stallings'stheoremwehavethestrongerresult: Theorem2.3.11. IfGisanitelypresentedgroupwithasdimG=1thenGisvirtuallyfree. Proof. LetGbeaf.p.groupwithasdim=1.Let)]TJ /F1 11.955 Tf 10.1 0 Td[(bethegraphofgroupsoftheDunwoody-Stallingstheorem.IfavertexgroupHisone-endedthenfromthetheoremwehaveprovenasdimH2.ButH
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SinceGisanitelygeneratedinnitegroupwehavethatasdimG6=0.SoasdimG=1.WewillprovethatGisnotvirtuallyfree.Supposeitwas.ThenthereexistsasubgroupSofGsuchthatSisfreeandtheindexjG:Sjisnite.LetN=NG(S)=fgsg)]TJ /F8 7.97 Tf 6.59 0 Td[(1:g2Gands2SbethenormalizerofS2G.WeknowthatNisanormalsubgroupofGandsincejG:SjisnitesoisjG:Nj.Recalltheshortexactsequence:0!(ZZ2)!G!Z!0anddenotetheimageofZZ2inGasH.WeclaimthatN\H=feg.Thisistruesinceifx2N\H=fegwehavethatsincex2HthereexistsayZZ2withf(y)=x.Butsincey2=e2ZZ2wehavex2=f(y)2=f(y2)=f(e)=e.Sincex2Nthereexistsg2Gandw2Ssuchthatx=gwg)]TJ /F8 7.97 Tf 6.58 0 Td[(1.Thuse=x2=gwg)]TJ /F8 7.97 Tf 6.59 0 Td[(1gwg)]TJ /F8 7.97 Tf 6.59 0 Td[(1=gw2g)]TJ /F8 7.97 Tf 6.59 0 Td[(1)e=w2.SincewbelongsinSwhichisfreewehavethatw=e.Thisimpliesx=gwg)]TJ /F8 7.97 Tf 6.59 0 Td[(1=geg)]TJ /F8 7.97 Tf 6.59 0 Td[(1=eprovingtheclaim.FromthesecondtheoremofgroupisomorphismsandsinceNisanormalsubgroupofGwehave:H H\N'HN NButH\N=fegandHN N
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CHAPTER3LIMITAPERIODICGROUPS 3.1MotivationThischapterdiscussestheideaoflimitaperiodicgroups.IfadiscretegroupGactsbyisometriesfreelyandcocompactlyonametricspaceXonecanstudyperiodicandaperiodictilingsofX.AtilingofXcanbedenedrstasatilingwithonetile,theVoronoicell(see( 14 )).UsinganitesetofcolorsonecanconsidertilingsofXbycolor.Thenusingthenotchingonecanswitchfromatilingbycolortoageometrictiling.ThestandardexamplehereisG=Z2andX=R2.NotethatthegroupGintheabovetilingsisinbijectionwiththetiles.Thus,constructionofageometrictilingonXcanbereducedtoacoloringofthegroupG.InthispaperwestudythecoloringsofdiscretegroupsGthatleadtolimitaperiodictilings.Letb2G,acoloringofagroupGisb-periodicifitisinvariantundertranslationbyb,i.e.,foreveryelementg2Gtheelementsgandbghavethesamecolor.Acoloringisaperiodicifitisnotb-periodicforanyb2Gnfeg.ThiscanberephrasedasThestabilizerofinthespaceofallcoloringsofGistrivial.Forinnitegroupsthereisastrongnotionofperiodicity:AcoloringisstronglyG-periodiciffortheorbitofundertheactionofG,wehavejOrbG()j<1.Thecorrespondingnegationcalled'weaklyaperiodic'meansthatOrbG()=G=StabG()ofisinnite.Acoloringiscalled(weakly)limitaperiodicifallcoloringsintheclosureoftheorbit OrbG()takeninanappropriatespaceofallcoloringsare(weakly)aperiodic.Inthischapterweconsiderthequestionraisedin( 14 ):Whichgroupsadmitlimitaperiodiccoloringsbynitelymanycolors?ThisisnotanobviousquestionevenforG=Z.In( 14 )itwasansweredpositivelyfortorsionfreehyperbolicgroups,Coxetergroups,andgroupscommensurabletothem.ThisquestioncanbestatedintermsofTopologicalDynamicalSystemstheory:LetGbeagroupandFbeaniteset.DoesthenaturalactionofGontheCantorsetFG 35

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admitaG-invariantcompactsubsetXFGsuchthattheactionofGonXisfree?ThedynamicalsystemreformulationofacorrespondingquestionaboutlimitweakaperiodiccoloringsasksaboutaG-invariantcompactsubsetXFGsuchthattheorbitsOrbG(x)areinniteforallx2X.ThiswasansweredafrmativelybyV.Uspenskii( 14 ).Inthischapterwegiveagrouptheoreticapproach.Wecallagroup`limitaperiodic'(LAforshort)ifitadmitsalimitaperiodiccoloringbynitelymanycolors.Weshowthatthesimplegroupconstructionsliketheproduct,theextension,theHNNextension,andthefreeproductpreservetheLAproperty.ToprovethesefactsweintroducethenotionofLAG-setandprovetheactiontheorem.WethendescribeaspecicG-set(thenaturalnumbers),whereGisaf.g.subgroupofthegroupofautomorphismsofZ,whichisnotlimitaperiodic.TofurtherinvestigatelimitaperiodicG-setsweintroducetheideaoflimitaperiodicsubgroupsanddiscusstheirproperties.Usingalgebraicresults,SuGao,SteveJacksonandBrandonSeward,( 18 )producedapaperthatprovesallcountablegroupstobelimitaperiodicbyusingonly2colors.Thispaperisextremelytechnical,whereasourproofsareelementaryand(hopefully)easytofollow.Alsotheirpaperislimitedtonitelygeneratedgroups. 3.2LimitAperiodicGroupsLetsstartwiththebasicdenitionsoncoloringsofagroup.LetGbeaf.g.group.Also,letFbeanitesetofelementswhichwecanthinkofascolors.AmapfromGtoFiscalledacoloringofG.WedenotebyFGthesetofallcoloringsfromGtoF.IfweconsiderFwiththediscretetopology,FGwiththeproducttopologybecomesatopologicalspacehomeomorphictotheCantorsetandGactsonFGwiththeleftaction,denedbytheformula(gf)(a)=f(g)]TJ /F8 7.97 Tf 6.58 0 Td[(1a)foreveryg,a2Gandf2FG.SinceFGismetrizable,afunctionbelongstotheclosureoftheorbitoff,2 OrbG(f),ifandonlyif=limk,fkgOrbG(f).Thisisequivalenttotheexistence 36

