ISOMETRY GROUPS OF NON-POSITIVELY CURVED SPACES: DISCRETE SUBGROUPS ‡ PIERRE-EMMANUEL CAPRACE* ANDNICOLASMONOD 9 Abstra t. We study latti es in non-positively urved metri spa es. Borel density is 0 established in that setting as well as a form of Mostow rigidity. A onverse to the (cid:29)at 0 torus theorem is provided. Geometri arithmeti ity results are obtained after a detour 2 throughsuperrigidityandarithmeti ityofabstra t latti es. Residual(cid:28)nitenessoflatti es n is also studied. Riemannian symmetri spa es are hara terised amongst CAT(0) spa es a admitting latti es in terms of theexisten e of paraboli isometries. J 8 ] R 1. Introdu tion G Latti es in semi-simple algebrai groups have a tantalisingly ri h stru ture; they in lude h. arithmeti groups and more generally S-arithmeti groups over arbitrary hara teristi s. t The nature of these groups is shaped in part by the fa t that they are realised as isometries a m of a anoni al non-positively urved spa e: the asso iated Riemannian symmetri spa e, or Bruhat(cid:21)Tits building, or a produ t of both types. [ Many other groups of rather diverse origins share this property to o ur as latti es in 1 v non-positively urved spa es, singular or not: 2 (cid:22) The fundamental group of a losed Riemannian manifold of non-positive se tionnal 2 urvature. Herethespa ea teduponistheuniversal overing,whi hisaHadamard 0 1 manifold. 1. (cid:22) Many Gromov-hy1perboli groups admit a properly dis ontinuous o ompa t a tion 0 on some CAT(− ) spa e by isometries. ACm′(o1n)gst theCe′(x1a)mTp(le4s)arising in this way 9 are hyperboli Coxeter groups [Mou88℄, 6 and 4 - small an ellation 0 2 7 : groups [Wis04℄, -dimensional -systoli groups [J‘06℄. It is in fa t a well known v openproblemofM.Gromovto onstru tanexampleofaGromov-hyperboli group i Γ X whi h isnot aCAT(0) group (see[Gro93, 7.B℄; alsoRemark2.3(2) inChapter III. r of [BH99℄). a (cid:22) In [BM00b℄, striking examples of (cid:28)nitely presented simple groups are onstru ted as latti es in a produ t of two lo ally (cid:28)nite trees. Tree latti es were previously studied in [BL01℄. (cid:22) AminimaladjointKa (cid:21)Moodygroupovera(cid:28)nite(cid:28)eld,asde(cid:28)nedbyJ.Tits[Tit87℄, BN is endowed with two -pairs whi h yield strongly transitive a tions on a pair of twinned buildings. When the order of the ground (cid:28)eld is large enough, the Ka (cid:21) Moody group is a latti e in the produ t of these two buildings [Rém99℄. (Γ,X) Subsumingalltheaboveexamples,wede(cid:28)neaCAT(0) latti easapair onsisting X Is(X) of a proper CAT(0) spa e with o ompa t isometry group and a latti e subgroup Keywordsandphrases. Latti e,arithmeti group,non-positive urvature,CAT(0)spa e,lo ally ompa t group. ‡*F.N.R.S. Resear h Asso iate. Supported in part bythe Swiss National S ien e Foundation. 1 2 PIERRE-EMMANUELCAPRACEANDNICOLASMONOD Γ < Is(X) , i.e. adis rete subgroup of (cid:28)nite invariant ovolume (the ompa t-open topology Is(X) makes a lo ally ompa t se ond ountable group whi h is thus anoni ally endowed (Γ,X) Γ Is(X) with Haar measures). We say that is uniformif is o ompa t in or, equiva- Γ X Γ lently, if the quotient \ is ompa t; that ase orresponds to being a CAT(0) group in the usual terminology. Amongst CAT(0) latti es, the most important, and also the best understood, notably through the work of G. Margulis, onsist undoubtedly of those arising from latti es in semi- simple groups over lo al (cid:28)elds. It is therefore natural to address two sets of questions. (a) What