Isometry groups of non-positively curved spaces: structure theory
We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and B…
Authors: P.-E. Caprace, N. Monod
ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y PIERRE-EMMANUEL CAPRA CE* AND NICOLAS MONOD ‡ Abstra t. W e dev elop the struture theory of full isometry groups of lo ally ompat non-p ositiv ely urv ed metri spaes. Amongst the disussed themes are de Rham deom- p ositions, normal subgroup struture and haraterising prop erties of symmetri spaes and BruhatTits buildings. Appliations to disrete groups and further dev elopmen ts on non-p ositiv ely urv ed latties are exp osed in a ompanion pap er [ CM08b ℄. 1. Intr odution Non-p ositiv ely urv ed metri spaes w ere in tro dued b y A. D. Alexandro v [ Ale57 ℄ and p opularised b y M. Gromo v, who alled them CA T(0) spaes. Their theory oers a wide gatew a y to a form of generalised dieren tial geometry , whose ob jets enompass Riemannian manifolds of non-p ositiv e setional urv ature as w ell as large families of singular spaes inluding Eulidean buildings and man y other p olyhedral omplexes. It has found a wide range of appliations to v arious elds, inluding semi-simple algebrai and arithmeti groups, and geometri group theory . A reurren t theme in this area is the in terpla y b et w een the geometry of a lo ally ompat CA T(0) spae X and the algebrai prop erties of a disrete group Γ ating prop erly on X b y isometries. This in teration is exp eted to b e esp eially ri h and tigh t when the Γ -ation is o ompat; the pair ( X, Γ) is then alled a CA T(0) group . The purp ose of the presen t pap er and its ompanion [ CM08b ℄ is to highligh t the rle of a third en tit y through whi h the in teration b et w een X and Γ transits: namely the full isometry group Is( X ) of X . The top ology of uniform on v ergene on ompata mak es Is( X ) a lo ally ompat seond oun table group whi h is th us anonially endo w ed with Haar measures. It therefore mak es sense to onsider latties in Is( X ) , i.e. disrete subgroups of nite in v arian t o v olume; w e all su h pairs ( X, Γ) CA T(0) latties (th us CA T(0) groups are preisely uniform CA T(0) latties). This immediately suggests the follo wing t w o-step programme: (I) T o dev elop the basi struture theory of the lo ally ompat group Is( X ) and dedue onsequenes on the o v erall geometry of the underlying prop er CA T(0) spae X . This is the main purp ose of the presen t pap er. (I I) T o study CA T(0) latties and th us in partiular CA T(0) groups b y building up on the struture results of the presen t pap er, using new geometri densit y and sup er- rigidit y te hniques. This is arried out in the subsequen t pap er [ CM08b ℄. W e no w pro eed to desrib e the main results of this rst part in more detail. First, in 1.A , w e presen t results in the sp eial ase of geo desially omplete CA T(0) spaes, i.e. spaes in whi h ev ery geo desi segmen t an b e extended to a bi-innite geo desi line Key wor ds and phr ases. Non-p ositiv e urv ature, CA T(0) spae, lo ally ompat group, lattie. *F.N.R.S. Resear h Asso iate. ‡ Supp orted in part b y the Swiss National Siene F oundation. 1 2 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD whi h need not b e unique. Imp ortan t examples of geo desially omplete spaes are pro vided b y BruhatTits buildings and of ourse Hadamard manifolds, e.g. symmetri spaes. The seond and longer part of the In tro dution, 1.B , will presen t results v alid for arbi- trary lo ally ompat CA T(0) spaes. In either ase, the en tire on ten ts of the In tro dution rely on more general, more detailed but probably also more um brous statemen ts pro v ed in the ore of the text. 1.A. Spaes with extensible geo desis. The onlusions of sev eral results b eome esp e- ially lear and p erhaps more striking in the sp eial ase of geo desially omplete CA T(0) spaes. Bey ond Eulidean buildings and Hadamard manifolds, w e reall that a omplete CA T(0) spae that is also a homology manifold has automatially extensible geo desis [ BH99 , I I.5.12℄. Note also that it is alw a ys p