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ofasequencefhkgGwithk=hkf.Thecondition=lim(hkf)impliesthatforeveryg2Gthereexistsak(g)2Nwith:(g)=hkf(g)forallkk(g).Thefollowingpropertywillbeusedalotinthefollowingparagraphs.It'sanelementarypropertyofthespaceconstructedabove. Property3.2.1. Givenasequenceoffunctionsfromacountablesettoanitesetthereexistsasubsequencethatconvergespoint-wise.Thuseverytimewehavethefamilyofmapsfhkfgforsomesequencehk2Gwecanconsidertheconvergingsubsequenceandchangetheindicestoproducealimit=lim(hkf). Denition3.2.2. LetGbeanitelygeneratedgroupandFanitesetofcolors,amapf:G!Fiscalledaperiodiciftheequationbf=fimpliesb=e.Iftheequationbf=fholdsforsomeb2Gwecallfb-periodicandbiscalledaperiodoff. Denition3.2.3. (LA1)LetGbeanitelygeneratedgroupandFanitesetofcolors,amapf:G!Fiscalledlimitaperiodicifandonlyifevery2 OrbG(f)isaperiodic. Denition3.2.4. (LA2)LetGbeanitelygeneratedgroupandFanitesetofcolors,amapf:G!Fwillbecalledlimitaperiodicifforeveryg2GnfegthereexistsanitesetSG,S=S(g),suchthatforeveryh2Gthereisac2Swithf(hc)6=f(hgc). Proposition3.2.5. Thesetwodenitionsareequivalentfornitelygeneratedgroups. Proof. Supposethatfsatisesthe(LA2)propertybutnotthe(LA1).Thenthereexistsa2 OrbG(f)suchthathasperiodg6=e.Theng)]TJ /F8 7.97 Tf 6.58 0 Td[(1isalsoaperiod.Letfhkgk2N2Gsuchthat=limhkf.ChoosethesetSforthatg.SinceSisnitewealsohavethatgSisnite.Fromthefactthat=limhkf,thereexistsann2Nsuchthatforallknandforallx2SSgSwehave(x)=hkf(x).WeapplyLA2forfwithgandh)]TJ /F8 7.97 Tf 6.59 0 Td[(1ntoobtainc2Ssuchthatf(h)]TJ /F8 7.97 Tf 6.58 0 Td[(1nc)6=f(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1ngc).Thiscontradictswiththefactthatg)]TJ /F8 7.97 Tf 6.59 0 Td[(1isaperiodfor:(c)=(hnf)(c)=f(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1nc)6=f(h)]TJ /F8 7.97 Tf 6.58 0 Td[(1ngc)=(hnf)(gc)=(gc)=(g)]TJ /F8 7.97 Tf 6.59 0 Td[(1)(c) 37

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Let'ssupposenowthatfsatisesthe(LA1)butnotthe(LA2).Thenthereexistsag2GsuchthatforeverynitesubsetSofGthereexistsanh2Gwiththeproperty:f(hc)=f(hgc)forallc2S.Fixthatg2G.TakeS1=fc2G:d(c,e)1gThedistancementionedistheoneinducedbythewordmetriconG.SinceGisf.g.jS1j<1,so,thereexistsanh12Gwithf(h1c)=f(h1gc)forallc2S1.TakeS2=fc2G:d(c,e)2g.AgainjS2j<1.Thenthereexistsanh22Gsuchthatf(h2c)=f(h2gc)forallc2S2.Continueforanyk2N.Thusweobtainasequencefhkgk2N2G.Takingasubsequencewemayassumethatthereisalimit:=limk!1h)]TJ /F8 7.97 Tf 6.59 0 Td[(1kf.Theclaimisthatisperiodicwithperiodg.Consideranarbitraryx2G.Namek1=d(x,e),thenx2Skforallkk1.Alsosinceisthelimitofh)]TJ /F8 7.97 Tf 6.59 0 Td[(1kfthereexistsak22Nsuchthatforallkk2:(x)=(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1kf)(x)Finallysinceisthelimitofhkfthereexistsak32Nsuchthatforallkk3:(gx)=(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1kf)(gx)Thus,forkmaxfk1,k2,k3gwehave:(x)=(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1kf)(x)=f(hkx)=f(hkgx)=(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1kf)(gx)=(gx)=g)]TJ /F8 7.97 Tf 6.59 0 Td[(1(x)Sincexwaschosenarbitrarily,wehavethathasg)]TJ /F8 7.97 Tf 6.58 0 Td[(1asaperiod.Thisisacontradictionsincebelongstothe OrbG(f)andfhasthe(LA1)property. 38

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Denition3.2.6. AnitelygeneratedgroupGwillbecalledlimitaperiodicifitadmitsalimitaperiodiccoloringf:G!FwithanitesetofcolorsF. Remark3.2.7. Thedenitionoflimitaperiodicgroupscaneasilybeextendedtoanygroupandnotonlynitelygeneratedones.Boththeproperty(LA1)and(LA2)applytogroupswithoutthef.g.hypothesis.Theirequivalencethoughdependsonthefactthatthegroupisnitelygenerated.Forusagroup(notnecessarilynitelygenerated)willbelimitaperiodicifitsatisesthe(LA1)property. Proposition3.2.8. Integersarealimitaperiodicgroup. Proof. WecolorZwithaMorse-Thuesequenceasin( 14 ),i.e.f:Z!f0,1gwithf(i)=m(jij)wherem:N!f0,1gistheMorse-Thuesequence( 31 ),(?).Itisshownin( 14 )thatthismapislimitaperiodic. Werecallthenotionofuniformaperiodicityfrom( 14 )andthecorrespondingnotation: Notation3.2.9. ConsideragroupGanddbethewordmetric.Wedenotethedisplace-mentofgathwith:dg(h)=d(gh,h)WithBr(h)wedenotetheballofradiusrwithcenterh.Finallykgk,thenormofg,isthedistancebetweengandenamely:kgk=d(g,e) Denition3.2.10. LetGbeanitelygeneratedgroup.Amapf:G!FwhereFisaniteset(ofcolors)willbecalleduniformlyaperiodicifthereexistsaconstant>0suchthatforeveryelementg2Gnfegandeveryh2G,thereexistsb2Bdg(h)(h)withf(gb)6=f(b). Denition3.2.11. AnitelygeneratedgroupiscalleduniformlyaperiodicifthereexistsannitesetFandafunctionf:G!Fsuchthatfisuniformlyaperiodic.Thenotionofuniformaperiodicitydoesnotdependonthechoiceofgenerators(see( 14 )). 39

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Proposition3.2.12. Iff:G!Fisuniformlyaperiodicthenfislimitaperiodic. Proof. WeshowthatfsatisesLA2.Supposenot.Thenletg2Gnfeg.DeneS=Bkgk(e)tobetheballwithcentereandradiuskgk.ClearlysinceGisnitelygenerated,Sisnite.Assumethatthereexistsanh2Gsuchthatforeveryc2Swehavef(hc)=f(hgc).Denotea=hgh)]TJ /F8 7.97 Tf 6.59 0 Td[(1.WeapplytheUAconditionforfwithaandhtoobtainbinBda(h)(h)withf(ab)6=f(b).Sinceb2Bda(h)(h)wehavethat:d(b,h)da(h)=d(hgh)]TJ /F8 7.97 Tf 6.59 0 Td[(1h,h)=d(hg,h)=d(g,e)=kgkwherethethirdequalitycomesfromthefactthatthemetricisleftinvariant.Noticethat:d(h)]TJ /F8 7.97 Tf 6.58 0 Td[(1b,e)=d(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1b,h)]TJ /F8 7.97 Tf 6.59 0 Td[(1h)=d(b,h)Thusd(h)]TJ /F8 7.97 Tf 6.58 0 Td[(1b,e)kgk.Thisimpliesthatc=h)]TJ /F8 7.97 Tf 6.59 0 Td[(1bbelongstoS.Sof(b)=f(h(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1b))=f(hc)=f(hgc)=f(hgh)]TJ /F8 7.97 Tf 6.59 0 Td[(1b)=f(ab)whichisclearlyacontradiction. 3.3G-SetsAndLimitAperiodicityThenotionoflimitaperiodicitycanbegeneralizedinthecaseofG-Sets.NamelyletXbeaset(mostofthetimesametricspace)andsupposethatGactsonXwithaleftaction,givingitthestructureofaG-set.Wewillusethenotationgx=gxforg2Gandx2X.FixanitesetF,whichwecanconsideragainascolors.DenotebyFXthesetofallmapsfromXtoF.ThenFXcanbecomeaG-setunderthefollowingaction:(gf)(x)=f(g)]TJ /F8 7.97 Tf 6.59 0 Td[(1x)forallx2X,g2Gandf2FX. 40