properties of these latti es are shared by all CAT(0) latti es? (b) What properties hara terise them within the lass of CAT(0) latti es? This arti le is devoted to the study of CAT(0) latti es and entres largely around the above questions, though we also address the general question of the interplay between the algebrai stru tureofaCAT(0)latti eandthegeometri propertiesoftheunderlying spa e. Some of the te hniques established in the present paper have been used in a subsequent investigation of latti es in produ ts of Ka (cid:21)Moody groups [CM08d℄. We shall now des ribe the main results of this arti le; for many of them, the ore of the text will ontain a stronger, more pre ise but perhaps more ponderous version. Our notation is standard, as re alled in the Notation se tion of the ompanion paper [CM08 ℄. We refer to the latter for terminology and shall quote it freely. . Geometri Borel density. As a link between the general theory exposed in [CM08 ℄ and the study of CAT(0) latti es, we propose the following analogue of A. Borel's density theorem [Bor60℄. X G Theorem 1.1. Let be a proper CAT(0) spa e, a lo ally ompa t group a ting ontin- X Γ < G X uously by isometries on and a latti e. Suppose that has no Eu lidean fa tor. G Γ If a ts minimally without (cid:28)xed point at in(cid:28)nity, so does . This on lusion fails for spa es with a Eu lidean fa tor. The theorem will be established more generally for losed subgroups with (cid:28)nite invariant ovolume. It should be ompared to (and an of ourse be gainfully ombined with) a similar density property of normal subgroups established as Theorem 1.10 in [CM08 ℄. Remark 1.2. Theorem1.1appliestogeneralproperCAT(0)spa es. Itimpliesinparti ular the lassi al Borel density theorem (see the end of Se tion 2). As with lassi al Borel density, we shall use this density statement to derive statements about the entraliser, normaliser and radi al of latti es in Se tion 2. A more elementary variant of the above theorem shows that a large lass of groups have rather restri ted a tions on proper CAT(0) spa es; as an appli ation, one shows: F X Any isometri a tion of R. Thompson's group on any proper CAT(0) spa e has a X (cid:28)xed point in , see Corollary 2.3. Theorem 1.1 also provides additional information about the totally dis- D j onne ted groups o urring in Theorem 1.6 in [CM08 ℄. . ISOMETRY GROUPS OF NON-POSITIVELY CURVED SPACES: DISCRETE SUBGROUPS 3 Latti es: Eu lidean fa tor, boundary, irredu ibility and Mostow rigidity. Re all that the Flat Torus theorem, originating in theRwnork of Gromoll(cid:21)WoZlfn[GW71℄ and Lawson(cid:21) Yau [LY72℄, asso iates Eu lidean subspa es to any subgroup of a CAT(0) group, see [BH99, ŸII.7℄. (In the lassi al setting, when the CAT(0) group is given by a ompa t non-positively urved manifold, this amounts to the seemingly more symmetri statement that su h a subgroup exists if and only if there is a (cid:29)at torus is the manifold.) 6.B 3 The onverse is a well known open problem stated by M. Gromov in [Gro93, Ÿ ℄; for manifolds see S.