ossible to artiially mak e a CA T(0) spae geo desially omplete b y gluing ra ys, though it is not alw a ys p ossible to preserv e prop erness (onsider a ompat but total set in an innite-dimensional Hilb ert spae). Deomp osing CA T(0) spaes in to pro duts of symmetri spaes and lo ally nite ell omplexes. Protot ypial examples of lo ally ompat CA T(0) spaes are mainly pro vided b y the follo wing t w o soures. Riemannian manifolds of non-p ositiv e setional urv ature, whose most prominen t represen tativ es are the Riemannian symmetri spaes of non-ompat t yp e. These spaes are regular in the sense that an y t w o geo desi segmen ts in terset in at most one p oin t. The full isometry group of su h a spae is a Lie group. P olyhedral omplexes of pieewise onstan t non-p ositiv e urv ature, su h as trees or Eulidean buildings. These spaes are singular in the sense that geo desis do bran h. The subgroup of the isometry group whi h preserv es the ell struture is totally disonneted. The follo wing result seems to indiate that a CA T(0) spae often splits as a pro dut of spaes b elonging to these t w o families. Theorem 1.1. L et X b e a pr op er ge o desi al ly omplete CA T(0) sp a e whose isometry gr oup ats o omp atly without xe d p oint at innity. Then X admits an Is( X ) -e quivariant split- ting X = M × R n × Y , wher e M is a symmetri sp a e of non- omp at typ e and the isometry gr oup Is( Y ) is total ly dis onne te d and ats by semi-simple isometries on Y (e ah fator may b e trivial). F urthermor e, the sp a e Y admits an Is( Y ) -e quivariant lo al ly nite de omp osition into onvex el ls, wher e the el l supp orting a p oint y ∈ Y is dene d as the xe d p oint set of the isotr opy gr oup Is( Y ) y . If X is regular, then Is( Y ) is disrete. In other w ords, the spae Y has bran hing geo desis as so on as Is( Y ) is non-disrete. W e refer to Theorem 1.6 and A ddendum 1.8 b elo w for a v ersion of the ab o v e without the assumption of extensibilit y of geo desis. W e emphasize that the `ells' pro vided b y Theorem 1.1 need not b e ompat; in fat if Is( Y ) ats freely on Y then the deomp osition in question b eomes trivial and onsists of a single ell, namely the whole of Y . Con v ersely the ell deomp osition is non-trivial pro vided Is( Y ) do es not at freely . The most ob vious w a y for the Is( Y ) -ation not to b e free is if Is( Y ) is not disrete. A strong v ersion of the latter ondition is that no op en sub gr oup of xes a p oint at innity ; this holds notably for symmetri spaes and Bruhat Tits buildings. A quite immediate onsequene of this ondition is that the ab o v e ells are ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 3 then neessarily ompat. W e shall sho w that m u h additional struture an b e deriv ed from it (see Setion 7.D b elo w). . Smo othness. The ell deomp osition of the third fator in Theorem 1.1 is deriv ed from the follo wing smo othness result for isometri ations of totally disonneted groups. Theorem 1.2. L et X b e a ge o desi al ly omplete pr op er CA T(0) sp a e X and G < I s ( G ) a total ly dis onne te d (lose d) sub gr oup ating minimal ly. The the p ointwise stabiliser in G of every b ounde d set is op en. This prop ert y , whi h is familiar from lassial examples, do es in gener al fail without geo desi ompleteness (see Remark 6.10 in [ CM08b ℄). It is an imp ortan t ingredien t for the onsiderations of Setion 7.D alluded to ab o v e, as w ell as for angle rigidit y results regarding b oth the Alexandro v angle (Prop osition 6.8 ) and the Tits angle (Prop osition 7.15 ). . A haraterisation of symmetri spaes and Eulidean buildings. In symmetri spaes and BruhatTits buildings, the stabilisers of p oin ts at innit y are exatly the para- b oli subgroups; as su h, they are o ompat. This o ompatness holds further for all Bass Serre trees, namely bi-regular trees. Com bining our results with w ork of B. Leeb [ Lee00 ℄ and A. Lyt hak [ Lyt05 ℄, w e establish a orresp onding haraterisation. Theorem 1.3. L et X b e a ge o desi al ly omplete pr op er CA T(0) sp a e. Supp ose that the