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Alsodenoteby:Fix(X)=fg2G:gx=x,8x2Xgthekerneloftheaction. Denition3.3.1. LetXbeaG-setandletf2FX.Wecallflimitaperiodicifandonlyifforevery2 OrbG(f)wehavethatisaperiodicmeaningthatifa=,thena2Fix(X).WearenowreadyforthedenitionoflimitaperiodicG-sets: Denition3.3.2. LetXbeaG-set.IfthereexistsanitesetFandamapf:X!FsuchthatfislimitaperiodicwesaythatXisalimitaperiodicG-set. Remark3.3.3. IfweconsideragroupGactingonitselfwithleftmultiplicationthenGislimitaperiodicasaG-setifandonlyifGislimitaperiodicasagroup,becauseunderthatactionFix(G)=feg.LetXbeaG-setandletGx=StabG(x)=fg2Gjgx=xgdenotethestabilizerofx2X.Thefollowingtheoremisthemaintheoreminthischapter.Allothertheoremsandcorollariesarebasedonthis. Theorem3.3.4. LetXbealimitaperiodicG-setandsupposethatGactstransitivelyonX.Fixx2XsuchthatX=OrbG(x).IfStabG(x)isalimitaperiodicgroupthenGisalimitaperiodicgroup. Proof. Let:X!F1bealimitaperiodicmapforXandlet :Gx!F2bealimitaperiodicmapforGx.WeknowthatthereexistsabijectionbetweenthesetofleftcosetsG=GxandtheorbitOrbG(x).FixasetofrepresentativesinGnamelyfai:i2IgforthequotientsG=Gx.Then(ajGx)=ajx.Denef=(f1,f2):G!F1F2byf1(g)=(gx)andf2(g)= (a)]TJ /F8 7.97 Tf 6.58 -.01 Td[(1jg)whereg2ajGx.Wewillprovethatthisfisalimitaperiodicmap.Supposethatthisisfalse. 41

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Thenthereexistsamap f2 OrbG(f)andanelementa2Gsuchthat fhasaasaperiod,i.e.,(a f)= fforallx2X.Let f=limhkfwithhk2G.Case1) Supposea=2Gx.Wewillshowthatf1(g)=(gx)foranyg.Thentheperiodicityoffimpliesperiodicityofitsrstcomponentandthusperiodicityof(gx)leadingtoacontradiction.Considerthelimit =limkhk.Recallthatwecanalwayschooseasubsequenceofhksuchthatthelimitexists.Forconveniencewekeepthesameindicesforthesubsequence.Clearly, 2 OrbG().Giveng2G,thereexistsk02Nsuchthatforallkk0wehave: (gx)=(hk)(gx)Alsothereexistsak12Nsuchthatforallkk1weget: f(g)=(hkf)(g).Letk2=maxfk0,k1gthenforallkk2wehave f(g)=f(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1kg).Therefore, f1(g)=f1(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1kg)=(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1kgx)=(hk)(gx)= (gx).Similarlythereexistsak32Nsuchthatforallkk3wehave: f1(a)]TJ /F8 7.97 Tf 6.58 0 Td[(1g)= (a)]TJ /F8 7.97 Tf 6.58 0 Td[(1gx)Since f1hasperiodawehave(a f1)(g)= f1(g).Hence,(a )(gx)= (a)]TJ /F8 7.97 Tf 6.59 0 Td[(1gx)= f1(a)]TJ /F8 7.97 Tf 6.58 0 Td[(1g)=(a f1)(g)= f1(g)= (a)]TJ /F8 7.97 Tf 6.59 0 Td[(1gx). 42

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SincegisarbitraryandGactsonXtransitivelywegetthatforeveryy2X,(a )(y)= (y).Sinceislimitaperiodicwehavethata2Fix(X).But:Fix(X)=\s2SStabG(s)Thus,a2Fix(X)StabG(x)=Gxcontradiction.Case2) Supposethata2Gx.WewillnowshowthattherestrictionofftothestabilizerGxhasthesecondcomponentequalto .ThenperiodicityoffimpliesperiodicityofitsrestrictiontothestabilizerGxandoftherstcomponentofthisrestriction,thusimpliesperiodicityof .LetfakgbeasequenceofelementsofGsuchthath)]TJ /F8 7.97 Tf 6.59 0 Td[(1kbelongstothecosetakGx.Thusk=hkakbelongstoGx.Takingasubsequencewemayassumethatthefollowinglimitexists: =limkk Noticethat 2 OrbGx( ).Leth2Gx.Thenthereexistsak02Nsuchthatforallkk0wehave (h)=(k )(h)Alsothereexistsak12Nsuchthatforallkk1weget: fjGx(h)=(hkf)(h)Thenforallkmaxfk0,k1gwehave: fjGx(h)=f(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1kh).Hence:( fjGx)2(h)= (a)]TJ /F8 7.97 Tf 6.59 0 Td[(1kh)]TJ /F8 7.97 Tf 6.59 0 Td[(1kh)= ((hkak))]TJ /F8 7.97 Tf 6.59 0 Td[(1h)=((hkak) )(h)=(k )(h)= (h)Noticenowthat:a =a fjGx= fjGx= 43

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Thus isperiodicwhichisacontradictionsince islimitaperiodic.Thisconcludestheproof. Thefollowinglemmaisanobviousconsequenceofthedenitions. Lemma3.3.5. LetXbealimitaperiodicG-set.SupposethatHisagroupactingonX,suchthattheactionofHfactorsthroughtheactionofG.ThenifXisalimitaperiodicH-set. Corollary3.3.6. IfGandHarelimitaperiodicgroupsand1!G)167(!E)167(!H!1isashortexactsequencethenEisalsolimitaperiodic. Proof. EactstransitivelyonE=G=Hbyleftmultiplication,thusbyLemma 3.3.5 HisalimitaperiodicE-space.NotethatStabE(eH)=Gislimitaperiodic.IfweapplyTheorem 3.3.4 wegetthecorollary. Fromthespecicshortexactsequence:1!G)166(!GH)166(!H!1weget: Corollary3.3.7. IfGislimitaperiodicandHislimitaperiodicthenGHislimitaperiodic.WearealsoabletoprovethatlimitaperiodicitypassestoHNN-extensions,namely: Corollary3.3.8. IfHisalimitaperiodicgroupand:H!HisagroupautomorphismthentheHNNextension?Hislimitaperiodic. Proof. WeknowthatifH=whereSisasetofgeneratorsandTisasetofrelationsthenG=?Hadmitsthefollowingpresentation?H=. 44