-T. Yau, problem 65 in [Yau82℄). Point (i) in the following result is a (very partial) answer; in the spe ial ase of o ompa t Riemannian manifolds, this was the main result of P. Eberlein's arti le [Ebe83℄. X G < Is(X) Theorem 1.3. Let be a proper CAT(0) spa e, a losed subgroup a ting X Γ < G minimally and o ompa tly on and a (cid:28)nitely generated latti e. Then: X n Γ (i) If the EΓu lidean fa tor ofΓ haZsndimΓe′nsion , then possesses a (cid:28)nite index sub- 0 0 group whi h splits as ≃ × . Moreover, the dimension of the Eu lidean fa tor is hara terised as the maximal Γ rank of a free Abelian normal subgroup of . G Γ (ii) has no (cid:28)xed point at in(cid:28)nity; the set of -(cid:28)xed points at in(cid:28)nity is ontained in the (possibly empty) boundary of the Eu lidean fa tor. Point (ii) is parti ularly useful in onju tion with the many results assuming the absen e of (cid:28)xed points at in(cid:28)nity in [CM08 ℄. In addition, it is already a (cid:28)rst indi ation that the mere existen e of a ((cid:28)nitely generated) latti e is a serious restri tion on a proper CAT(0) spa e even within the lass of o ompa t minimal spa es. We re all that E. Heintze [Hei74℄ produ ed simply onne ted negatively urved Riemannian manifolds that are homogeneous (in parti ular, o ompa t) but have a point at in(cid:28)nity (cid:28)xed by all isometries. Sin e a CAT(0) latti e onsists of a group and a spa e, there are two natural notions of irredu ibility: of the group or of the spa e. In the ase of latti es in semi-simple groups, the two notions are known to oin ide by a result of Margulis [Mar91, II.6.7℄. We prove that this is the ase for CAT(0) latti es as above. Γ Theorem 1.4. In the setting of Theorem 1.3, is irredu ible as an abstra t group if and Γ Γ X = X X 1 1 1 2 only if for any (cid:28)nite index subgroup and any -equivariant de omposition × X Γ Is(X ) i 1 i with non- ompa t, the proje tion of to both is non-dis rete. The ombinationofTheorem1.4,Theorem1.3andofanappropriateformofsuperrigidity allow us to give a CAT(0) version of Mostow rigidity for redu ible spa es (Se tion 4.E). . Geometri arithmeti ity. We now expose results giving perhaps unexpe tedly strong on lusions for CAT(0) latti es (cid:22) both for the group and for the spa e. These results were announ ed in[CM08e℄inthe aseofCAT(0) groups; thepresentsettingof(cid:28)nitelygenerated latti es is more general sin e CAT(0) groups are (cid:28)nitely generated ( f. Lemma 3.3 below). g inf d(gx,x) x∈X We re all that an isometry is paraboli if the translation length is not a hieved. For general CAT(0) spa es, paraboli isometries are not well understood; in fa t, ruling out their existen e an sometimes be the essential di(cid:30) ulty in rigidity statements. (Γ,X) X Theorem 1.5. Let be an irredu ible (cid:28)nitelygenerated CAT(0) latti e with geodesi- X ally omplete. Assume that possesses some paraboli isometry. 4 PIERRE-EMMANUELCAPRACEANDNICOLASMONOD Γ X If is residually (cid:28)nite, then is a produ t of symmetri spa es and Bruhat(cid:21)Tits build- Γ X ings. In parti ular, is an arithmeti latti e unless is a real or omplex hyperboli spa e. Γ X If is not residually (cid:28)nite, then still splits o(cid:27) a symmetri spa e fa tor. Moreover, Γ Γ Γ/Γ D D the (cid:28)nite residual of is in(cid:28)nitely generated and is an arithmeti group. (Re allthatthe(cid:28)niteresidualofagroupistheinterse tionofall(cid:28)niteindexsubgroups.) We single out a purely geometri onsequen e. (Γ,X) X Corollary 1.6. Let be a (cid:28)nitely generated CAT(0) latti e with geodesi ally om- plete. X X = M X′ M ∼ Then possesses aparaboli isometryifandonlyif × , where isasymmetri spa e of non- ompa t type. Without the assumption of geodesi ompleteness, we still obtain an arithmeti ity state- ment when the underlying spa e admits some paraboli isometry that is neutral, i.e. whose displa ement length vanishes. Neutral paraboli isometries are even less understood, not even for their dynami al