A characterization of hyperbolic spaces

arXiv:math/0410035v3[math.GR]19May2006ACHARACTERIZATIONOFHYPERBOLICSPACESINDIRACHATTERJIANDGRAHAMNIBLOAbstract.WeshowthatageodesicmetricspaceishyperbolicinthesenseofGromovifandonlyifintersectionsofballshaveboundedeccentricity.Inparticular,R-treesarecharacterizedamonggeodesicmetricspacesbythepropertythattheintersectionofanytwoballsisalwaysaball.BothGromovhyperbolicityandCAT(κ)geometrycanbecharacterisedintermsofthegeometryofthein-tersectionofballs.IntroductionItiswellknownthatinanR-treetheintersectionofanytwomet-ricballsisitselfametricball.InthispaperwewillshowthatthisisactuallyacharacterizationofR-trees,andthat,moregenerally,thegeometryoftheintersectionofballsencodesinformationaboutthecurvatureofageodesicmetricspace.Recallfrom[1],[2],[3]or[4]thatageodesicmetricspaceishyperbolic(inthesenseofGromov)ifthereisaconstantδ≥0suchthatforanygeodesictriangle,anyonesideiscontainedintheδ-neighborhoodoftheunionofthetwoothersides.Weprovethefollowingcharacterizationofhyperbolicity.Theorem1.Ageodesicmetricspace(X,d)ishyperbolicifandonlyiftheintersectionofanytwometricballsisatuniformlyboundedHaus-dorffdistancefromaball.StudyingcurvatureintermsofthegeometryoftheintersectionofmetricballsturnsouttobeverynaturalandbothGromovhyperbolic-ityandthenotionofCAT(κ)geometrymaybecharacterisedinthesesterms(seeSection4).TrackingconstantsintheproofofTheorem1,onecanshowthatthehyperbolicityconstantdependsonlyontheeccentricityconstant.AsPierrePansupointedouttousitisathenaneasyobservationtodeducethatthehyperbolicityboundvarieslinearlywiththeeccentricityboundsoweobtain:Corollary2.AgeodesicmetricisanR-treeifandonlyiftheinter-sectionofanytwoballsisaball.Date:February1,2008.12INDIRACHATTERJIANDGRAHAMNIBLOThischaracterisationofR-treeswasfirstconjecturedinanearlyver-sionofthispaperandWengerhasindependentlyestablishedthecon-jectureusingverydifferentmethods,see[9].Ourapproachisentirelyself-contained.Thefollowingnotioniscrucialforourpurposes.Definition3.WesaythatasetShaseccentricitylessthanδ(forsomeδ≥0)ifthereisR≥0suchthatB(c,R)⊆S⊆B(c′,R+δ)forsomec,c′∈X.Byconventiontheeccentricityoftheemptysetis0.Weshallseethattheintersectionofballshavinguniformlyboundedeccentricityisalsoequivalenttohyperbolicity(Proposition15andLemma18).Thepaperisorganizedasfollows.InSection1wediscussthegeom-etryof(1,q)-quasigeodesicsfollowinganideaofPapasogluin[6]andPomroyin[7],whichisanimportantstepintheproof.Section2dis-cussesdivergencefunctionsandaquantitativeversionofatheoremin[6]andaclassicalargumentimplyinghyperbolicity.Section3collectstheproofsofTheorem1andCorollary2.Theideaistoshowthathyperbolicityisequivalenttointersectionsofballshavinguniformlyboundedeccentricity.Onetechnicaldifficultyliesinthefactthatthecentreandradiusofaballarenot,ingeneral,welldefined.WhentheyareanelementaryproofcanbegivenasinSection4.Pomroy’sworkappearedinhisWarwickUniversityMastersdissertationbuthasneverbeenpublished.IntheAppendixwetakethisopportunitytoplacehismaintheoremontherecordwithourownvariationontheproof.Thefiguresmaybefoundatthebackofthepaper,insertedpriortothebibliography.Acknowledgements:TheauthorsareextremelyindebtedtoPierrePansuforpointingouttousthatourboundonthehyperbolicitycon-stantimpliedalinearboundviascaling.WewouldalsoliketothankTheoBuehlerandDeborahRuossforpointingoutanerrorinanearlydraftofthispaperandChrisHruskaforthereference[5].FinallywewouldliketothankHamishShortforhiscommentsontheclassicalargumentprovingthatnon-lineardivergenceofgeodesicsimplieshy-perbolicity.1.BigonsingeodesicmetricspacesInthissectionweestablishpreliminaryresultsconcerningthege-ometryofgeodesicsandquasi-geodesicsinageodesicmetricspace.ACHARACTERIZATIONOFHYPERBOLICSPACES3Westartwiththesimpleobservationthatiftwogeodesicsaresyn-chronouslyfarapart,thentheyareasynchronouslyatleasthalfasfarapartaswell.Lemma4.Letγandγ′begeodesicswithγ(0)=γ′(0)=e.Ifthereexistst≥0suchthatd(γ(t),γ′(t))≥Kthend(γ(t),γ′(s))≥K/2foralls.Proof.Supposethatthereisans≤tsuchthatd0=d(γ′(s),γ(t))K/2.Thensinceγisageodesicwehavet=d(γ(0),γ(t))≤d(e,γ′(s))+d(γ′(s),γ(t))=s+d0.Henced0≥t−s.ButbyhypothesisandthetriangleinequalityK≤d(γ′(t),γ(t))≤d(γ′(t),γ′(s))+d(γ′(s),γ(t))=t−s+d0≤2d0K.Thisisacontradiction.Similarly,ifthereisanstsuchthatd0=d(γ′(s),γ(t))K/2.Thensinceγisageodesicwehaves=d(γ′(0),γ′(s))≤d(e,γ(t))+d(γ(t),γ′(s))=t+d0.Henced0≥s−t.ButbyhypothesisandthetriangleinequalityK≤d(γ′(t),γ(t))≤d(γ′(t),γ′(s))+d(γ′(s),γ(t))=s−t+d0≤2d0K.Thisisagainacontradiction.Definition5.Foraconstantq≥0a(1,q)-quasigeodesicisacontin-uousmapγ:[0,d]→Xsuchthatγ(0)=γ′(0),γ(d)=γ′(d)andforallt∈[0,d]|t−t′|−q≤d(γ(t),γ(t′))≤|t−t′|+q.Thepointsγ(0)andγ(d)aresaidtobetheendpointsofγ.A(1,q)quasi-geodeSIcbigonisapairγ,γ′of(1,q)quasi-geodesicbigonswhichhavethesameendpoints.Theimagesofthequasigeodesicsγandγ′inXarecalledthesidesoftheq-bigon.ForK≥0,wesaythataq-bigonisK-fatifthereares,tsuchthatd(γ(s),γ′(t))≥KandthatitisK-thinifitisnotK-fat.Thefollowingremarkprovidesaneasymechanismforconstructing(1,q)quasi-geodesics.Lemma6.LetK≥0andx,yinX.AK-pathfromxtoyisacontinuouspathμfromxtoysuchthat,foranyz=μ(t)fo