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NotethatGactstransitivelyonthesetG=HofallleftcosetsofH.NotethatG=H=ftiHji2Zg=Zwhichweshowedtobealimitaperiodicset.Thus,GactsonZbytranslationswithStabG(fHg)=H( 35 ).ByLemma 3.3.5 itisalimitaperiodicG-set.Theorem 3.3.4 completestheproof. Weturnnowtothequestionofthefreeproductoflimitaperiodicgroups.Let'sestablishsomenotationsrst.LetAandBbetwogroups.WewillconstructasetXsuchthatthefreeproductG=A?BactsonXfreelyandtransitivelyandXisalimitaperiodicG-set.LetT0betheBass-SerretreeassociatedwithA?B.WerecallthattheverticesofT0areleftcosetsG=A[G=BandtheverticesofthetypexA,xBandonlythemformanedge[xA,xB]inT0.ThustheedgesofT0areinthebijectionwithG.NotethatGactsonT0byleftmultiplication.LetTbethebarycentricsubdivisionofT0andletXbethesetofthebarycenters.WewillidentifythetreeTwiththesetofitsvertices.ThenthegroupGactsbyisometriesonTyieldingthreeorbitsontheverticesX=OrbG(e),G=AandG=B.WeregardTasarootedtreewiththeroote.Letkxk=dT(x,e)denotethedistancetotheroot. Lemma3.3.9. LetA,BbetwogroupsandletG=A?B,X,Tdenedasabove.IfAandBarelimitaperiodic,thenTisalimitaperiodicG-set. Proof. Let:G!Awith(w)=(a1b1a2b2...anbn)=a1a2...anand:G!Bwith(w)=(a1b1a2b2...anbn)=b1b2...bn.Clearlybothandaregrouphomomorphisms.LetalsofA:A!FAbethelimitaperiodicmapforthegroupAandfB:B!FBbethelimitaperiodicmapforthegroupB.Alsolet:Z!f0,1,2gbethevariationoftheMorse-ThuesequencewhichhasnowordsWW(seeforexample( 14 )).AlsoxetobethevertexrepresentingtheidentityelementinT.ThenconsideracoloringofTasfollows:f:T!f0,1,2gf0,1,2g(FA[fg)(FB[fg)=F 45

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wheref:=(f0,f1,f2,f3)with:f0(x)=(kxk),f1(x)=kxkmod3,f2(x)=fA((x))ifx2Xandf2(x)=ifx2T)]TJ /F3 11.955 Tf 12.08 0 Td[(X.Finallyletf3(x)=fB((x))ifx2Xandf3(x)=ifx2T)]TJ /F3 11.955 Tf 11.95 0 Td[(X.ThegroupGactsonthespaceofcoloringsFTasfollows:(gf)(x)=((gf0)(x),(gf1)(x),((g)f2)(x),((g)f3)(x)=(f0(g)]TJ /F8 7.97 Tf 6.59 0 Td[(1x),f1(g)]TJ /F8 7.97 Tf 6.58 0 Td[(1x),f2()]TJ /F8 7.97 Tf 6.58 0 Td[(1(g)x),f3()]TJ /F8 7.97 Tf 6.58 0 Td[(1(g)x)).Supposethatfisnotalimitaperiodicmap.Thenthereexistsacoloring =( 0, 1, 2, 3)suchthat 2 OrbG(f)and hasaperiodb2GnFix(T).Let =limgkf.Then A=lim(gk)fAhasperiod(b).Indeed,foreveryx2AG=Xforlargeenoughk,((gk)fA)(x)=fA((g)]TJ /F8 7.97 Tf 6.59 0 Td[(1kx))=f2(g)]TJ /F8 7.97 Tf 6.58 0 Td[(1kx)=(gkf2)(x)=(gkf2)(bx)=f2(g)]TJ /F8 7.97 Tf 6.58 0 Td[(1kbx)=fA((g)]TJ /F8 7.97 Tf 6.58 0 Td[(1kbx))=((gk)fA)((b)x).Similarly B=lim(gk)fBhasaperiod(b).Thus,(b)=eAand(b)=eB.Noticethat( 0, 1)2 OrbG(f0,f1).Denote=( 0, 1)and=(f0,f1).Thenisacoloringofasimplicialtree(T)onwhichGactsbyisometries.Moreover2 OrbG().Fromproposition4,page318in( 14 )wehavethatb6=forallb2Gwithunboundedorbitfbkx0jk2Zg.Thisclearlyimpliesthatb 6= foreveryb2Gwithunboundedorbit.Ontheotherhand hasperiodbandthuswehavefbkx0jk2Ngisbounded.ThisimpliesthatbxesapointinT.Callthatpointx1.SincetheactionofGonXisfree,x1=2X.Thus,x12G=Aorx12G=B.Assumethelater,x1=wBforsomew2G.SincebxeswB,b=wb)]TJ /F8 7.97 Tf 6.59 0 Td[(1forsomeb02BnfeBg.Then(b)=(w)b0(w))]TJ /F8 7.97 Tf 6.59 0 Td[(16=eB.Contradiction. Lemma3.3.10. LetG=A?B,T,X,f,Fasabove.ThenXisalimitaperiodicG-set. Proof. NotethatintherootedtreeTeveryvertexx6=ehasauniquepredecessordenotedpred(x).Wedenef0:X!FFasf0(x)=(fjX(x),^f)where^f(x)= 46

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f(pred(x)).Weshowthatf0islimitaperiodic.Supposethatf0isnotlimitaperiodic.Thenthereexistsasequencefgkg2Gs.t. 0=limkgkf0andb2FixG(X)withb 0= 0.Wemayassumethatthereisthelimit =limgkf.InviewofLemma 3.3.9 itsufcestoshowthat isb-periodic.Itisb-periodiconX,soitsufcetocheckthatitisb-periodiconTnX.Letz2TnX.Wecheckthat (bz)= (z).SincetherooteliesinX,wemayassumethatz6=e.Letx0=pred(z)andletx1besuchthatz=pred(x1).Wenotethatx1isnotunique.Sowexone.Notethatx0,x12X.Thereisk0suchthatforkk0 0(xi)=(gkf0)(xi), 0(bxi)=(gkf0)(bxi),i=0,1and (z)=(gkf)(z), (bz)=(gkf)(bz).Fixkk0.SinceGactsonTbyisometriesthedistancefromg)]TJ /F8 7.97 Tf 6.59 0 Td[(1kztog)]TJ /F8 7.97 Tf 6.58 0 Td[(1kxi,i=0,1equals1.Wedenotea
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g)]TJ /F8 7.97 Tf 6.59 0 Td[(1kbx0
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Proposition3.3.12. ConsidertheautomorphismgroupoftheintegersAut(Z).Lets:Z!Zwiths(n)=n+1andt:Z!Zwitht(0)=1,t(1)=0andt(n)=nifn6=1andn6=0.LetS=thenNisnotalimitaperiodicS-set. Proof. SupposethatNwasalimitaperiodicS-set.ThenletFbeasetofcolorswithjFj<1andamapf:N!FsuchthatfisalimitaperiodicmapundertheactionofS.SincejFj<1andjNj=1thereexistsatleastonea2Fsuchthatinnitelymanyan2Nhavef(an)=a.ChooseastrictlyincreasingsequenceinNsuchthatf(an)=aforalln2N.Consider(i,ai)tobethetranspositionthattakesitoai.Withsandtwecanconstructallthetranspositions.Thusall(i,ai)belongtoS.ConstructthefollowingelementsinSwherethetranspositionsareappliedfromlefttoright:h1=(1,a1)h2=(1,a1)(2,a2)...hn=(1,a1)(2,a2)...(n,an)...Clearlyifnk,n,k2Nwehavethathnk=ak.Considernowthesequencefh)]TJ /F8 7.97 Tf 6.58 0 Td[(1nfgandtakeaconvergingsubsequence.Forconvenienceinnotationletuskeepthesameindicesforthesubsequence.Thusif: =limn!1h)]TJ /F8 7.97 Tf 6.58 0 Td[(1nfwehavethat 2 OrbS1(f).Thus hastobeaperiodic.Theclaimisthat (k)=aforallk2Nandthus isclearlyperiodicwhichwillleadtoacontradiction.Letk2Nthenthereexistsann1suchthatforallnn1wehavethat (k)=(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1nf)(k).Thusforn=maxfk,n1gwehavethat: (k)=(h)]TJ /F8 7.97 Tf 6.59 0 Td[(1nf)(k)=f(hnk)=f(ak)=a. 49