properties (whi h an be ompletely wild at least in Hilbert spa e [Ede64℄); as for familiar examples, they are provided by unipotent elements in semi- simple algebrai groups. (Γ,X) X Theorem 1.7. Let be an irredu ible (cid:28)nitely generated CAT(0) latti e. If admits any neutral paraboli isometry, then either: Is(X) (i) is a rank one simple Lie group with trivial entre; or: Γ Γ Γ/Γ Γ D D D (ii) has a normal subgroup su h that is an arithmeti group. Moreover, is either (cid:28)nite or in(cid:28)nitely generated. We turn to another type of statement of arithmeti ity/geometri superrigidity. Having established an abstra t arithmeti ity theorem (presented below as Theorem 1.9), we an appeal to our geometri results and prove the following. (Γ,X) X Theorem 1.8. Let be an irredu ible (cid:28)nitelygenerated CAT(0) latti e with geodesi- Γ ally omplete. Assume that possesses some faithful (cid:28)nite-dimensional linear representa- ( = 2,3) tion in hara teristi 6 . X Γ X If is redu ible, then is an arithmeti latti e and is a produ t of symmetri spa es and Bruhat(cid:21)Tits buildings. Se tion 6 ontains more results of this nature but also demonstrates by a family of exam- ples that some of the intri a ies in the more detailed statements re(cid:29)e t indeed the existen e (Γ,X) of more exoti pairs . . Abstra t arithmeti ity. When preparing for the proof of our geometri arithmeti ity statements, we are led to study irredu ible latti es in produ ts of general topologi al groups in the abstra t. Building notably on ideas of Margulis, weQstZablish the following arithmeti - ity statement (for whi h we re all that the quasi- entre of a topologi al group is the subset of elements with open entraliser). Γ < G = G G 1 n Theorem 1.9. Let ×···× be an irredu ible (cid:28)nitely generated latti e, where G i ea h is any lo ally ompa t group. Γ If admits a faithful Zariski-dense representation in a semi-simple group over some (cid:28)eld = 2,3 R G QofZ h(aGr)a teristi 6 , then the aΓmeRnable radi al of is ompa t aGnd the quasi- entre is virtually ontained in · . Furthermore, upon repla ing by a (cid:28)nite index ISOMETRY GROUPS OF NON-POSITIVELY CURVED SPACES: DISCRETE SUBGROUPS 5 G/R G+ QZ(G/R) G+ subgroup, the quotient splits as × where is a semi-simple algebrai Γ G+ group and the image of in is an arithmeti latti e. QZ(G/R) Inparti ular, thequasi- entre isdis rete. Inshorterterms,thistheoremstates G that up to a ompa t extension, is the dire t produ t of a semi-simple algebrai group Γ by a (possibly trivial) dis rete group, and that the image of in the non-dis rete part is an arithmeti group. The assumption on the hara teristi an be slightly weakened. In the ourse of the proof, we hara terise all irredu ible (cid:28)nitely generated latti es in G = S D S D produ ts of the form × where is a semi-simple Lie group and a totally D dis onne ted group (Theorem 5.18). In parti ular, it turns our that must ne essarily be lo allypro(cid:28)nitebyanalyti . The orresponding questionforsimplealgebrai groupsinstead of Lie groups is also investigated (Theorem 5.20). . Unique geodesi extension. Complete simply onne ted Riemannian manifolds of non- positive urvature, sometimes also alled Hadamard manifolds, form a lassi al family of proper CAT(0) spa es to whi h the pre eding results may be applied. In fa t, the natural lass to onsider in our ontext onsists of those proper CAT(0) spa es in whi h every geo- desi segment extends uniquely to a bi-in(cid:28)nite geodesi line. Clearly, this lass