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Thisconcludestheproof. 3.4LimitAperiodicSubgroupsThefactthatthereexistlimitaperiodicG-setsandothersthatarenotlimitaperiodic,whenGactstransitivelyonthem,turnsouttobeaquestionaboutthesubgroupsofG,WeintroducethedenitionoflimitaperiodicsubgrouptoinvestigatethelimitaperiodicG-sets.Fromnowonourgroupswillbenitelygeneratedunlessotherwisestated. Denition3.4.1. LetGbeagroupandHasubgroupofG.ConsiderthesetofleftquotientsG H.Gactsonthissetbyleftmultiplication.WewillcallHalimitaperiodicsubgroupofG,ifG HisalimitaperiodicG-set. Remark3.4.2. SupposethatGactstransitivelyonaspaceXandxx2XwithX=OrbG(x).ThenitisclearthatXisalimitaperiodicG-setifandonlyifStab(x)isalimitaperiodicsubgroupofG.InotherwordslimitaperiodicG-setscorrespondtolimitaperiodicsubgroups. Proposition3.4.3. SupposeNisanormalsubgroupofagroupG.IfA'G NisalimitaperiodicgroupthenNisalimitaperiodicsubgroupofG. Proof. Noticethat1!N!G!A!1isashortexactsequence,SinceAisalimitaperiodicgroupitisalimitaperiodicA-set.SincetheactionofGonAfactorsthroughAwegetthatAisalimitaperiodicG-set.ThusNisalimitaperiodicsubgroupofG. Proposition3.4.4. LetXbeanynitespaceandGanygroupactingonXtransitively.ThenXisalimitaperiodicG-set. Proof. LetX=fx1,x2,...,xng.Considerthecoloringf:X!f1,2,...,ngwithf(xi)=i.Thismapisclearlyalimitaperiodicmap. Thuswegetthefollowingtwoobviouscorollaries: Corollary3.4.5. Allnitegroupsarelimitaperiodic. 50

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Corollary3.4.6. LetGbeagroupandHisasubgroupofGofniteindex.ThenHisalimitaperiodicsubgroup. Proof. SinceHhasniteindexinG,X=G HisanitespaceandthusalimitaperiodicG-set. Proposition3.4.7. LetHbeasubgroupofacountablegroupG.SupposethatHcontainsasubgroupN,whichisalimitaperiodicsubgroupofG.IfjH Nj=nthenHisalimitaperiodicsubgroupofG. Proof. SinceNisofniteindexinGitisalimitaperiodicsubgroup.ThusthereexistsanitesetFandalimitaperiodicmapf:G N!F.Leta1,a2,...anbeasetofrepresentativesforH N.Deneamap:G H!Fnwith(bH)=(f(ba1N),f(ba2N),...f(banN))Supposethatwasnotlimitaperiodic.Thenthereexistsasequencefgkg2Gandac2G,x=2FixG(G H)suchthatfor =limgkwehave =c .Let!=limgkf(chooseaconvergingsubsequencesuchthatthelimitexistsandrenametheindices).Fromthedenitionof andthefactthat =c weget!=c!.But!2 OrbG(f).Sincefislimitaperiodicwehavethatc2FixG(G N).ClearlyFixG(G N)FixG(G H).Thusc2FixG(G H)whichisacontradiction.ThusislimitaperiodicandG HisalimitaperiodicG-setwhichforcesHtobealimitaperiodicsubgroupofG. Corollary3.4.8. IfGisalimitaperiodicgroupandHisanitesubgroupofGthenHisalimitaperiodicsubgroupofG. Proof. ConsiderN=feg.ThenNisalimitaperiodicsubgroupofGsinceG N'GandGisalimitaperiodicgroupandthusalimitaperiodicG-set.AlsoNHandjH Nj=jHjisnite.ThusfromthepreviouscorollarywegetthatHisalimitaperiodicsubgroupofG. 51

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CHAPTER4COMPACTIFICATIONSONHYPERBOLICSPACES 4.1ShortHistoryOverviewRecallthattheHigsoncompacticationisdenedasfollows. Denition4.1.1. LetXbemetricspacewithbasepointx0andf:X!Cacontinuousfunctiontothecomplexnumbers(orrealnumbers).DeneVr(f):X!R+byVr(f)(x)=supfjf(y))]TJ /F3 11.955 Tf 11.96 0 Td[(f(x)j:y2Br(x)Wesaythatfisslowlyoscillating,orthatitsatisespropertyH,ifforeachr>0,Vr(f)!0atinnity,meaningthatforeach>0thereexistsaD>0suchthatforallx2X)]TJ /F3 11.955 Tf 11.95 0 Td[(BD(x0)wehaveVr(f)(x)<.DeneCh(X)tobethesetofallboundedcontinuousfunctionsf:X!CwithpropertyH.ThisformsaunitalC)]TJ /F3 11.955 Tf 12.21 0 Td[(algebra.ItfollowsfromtheGNS-constructionthatCh(X)isthealgebraofcontinuousfunctionsonsomecompacticationofX.WecallthistheHigsoncompactication.AlsothespaceHnisdenedasfollows: Construction4.1.2. ConsiderRn+1endowedwiththesymmetricbilinearform=(ni=1uivi))]TJ /F3 11.955 Tf 11.95 0 Td[(un+1vn+1Denethehyperbolicn-spacetobethesetHn=fu2Rn+1:=)]TJ /F9 11.955 Tf 9.3 0 Td[(1,un+1>0gwhichistheuppersheetofthehyperboloidfu2Rn+1:=)]TJ /F9 11.955 Tf 9.29 0 Td[(1g.Ametriccanbedenedasfollows.LetA,B2Hn.Thendened(A,B)tobtheuniquenon-negativenumbersuchthatcosh(d(A,B)=)]TJ /F5 11.955 Tf 14 0 Td[(.Itiseasytoshow( 5 )that(Hn,d)isageodesicmetricspace. 52