ontains all Hadamard manifolds, but it presumably ontains more examples. It is, however, somewhat restri ted with respe t to the main thrust of the present work sin e it does not allow for, say, simpli ial omplexes; a ordingly, the on lusions of the theorem below are also more stringent. X Theorem1.10. Let beaproper CAT(0) spa e withuniquely extensible geodesi s. Assume Is(X) that a ts o ompa tly without (cid:28)xed points at in(cid:28)nity. X X Is(X) (i) If is irredu ible, then either is a symmetri spa e or is dis rete. Is(X) Γ (ii) If possesses a (cid:28)nitely generated non-uniform latti e whi h is irredu ible as X an abstra t group, then is a symmetri spa e (without Eu lidean fa tor). Is(X) Γ Γ (iii) Suppose that possesses a (cid:28)nitely generated latti e (if is uniform, this is Γ X equivalent to the ondition that is a dis rete o ompa t group of isometries of ). Γ X X If is irredu ible (as an abstra t group) and is redu ible, then is a symmetri spa e (without Eu lidean fa tor). Inthespe ial aseofHadamardmanifolds,statement(i)wasknownundertheassumption Is(X) Is(X) that satis(cid:28)es the duality ondition (without assuming that a ts o ompa tly without (cid:28)xed points at in(cid:28)nity). This is due to P. Eberlein (Proposition 4.8 in [Ebe82℄). Likewise, statement (iii) for manifolds is Proposition 4.5 in [Ebe82℄. More re ently, Farb(cid:21)Weinberger [FW06℄ investigated analogous questions for aspheri al manifolds. . Latti es and the de Rham de omposition. In [CM08 ℄, we proved a (cid:16)de Rham(cid:17) de omposition X′ = X X Rn Y Y ∼ 1 p 1 q (1.i) ×···× × × ×···× X Is(X) for proper CAT(0) spa es with (cid:28)nite-dimensional Tits boundaryXa′nd sXu h that has no (cid:28)xed point at in(cid:28)nity, see Addendum 1.8 in [CM08 ℄. (Here ⊆ is the anoni al 6 PIERRE-EMMANUELCAPRACEANDNICOLASMONOD X′ = X X minimalinvariantsubspa e,andwere allthat e.g. when isgeodesi ally omplete and admits a o ompa t latti e by Lemma 3.13 in [CM08 ℄.) It turns out that this de Rham de omposition is an invariant of CAT(0) groups in the following sense (see Corollary 4.14). X Γ < Is(X) Theorem 1.11. Let be a proper CAT(0) spa e and be a group a ting properly dis ontinuously and o ompa tly. Γ Then any other su h spa e admitting a proper o ompa t -a tion has the same number of fa tors in (1.i) and the Eu lidean fa tor has same dimension. ISOMETRY GROUPS OF NON-POSITIVELY CURVED SPACES: DISCRETE SUBGROUPS 7 Contents 1. Introdu tion 1 2. An analogue of Borel density 8 2.A. Fixed points at in(cid:28)nity 8 2.B. Geometri density for subgroups of (cid:28)nite ovolume 9 2.C. The limit set of subgroups of (cid:28)nite ovolume 11 3. CAT(0) latti es, I: the Eu lidean fa tor 12 3.A. Preliminaries on latti es 12 3.B. Variations on Auslander's theorem 13 3.C. Latti es, the Eu lidean fa tor and (cid:28)xed points at in(cid:28)nity 14 4. CAT(0) latti es, II: produ ts 18 4.A. Irredu ible latti es in CAT(0) spa es 18 4.B. The hull of a latti e 20 4.C. On the anoni al dis rete kernel 21 4.D. Residually (cid:28)nite latti es 22 4.E. Strong rigidity for produ t spa es 23 5. Arithmeti ity of abstra t latti es 24 5.A. Superrigid pairs 26 5.B. Boundary maps 29 5.C. Radi al superrigidity 30 5.D. Latti es with non-dis rete ommensurators 31 5.E. Latti es in produ ts of Lie and totally dis onne ted groups 32 5.F. Latti es in general produ ts 34 6. Geometri arithmeti ity 37 6.A. CAT(0) latti es and paraboli isometries 37 6.B. Arithmeti ity of linear CAT(0) latti es 39 6.C. A family of examples 40 7. A few