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N.HigsonintroducedtheHigsoncompacticationtostudythecoarseBaum-Connesconjecture.HenoticedthatiftheHigsoncompacticationoftheclassifyingspaceofadiscretegroupisacyclic,thenthecoarseBaum-Connesconjectureholdstrueforthegroup.TheacyclicitystatementoftheHigsoncompacticationgotthenametheHigsonconjecture( 33 ).J.KeeslingnoticedthattheHigsonconjectureisfalseindimensiononepracticallyforallspaces( 27 ).Inthecasewhentheclassifyingspaceisann-manifold(likeRnorHn)theHigsonconjectureindimensionnwasofgreatimportance.Nevertheless,A.DranishnikovandS.Ferry( 12 )showedthatthecohomologyoftheHigsoncompacticationofRnisnontrivialinalldimensionssmallerthann.TheyalsoprovedthatthecohomologyoftheHigsoncompacticationforH2,thehyperbolicplane,istrivialindimension2.ThenwiththeadditionofS.Weinberger(( 13 ))theyprovedthatthecohomologygroupswithZpcoefcientsoftheHigsoncompacticationofHnaswellasofRnaretrivial.In( 12 )amethodwasdevelopedinordertoshowthatHn( H4)=0andHn( H8)=0forn>1usingspecicbundlesofspheres.Usingthesametechniqueandmethodsfrom( 24 )wewereabletoshowthat: Theorem4.1.3. Hn( Hn)=0forallevenn.Stillthegeneralquestionremainsanopenproblem. 4.2TheCoHigsonCompacticationTheCoHigsonCompacticationwasstudiedbyNickWright1.Inparticular,WrightprovedaversionoftheCoarseBaum-ConnesconjecturefortheCocoarsestructure.WedenetheCocoarsestructurebelow.InthissectionweshowthattheHigsonconjecturefortheCoHigsoncompacticationholdstrueforHn.ThisisalsotrueforthesublinearHigsoncompactication( 9 )forHnsinceasimilarpropositionto 4.2.8 holds. 1Alinktohisthesisishttp://www.personal.soton.ac.uk/wright/papers/thesis.pdf 53

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JohnRoeinhisbookLecturesoncoarsegeometry( 34 )gavethedenitionofacoarsestructureandusedittodeneamongotherthingscertaincompacticationsofspacesusingthosecoarsestructures.Herearethebasicdenitionsfromhisbook. Denition4.2.1. LetXbeasetandconsidertheproductXX.AcollectionofsetsE=fEgP(XX)isacoarsestructureinthesenseofRoeifitsatisesthefollowingconditions: ItcontainsthediagonalD=f(x,x):x2Xg. IfA2E,BAthenB2E. IfA2EthenthesetA)]TJ /F8 7.97 Tf 6.59 0 Td[(1=f(y,x):(x,y)2AgisalsocontainedinE.thissetiscalledthesymmetricinverse. IfA,B2EthenthesetAB=f(x,y):9z2Xwith(x,z)2Aand(z,y)2BgisalsocontainedinE.ThissetiscalledtheproductofAandB. IfA,B2EthenA[B2E. Denition4.2.2. LetXbeasetandEbeacoarsestructureonit.AsetBXiscalledcoarselybounded(orjustboundedifthecontextisclear)ifBB2E. Denition4.2.3. AspaceXiscalledapropercoarsespaceifitisHausdorff,paracom-pactandthereexistsacoarsestructureoneX,Esuchthat: SomeneighborhoodofthediagonalD=f(x,x):x2XiscontainedinE. EverycoarselyboundedsubsetofXisrelativelycompact. Denition4.2.4. CoCoarsestructureLet(X,d)beametricspace.DeneEtobethecollectionofallsetsEXXsuchthatdjEtendstozeroatinnity.Thismeansthatforall>othereexistsacompactsetKXsuchthatd(x,y)forall(x,y)2E)]TJ /F3 11.955 Tf 11.96 0 Td[(KK.ThisiscalledtheCocoarsestructure. Example4.2.5. LetX=R.ThenthesetA=f(x,y)2R2:1x,1y,x)]TJ /F8 7.97 Tf 13.5 4.71 Td[(1 xyx+1 xgisinCo(R)sinceifx!1,theny!x,sod(x,y)!0.In( 34 )itisproventhatCoisapropercoarsestructureifXisapropermetricspace. 54

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Figure4-1. AnelementofCo(R) Denition4.2.6. Let(X,d)beametricspacewiththeCocoarsestructure.Letf:X!Cbeabounded,continuousfunction.Denotebydfthefunction:df(x,y)=f(y))]TJ /F3 11.955 Tf 11.95 0 Td[(f(x):XX!C.WewillsaythatfisaHigsonfunctionifforeverycontrolledsetEtherestrictionofdftoEvanishesatinnity.LetCh0(X)denotethesetofallbounded,continuousHigsonfunctionswiththeCocoarsestructure.ThisformsaunitalC)]TJ /F3 11.955 Tf 18.21 0 Td[(algebra.ItfollowsfromtheGNS-constructionthatCh0(X)isthealgebraofcontinuousfunctionsonsomecompacticationofX.NoticethatifCl(X)=ff:X!CjfisboundedandlipschitzgthenClCh0(X).FurthermoreifCosc(X)=ff:X!CjfisboundedcontinuousandslowlyoscillatinggthenCosc(X)Ch0(X). Denition4.2.7. ThecompacticationhoXcharacterizedbythepropertyC(hoX)=Cho(X)iscalledtheCoHigsoncompactication.ItsboundaryhoX)]TJ /F3 11.955 Tf 11.06 0 Td[(Xisdenotedby0XanditscalledtheCoHigsonCorona. 55

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Weusenotationkxk=dX(x,x0)whereXisametricspaceswithabasedpoint.Werecallthatamapf:X!Ybetweenmetricspacesiscalleduniformlycontinuousifthereisamonotonefunction!:[0,1]![0,1]calledmodulusofcontinuitywithlimt!0!(t)=0suchthatdY(f(x),f(x0))!(dX(x,x0))forallx,x02X( 16 ) Theorem4.2.8. LetXbeapropergeodesicmetricspace.Thenf2Ch0(X)ifandonlyifitisuniformlycontinuousandbounded. Proof. Letf2Ch0(X)andassumethatfisnotuniformlycontinuous.Hencethereis>0andsequencesxnandynwithd(xn,yn)<1=nandwithjf(xn))]TJ /F3 11.955 Tf 12.08 0 Td[(f(yn)j.Clearly,E=f(xn,yn)gisacontrolledset.WeobtainacontradictionwiththeconditiondfjE!0.Letf:X!Cbeauniformlycontinuousboundedfunctionwithamodulusofcontinuity!andletEbeacontrolledsetforCo.Thentheinequalityjf(x))]TJ /F3 11.955 Tf 12.61 0 Td[(f(x0)j!(dX(x,x0))impliesthatdfjE!0. WenotethatinthedenitionoftheCoHigsoncompacticationthecomplexnumbersCcanbereplacedbytherealsR.Wemakeuseofthefollowingfact( 32 ). Theorem4.2.9. Everyuniformlycontinuousboundedfunctiononametricspacef:X!RisauniformlimitofLipschitzfunctions.Thisisaspecialcaseofageneraltheoremforreal-valuedfunctionsthatadmitaconcavemodulusofcontinuity2. 4.3`1-cohomologyLetAbeanormedabeliangroupandXaCWcomplex.LetEn(X)denotethesetofn-cellsinXandCn(X,A)=Hom(En(X)Z,A)denotethegroupofcellularn-cochainsonXwithvalueinA.LetCn(1)(X,A)=f2Cn(X,A)j9b:j(e)jb8e2En(X)g 2(seehttp://en.wikipedia.org/wiki/Modulusofcontinuity) 56