questions 42 Referen es 44 8 PIERRE-EMMANUELCAPRACEANDNICOLASMONOD 2. An analogue of Borel density Before dis ussing our analogue of Borel's density theorem [Bor60℄ in Se tion 2.B below, we present a more elementary phenomenon based on o-amenability. H G 2.A. Fixed points at in(cid:28)nity. Re all that a subgroup of a topologi al group is G o-amenable if any ontinuous a(cid:30)ne -a tion on a onvex ompa t set (in a Hausdor(cid:27) H lo ally onvex topologi al ve tor spa e) has a (cid:28)xed point whenever it has an -(cid:28)xed point. The arguments of Adams(cid:21)Ballmann [AB98℄ imply the following preliminary step towards Theorem 2.4: G Proposition 2.1. Let be a topologi al group with a ontinuous isometri a tion on a X G proper CAT(0) spa e without Eu lidean fa tor. Assume that the -a tion is minimal and ∂X does not have a global (cid:28)xed point in . G ∂X Then any o-amenable subgroup of still has no global (cid:28)xed point in . H < G ξ ∂X GProof. Supposefora ontradi tionµthata∂X o-amenable subgroup (cid:28)xes f∈: X . TRhen preserves a probability measure on and we obtain a onvex fun tion → by integrating Busemann fun tions against thismeasure; asin [AB98℄, the o y le equation for f G Busemann fun tions (see Ÿ2 in [CM08 ℄) imply that is -invariant up to onstants. The f µ arguments therein show that is onstant and that is supported on (cid:29)at points. However, in the absen e of a Eu lidean fa tor, thGe set of (cid:29)at points has a unique ir um entre whe(cid:3)n non-empty [AB98, 1.7℄; this provides a -(cid:28)xed point, a ontradi tion. Combining the above with the splitting methods used in Theorem 4.3 in [CM08 ℄, we re ord a onsequen e showing that the exa t on lusions of the Adams(cid:21)Ballmann theo- G rem [AB98℄ hold under mu h weaker assumptions than the amenability of . G Corollary 2.2. Let be a topologi al group with a ontinuous isometri a tion on a proper X G CAT(0) spa e . Assume that ontains two ommuting o-amenable subgroups. G X Then either (cid:28)xes a point at in(cid:28)nity or it preserves a Eu lidean subspa e in . We emphasise that one an easily onstru t a wealth of examples of highly non-amenable Q groups saGtisf=yinZg t⋉heseassuQmptions. For instan e, given any group , therestri ted wHreat=h produ t n∈Z ontains the pair of ommuting o-amenable groups + Q H =L Q H n≥0 and − n<0 , see [MP03℄. (In fa t, one an even arrange for ± to be L L Z onjugated upon repla ing by the in(cid:28)nite dihedral group.) Forsimilarreasons,wededu ethefollowing(cid:28)xed-pointpropertyforR.Thompson'sgroup F := g ,i N g−1g g = g j > i ; i ∈ | i j i j+1 ∀ (cid:10) (cid:11) this(cid:28)xed-point resultexplains why the strategy proposed in[Far08℄to disprove amenability F of with the Adams(cid:21)Ballmann theorem annot work. F X Corollary 2.3. Any -a tion by isometries on any proper CAT(0) spa e has a (cid:28)xed X point in . G Proof of Corollary 2.2. We assume that has no (cid:28)xed point at in(cid:28)nity. By Proposition 4.1 G in [CM08 ℄, there is a minimal non-empty losed onvex -invariant subspa e. Upon on- X sidering the Eu lidean de omposition [BH99, II.6.15℄ of the latter, we an assume that is G G X -minimal and without Eu lidean fa tor and need to show that (cid:28)xes a point in . H < G ± Let be the ommuting o-amenable groups. In view of Proposition 2.1, both H = H H + − a t without (cid:28)xed point at in(cid:28)nity. In parti ular, we have an a tion of × without (cid:28)xed point at in(cid:28)nity and the splitting theorem from [Mon06℄ provides us with a ISOMETRY GROUPS OF NON-POSITIVELY CURVED SPACES: DISCRETE SUBGROUPS 9 X X X H ∂X + − + anoni al subspa e × ⊆ with omponent-wise and minimal -a tion. All of H X − is (cid:28)xed by , whi h means that this boundary is empty. Sin e is proper, it follows X H + + that is bounded and hen e redu ed to a point by minimality. Thus (cid:28)xes a point in X X G µ X µ ⊆ and o-amenability implies that (cid:28)xes a probability measure on . If were ∂X G supported on , the proof of Proposition 2.1 would provide a -(cid:28)xed point at in(cid:28)nity, µ(X) > 0 B X whi h is absurd. Therefore . Now hoose a bounded set ⊆ large enough so µ(B) > µ(X)/2 G B B G that . Then any -translate of must meet . It follows that has(cid:3)a bounded orbit and hen e a (cid:28)xed point as laimed. F Proof of Corollary 2.3. We refer to [CFP96℄ for a detailed introdu tion to the group . F In parti ular, an be realised as the group of all orientation-preserving pie ewise a(cid:30)ne [0,1] 2n nhomZeomorphisms of thAe int[e0r,v1a]l that haFve <dyFadi breakpoints and slopes A with A ∈ . Given a subset ⊆ we denote by the subgroup supported on . A F F A We laimthatwhenever hasnon-emptyinterior, is o-amenablein . Theargument is analogous to [MP03℄ and to [GM07, Ÿ4.F℄; indeed, in view of the alternative de(cid:28)nition of F g F g A [1/n,1 1/n] n n just re Falglned, one anF hoose a sequen e { } in su h that ontainsF/F − and thus A ontains [1/n,1−1/n]. Consider the ompa t spa e of means on A, namely ℓ∞(F/F ) A (cid:28)nitely additive measureµs, endowed with the weak-* topologyg−fr1oFm the dual of . n A Any a umFu′lation point Fof the sequen e ofFD′ ira masses at will be invariant under [1/n,1−1/n] tFheun2iZon 2Zofthegroups 0.,N1ow isthekerneFlo′fthederivativehomFomorphism →F′ × at theµpair of points { }. In partFi ular, is o-amenFab/lFe in and thus A the -invarian e of implies that there is also a -invariant mean on , whi h is one of the hara terisations of o-amenability [Eym72℄. X F Let now be any proper CAT(0) spa e with an -a tion by isometries. We an assume F X that has no (cid:28)xed point at in(cid:28)nity and therefore we an also assume that is minimal by Proposition 4.1 in [CM08 ℄. The above laim provides us with many pairs of ommuting o-amenable subgroXup=s uRpnon takingndisjoint sets of non-empty interior. Therefore, CoFrol- ∼ lary2.2showsthat forsome . Inparti ulartheisometrygroupislinear. Sin e is g g 0 1 (cid:28)nitely generated (by and in the above presentation, ompare also [CFP96℄), Mal ev's F F theorem [Mal40℄ implies that the image of Fis′residually (cid:28)nite. The derived subgroup of (whi h in identally oin ides with the group introdu ed above) being simple [CFP96℄, it Rfolnlows that it a ts trivially. It remains onRlyn to observe that two ommuting isometries o(cid:3)f always have a ommon (cid:28)xed point in , whi h is a matter of linear algebra. Theabovereasoning anbeadapted to yieldsimilar resultsfor bran h groupsandrelated groups; we shall address these questions elsewhere. 