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beasubgroupgeneratedbyboundedcochains.Ifonetakesthe`1normonCn(X)=En(X)Zthethegroupofboundedcochainsconsistsofhomomorphisms:Cn(X)!Aboundedwithrespecttothenorms.WedenotethecorrespondingcohomologygroupsbyH(1)(X;A).Theboundedvaluecohomologyfortheaugmentedchaincomplex!Cn(X)!!Co(X)!Z!0arecalledthereduced`1cohomologyanddenotedbyH(1)(X;A).IfagroupAisnitelygeneratedthen,clearly,H(1)(X;A)doesnotdependonthechoiceofthenormonA.( 30 ) Proposition4.3.1. Thebarycentricsubdivisionchainmap:C(X)!C(1X)ofasimplicialcomplexXinducesanisomorphismforthe`1-cohomology. Proposition4.3.2. Supposethatasimplicialmapf:X!Ybetweenuniformlycontractiblesimplicialcomplexesisaquasi-isometry.Thenthemapf:H(1)(Y)!H(1)(X)isanisomorphism.ThefollowingPropositionwasprovenin( 22 ). Proposition4.3.3. LetXbetheuniversalcoverofK(,1)withniteskeletonsK(,1)(n)foralln.ThentheinclusionZ!RinducesanisomorphismHi(1)(X;Z)!Hi(1)(X;R)fori0. Proof. TheresultfollowsfromthefactsthatthegroupS1=R)=Zisboundedwithrespecttothequotientnorm,theequalityHi(1)(X;S1)=Hi(X;S1)=0fori0,andtheCoefcientLongExactsequence. 57

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Foragroup)]TJ /F1 11.955 Tf 10.1 0 Td[(withnitecomplexK(,1)(n)wedeneHi(1)(;A)=Hi(1)(X;A)whereXistheuniversalcover.Itwasshownin( 22 )thatthecohomologygroupdoesnotdependonchoiceofK(,1)withthatnitenesscondition.Wenotethatforahyperbolicgroup)]TJ /F1 11.955 Tf 10.1 0 Td[(thereisacomplexK(,1)withK(,1)(n)niteforalln( 5 ). Theorem4.3.4. Foreveryhyperbolicgroup)]TJ /F11 11.955 Tf 6.78 0 Td[(,Hi(1)(\=0foralli>1.Thistheoremwasprovenin( 29 )forcoefcientsinR.InviewoftheabovePropositionitholdstrueforcoefcientsinZ. 4.4MainResultThefollowingtheoremiscrucialtotheproofofthemaintheorem. Theorem4.4.1. Letf,g:X!KtwoLipschitzmapsofasimplicialcomplexXtoasimplyconnectednitecomplexK.SupposethatthereisaLipschitzhomotopyH:X(n)I!KwithHjX(n)f0g=fjX(n)f0gandHjX(n)f1g=gjX(n)f1g.ThenthereisaLipschitzmapH:X(n+1)I!KthatcoincideswithHonX(n)]TJ /F8 7.97 Tf 6.59 0 Td[(1)IandwithHjX(n+1)f0g=fjX(n+1)f0gandHjX(n+1)f1g=gjX(n+1)f1gifandonlyiftheobstructionclass(f,g)2Hn+1(1)(X;n+1(K))iszero.Asimilartheoremwasprovedin( 6 )chapter7.Namely: Theorem4.4.2. Letf0:X!Kandf1:X!KbetwomapsfromasimplicialcomplexXtoasimplyconnectednitecomplexY.SupposethatthereisahomotopyH:X(n)I!KwithHjX(n)f0g=fjX(n)f0gandHjX(n)f1g=gjX(n)f1g 58

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ThenthereisamapH:X(n+1)I!KthatcoincideswithHonX(n)]TJ /F8 7.97 Tf 6.59 0 Td[(1)IandwithHjX(n+1)f0g=fjX(n+1)f0gandHjX(n+1)f1g=gjX(n+1)f1gifandonlyiftheobstructionclass(f,g)2Hn+1(X;n+1(K))iszero.Soforourtheoremtobetrueweneedtoshowthat: 1. Theproducedhomotopyintheproofofthetheoremislipschitz. 2. IfF:X!KisaLipschitzmapsimplicialcomplexXtoasimplyconnectednitecomplexKthenthecorrespondingobstructioncocycleisbounded.TherstpropertyiseasilymetsincethehomotopyinvolvedintheproofisthehomotopyH:CI!CwhereC=(D2f0g)[(S1I)(theopencan)suchthat i) H(x,0)=idC ii) H(x,1)=x0(aconstantpoint) iii) H(x0,t)=x0forallt2[0,1]ItisnotsohardtoshowthatHhastheextraproperty: iv) Hislipschitzsinceitisacontractionofaboundedsettoapoint.(Thecontractioncanbedoneusinglinearspeed)Toprovethesecondpropertyweneedthefollowingtheorem: Theorem4.4.3. Letf:Sn!Kbea-lipschitzmapformthen-dimensionalspheretoanitesimplicialcomplexK.Supposethatn(K)isanormedabeliangroup.Then9b>0suchthatkfkb. Proof. ConsiderthespaceLip(Sn,K)=fg:Sn!K,lipschitzmapsgThisspaceiscompact.Thenconsiderthemaps 59

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Lip(Sn,K))166(![Sn,K]iso)166(!Hk.k)166(!Zwhere(g)=[g].Clearlyifisthecompositionofthosethreemapsitiscontinuous.SinceLip(Sn,K)iscompact(Lip(Sn,K))iscompactandthusboundedSo9b>0suchthatforallg2Lip(Sn,K)wehave(g)b)kgkb Thusfromthedenitionoftheobstructioncocycleandthetheoremaboveweget: Corollary4.4.4. Letf:X!Kbealipschitzmapthenthecorrespondingobstructionclass(f)isbounded.Thusn(f)2Hn+1(1)(X;n+1(K))Wearenowreadytoprovethemaintheoremofthissection. Theorem4.4.5. LetHnbethendimensionalhyperbolicspaceandh0Hn,itsCoHigsoncompactication.ThenHk(h0Hn)=f0gforn>1. Proof. RecallthatHk(h0Hn)=[hoHn,K(Z,n)].Solet[f]2Hk(h0Hn).Thisinducesamapf:h0Hn!K(Z,k)=Y.Considertherestrictionf:Hn!Y.Sincef2Ch0thereexistsalipschitzmapg:Hn!Y-closetof.SincegislipschitzitextendstohoHn.LetK=HnandconsiderHi:K(i)I!Yafamilyofhomotopiessuchthat:HijKf0g=g,HijKf1g=c(aconstantfunction)andHiislipschitz.Suchafamilyofmapsexistsfromtheorem 4.4.1 sincefromtheorem 4.3.4 wehavethatHn+1(1)(Hk;n+1(K))=f0g.Considerthenh:Hn!Lip(I,Y)YIdenedbythoseHi's.ByAscoliArzelaTheorem,Lip(I,Y)isprecompactsoitliesinacompactmetricspace.Thenh:Hn!C(compactmetricspace).Thisimpliesthath:hoHn!YIandthusweobtainH:hoHnI!Yahomotopybetweengandaconstantmapso[g]=[0].Noticethatf,garecloseonHnthus[f]=[g]=[0].Thisconcludestheproof,thechapterandthethesisitself. 60