2.B. Geometri density for subgroups of (cid:28)nite ovolume. The following geometri density theorem generalises Borel's density (see Proposition 2.8 below) and ontains Theo- rem 1.1 from the Introdu tion. G Theorem 2.4. Let be a lo ally ompa t group with a ontinuous isometri a tion on a X proper CAT(0) spa e without Eu lidean fa tor. G ∂X If a ts minimally and without global (cid:28)xed point in , then any losed subgroup with G (cid:28)nite invariant ovolume in still has these properties. Remark 2.5. For a related statement without the assumption on the Eu lidean fa tor of X or on (cid:28)xed points at in(cid:28)nity, see Theorem 3.14 below. Γ < G Proof. Retain the notation of the theorem and let be a losed subgroup of (cid:28)nite Γ invariant ovolume. In parti ular, is o-amenable and thus has no (cid:28)xed points at in(cid:28)nity 10 PIERRE-EMMANUELCAPRACEANDNICOLASMONOD by Proposition 2.1. By Proposition 4.1 in [CM08 ℄, there is a minimal non-empty losed Γ Y X Y = X x X 0 onvex -infva:rXiant sRubset ⊆ and it remains to show . Choose a point ∈ and de(cid:28)ne → by f(x)= d(x,gY) d(x ,gY) dg. Z − 0 G/Γ(cid:0) (cid:1) d(x,x ) f 0 This integral onverges be ause the integrand is bounded by . The fun tion is ontinuous, onvex (by [BH99, II.2.5(1)℄) and (cid:16)quasi-invariant(cid:17) in the sense that it satis(cid:28)es f(hx) = f(x) f(hx ) h G. 0 (2.i) − ∀ ∈ G f Sin e a ts minimally and without (cid:28)xed point at in(cid:28)nity, this implies that is onstant ∂X (see Se tion 2 in [AB98℄; alternatively, when is (cid:28)nite-dimensional, it follows from The- f oGr′em 1.10 in [CM08 ℄ sin e (2.i) implies that is invariant under the derived subgroup ). d(x,gY) g x X In parti ular, is a(cid:30)ne for all . It follows that for all ∈ the losed set Y = z X : d(z,Y) = d(x,Y) x ∈ (cid:8) (cid:9) Y d(z,Y)= d(y,Y ) z Y x x is onvex. We laim that it isparallel to in the sensethat for all ∈ y Y d(z,Y) z Y x and all ∈ . Indeed, on the one hand is onstant over ∈ by de(cid:28)nition, and on d(y,Y ) Y d(,Y ) Γ x x the other hand is onstant by minimality of sin e · is a onvex -invariant Y Γ Y x fun tion. In parti ular, is -equivariantly isometri to via nearest point proje tion Y Γ x ( ompare [BH99, II.2.12℄) and ea h is -minimal. At this point, Remarks 39 in [Mon06℄ Γ show that there is an isometri -invariant splitting X = Y T. ∼ × T s,t T It remains to show that the (cid:16)spa e of omponents(cid:17) isredu ed to apoint. Let thus ∈ m Y 0 and let be their midpoint. Applying the above reasoning to the hoi e of minimal set Y m Y 0 orresponding to ×{ }, we dedu e again that the distan e to is an a(cid:30)ne fun tion on X d(,m) T . However, this fun tion is pre isely the distan e fun tion · in omposed with the X T [s,t] proje tsio=nt → . Being non-negative and a(cid:30)ne on , it vanishes on that segment an(cid:3)d hen e . Γ G Remark 2.6. When is o ompa t in , the proof an be shortened by integrating just d(x,gY) f in the de(cid:28)nition of above. X G = Corollary 2.7. Let be a proper CAT(0) spa e without Eu lidean fa tor su h that Is(X) Γ < G a ts minimally without (cid:28)xed point at in(cid:28)nity, and let be a losed subgroup with (cid:28)nite invariant ovolume. Then: Γ (i) has trivial amZena(bΓle) radi al. G (ii) ThΓe entraliser is trivial. N (Γ) G (iii) If is (cid:28)nitely generated, then is has (cid:28)nite index in its normaliser and the G latter is a (cid:28)nitely generated latti e in . Proof. (i) and (ii) follow by the same argument as in the proof of Theorem 1.10 in [CM08 ℄. Γ For (iii) we follow [Mar91, Lemma II.6.3℄. Sin e is losed and ountable, it is dis rete G by Baire's ategory theorem and thus is a latti e in .NSin( Γe)it is (cid:28)nitely generatAeud,t(iΓts) G automorphismgroupis ountable. ByN(ii)(,Γt)henormaliser G mapsinje tivelyto G andhen eis ountable aswell. ThusΓ , being losedin , isdis retebyaΓpplyNing(BΓa)ire G again. Sin eNit (oΓn)tains the latti e , it is itself a latti e and the index of in (cid:3)is G (cid:28)nite. Thus is (cid:28)nitely generated.