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REFERENCES [1] G.BellandA.Dranishnikov,Onasymptoticdimensionofgroups,Algebr.Geom.Topol.1(2001),57(electronic).MR1808331(2001m:20062) [2] G.C.BellandA.N.Dranishnikov,AHurewicz-typetheoremforasymptoticdi-mensionandapplicationstogeometricgrouptheory,Trans.Amer.Math.Soc.358(2006),no.11,4749(electronic).MR2231870(2007e:20092) [3] G.C.Bell,A.N.Dranishnikov,andJ.E.Keesling,Onaformulafortheasymptoticdimensionoffreeproducts,Fund.Math.183(2004),no.1,39.MR2098148(2005g:20064) [4] GregoryC.Bell,Asymptoticpropertiesofgroupsactingoncomplexes,Proc.Amer.Math.Soc.133(2005),no.2,387(electronic).MR2093059(2005h:20101) [5] MartinR.BridsonandAndreHaeiger,Metricspacesofnon-positivecurvature,GrundlehrenderMathematischenWissenschaften[FundamentalPrinciplesofMathematicalSciences],vol.319,Springer-Verlag,Berlin,1999.MR1744486(2000k:53038) [6] JamesF.DavisandPaulKirk,Lecturenotesinalgebraictopology,GraduateStudiesinMathematics,vol.35,AmericanMathematicalSociety,Providence,RI,2001.MR1841974(2002f:55001) [7] A.Dranishnikov,Cohomologicalapproachtoasymptoticdimension,Geom.Dedicata141(2009),59.MR2520063(2011a:20114) [8] A.DranishnikovandJ.Smith,Asymptoticdimensionofdiscretegroups,Fund.Math.189(2006),no.1,27.MR2213160(2007h:20041) [9] A.DranishnikovandJ.Smith,Onaymptoticassouadnagatadimension,TopologyAppl.154(2007). [10] A.N.Dranishnikov,Asymptotictopology,UspekhiMat.Nauk55(2000),no.6(336),71.MR1840358(2002j:55002) [11] ,Onhypersphericityofmanifoldswithniteasymptoticdimension,Trans.Amer.Math.Soc.355(2003),no.1,155(electronic).MR1928082(2003g:53055) [12] A.N.DranishnikovandS.Ferri,OntheHigson-Roecorona,UspekhiMat.Nauk52(1997),no.5(317),133.MR1490028(98k:58214) [13] A.N.Dranishnikov,S.C.Ferry,andS.A.Weinberger,AnetaleapproachtotheNovikovconjecture,Comm.PureAppl.Math.61(2008),no.2,139.MR2368371(2008j:57054) 61

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[14] AlexanderDranishnikovandViktorSchroeder,AperiodiccoloringsandtilingsofCoxetergroups,GroupsGeom.Dyn.1(2007),no.3,311.MR2314048(2009h:20045) [15] M.J.Dunwoody,Theaccessibilityofnitelypresentedgroups,Invent.Math.81(1985),no.3,449.MR807066(87d:20037) [16] A.V.Emov,Modulusofcontinuity,encyclopaediaofmathematics,Springer,2001. [17] KojiFujiwaraandKevinWhyte,Anoteonspacesofasymptoticdimensionone,Algebr.Geom.Topol.7(2007),1063.MR2336248(2008e:20066) [18] SuGao,SteveJackson,andBrandonSeward,Acoloringpropertyforcountablegroups,Math.Proc.CambridgePhilos.Soc.147(2009),no.3,579.MR2557144(2010j:03056) [19] T.Gentimis,Oncohomologyofthehigsoncompacticationofhyperbolicgroups,preprint. [20] ,OnlimitaperiodicG-sets,preprintunderreview. [21] ,Asymptoticdimensionofnitelypresentedgroups,ProceedingsoftheAmericanMathematicalSociety136(2008),no.12,4103. [22] S.M.Gersten,Cohomologicallowerboundsforisoperimetricfunctionsongroups,Topology37(1998),no.5,1031.MR1650363(2000c:20063) [23] M.Gromov,Asymptoticinvariantsofinnitegroups,Geometricgrouptheory,Vol.2(Sussex,1991),LondonMath.Soc.LectureNoteSer.,vol.182,CambridgeUniv.Press,Cambridge,1993,pp.1.MR1253544(95m:20041) [24] AllenHatcher,Algebraictopology,CambridgeUniversityPress,Cambridge,2002.MR1867354(2002k:55001) [25] WitoldHurewiczandHenryWallman,DimensionTheory,PrincetonMathematicalSeries,v.4,PrincetonUniversityPress,Princeton,N.J.,1941.MR0006493(3,312b) [26] TadeuszJanuszkiewiczandJacekSwiatkowski,Fillinginvariantsofsystoliccom-plexesandgroups,Geom.Topol.11(2007),727.MR2302501(2008d:20079) [27] J.Keesling,Theonedimensionalcechcohomologyofthehigsonccompacticationanditscorona. [28] RogerC.LyndonandPaulE.Schupp,Combinatorialgrouptheory,ClassicsinMathematics,Springer-Verlag,Berlin,2001,Reprintofthe1977edition.MR1812024(2001i:20064) [29] IgorMineyev,Higherdimensionalisoperimetricfunctionsinhyperbolicgroups,Math.Z.233(2000),no.2,327.MR1743440(2001d:20042) 62

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[30] D.Morris,Introductiontoarithmeticgroups,preprint2008. [31] MarstonMorseandGustavA.Hedlund,Unendingchess,symbolicdynamicsandaprobleminsemigroups,DukeMath.J.11(1944),1.MR0009788(5,202e) [32] I.Rihaoui,Approximationpardesfonctionslipschitziennesetcrit`eredeconver-genceetroitedunesuitedeprobabilites.(french.englishsummary),approximationbylipschitzfunctionsandaweakconvergencecriterionforaprobabilitysequence,1984,pp.514. [33] JohnRoe,CoarsecohomologyandindextheoryoncompleteRiemannianmani-folds,Mem.Amer.Math.Soc.104(1993),no.497,x+90.MR1147350(94a:58193) [34] ,Lecturesoncoarsegeometry,UniversityLectureSeries,vol.31,AmericanMathematicalSociety,Providence,RI,2003.MR2007488(2004g:53050) [35] Jean-PierreSerre,Trees,SpringerMonographsinMathematics,Springer-Verlag,Berlin,2003,TranslatedfromtheFrenchoriginalbyJohnStillwell,Corrected2ndprintingofthe1980Englishtranslation.MR1954121(2003m:20032) [36] J.Smith,Onasymptoticdimensionofcountableabeliangroups,TopologyAppl.153(2006),no.12,2047.MR2237596(2007g:20044) [37] G.Yu,Thenovicovconjectureforgroupswithniteasymptoticdimension,Ann.ofMath.147(1998),325. 63

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BIOGRAPHICALSKETCH AthanasiosGentimiswasbornonDecember10,1980inAthens,Greece.Theoldestof2children,hegrewupinHalandri,Athens,graduatingfromthe4thGeneralHighschoolin1998.Bothhisparentsaremathematiciansandhisbrotherisaphysicist.HeearnedhisB.S.inMathematicsandhisM.S.inTheoreticalMathematicsfromtheKapodistriakoUniversityofAthenson2002and2004respectively.Upongraduationin2004withhisM.S.hepursuedacareerinteachingmathematics,whilecontinuinghisresearch.In2006hegotascholarshipforaPh.D.programinmathematicsattheUniversityofFloridawhichhecompletedwithhonorsinAugust2011.UponcompletionofhisPh.D.program,AthanasioswillpursueacareerinteachingstartingattheUniversityofFloridaasanadjunctprofessor.AthanasiosisengagedtoMariaBampasidou. 64