stabiliser of every p oint at innity ats o omp atly on X . Then X is isometri to a pr o dut of symmetri sp a es, Eulide an buildings and BassSerr e tr e es. The Eulidean buildings app earing in the preeding statemen t admit an automorphism group that is strongly transitiv e, i.e. ats transitiv ely on pairs ( c, A ) where c is a ham b er and A an apartmen t on taining c . This prop ert y haraterises the BruhatTits buildings, exept p erhaps for some t w o-dimensional ases where this is a kno wn op en question. The ab o v e haraterisation is of a dieren t nature and indep enden t of the haraterisa- tions using latties that will b e presen ted in [ CM08b ℄. . A tions of simple algebrai groups. Both for the general theory and for the geometri sup errigidit y/arithmetiit y statemen ts of the ompanion pap er [ CM08b ℄, it is imp ortan t to understand ho w algebrai groups at on CA T(0) spaes. Theorem 1.4. L et k b e a lo al eld and G b e an absolutely almost simple simply onne te d k -gr oup. L et X b e a CA T(0) sp a e (not r e du e d to a p oint) on whih G = G ( k ) ats ontinuously and o omp atly by isometries. Then X is isometri to X model , the R iemannian symmetri sp a e or BruhatTits building asso iate d with G . A stronger and m u h more detailed statemen t is pro vided b elo w as Theorem 7.4 . In partiular, a mo diation of the statemen t holds without extensibilit y of geo desis and the 4 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD o ompatness assumption an b e relaxed. Ho w ev er, w e also sho w there b y means of t w o examples that some assumptions remain neessary . (As a p oin t of terminology , w e do not ho ose a partiular saling fator on X model , so that the isometry of our statemen t ould b eome a homothet y for another mo del.) . 1.B. General ase. When dealing with CA T(0) spae in the highest p ossible lev el of gen- eralit y , one has to fae sev eral te hnial diulties aused b y the exibilit y of the CA T(0) ondition. F or example, giv en a CA T(0) spae X , there are man y w a ys to deform it in order to onstrut another spae Y , non-isometri to X , but with the prop ert y that X and Y ha v e isomorphi isometry groups or/and iden tial b oundaries. Amongst the simplest on- strutions, one an form (p ossibly w arp ed) pro duts with ompat CA T(0) spaes or gro w hair equiv arian tly along a disrete orbit. Mu h wilder (non-quasi-isometri) examples an b e onstruted for instane b y onsidering w arp ed pro duts with the v ery v ast family of CA T(0) spaes ha ving no isometries and a unique p oin t at innit y . In order to address these issues, w e in tro due the follo wing terminology . Minimalit y . . . . ís a ti n eÒpoi sfara âgk¸ m ia, aÎt t a ˜ ut a kaÈ falkra âgk¸ m ia diexèr q et ai. Sunèsi o Kurena ou, Falkra âgk¸ m i o n . 1 An isometri ation of a group G on a CA T(0) spae X is said to b e minimal if there is no non-empt y G -in v arian t losed on v ex subset X ′ ( X ; the spae X is itself alled minimal if its full isometry group ats minimally . A CA T(0) spae X is alled b oundary-minimal if it p ossesses no losed on v ex subset Y ( X su h that ∂ Y = ∂ X . Here is ho w these notions relate to one another. Prop osition 1.5. L et X b e a pr op er CA T(0) sp a e. (i) Assume ∂ X nite-dimensional. If X is minimal, then it is b oundary-minimal. (ii) Assume Is( X ) has ful l limit set. If X is b oundary-minimal, then it is minimal. (iii) If X is o omp at and ge o desi al ly omplete, then it is b oth minimal and b oundary- minimal. W e emphasize that it is not true in general that a minimal CA T(0) spae is geo desially omplete, ev en if one assumes that the isometry group ats o ompatly and without global xed p oin t at innit y . . Group deomp ositions. W e no w turn to struture results on the lo ally ompat isometry group Is( X ) of a prop er CA T(0) spae X in a broad generalit y; w e shall mostly ask that no p oin t at innit y b e xed sim ultaneously b y all isometries of X . This non-degeneray assumption will b e sho wn to hold automatially in the presene of latties (see Theorem 3.11 in [ CM08b ℄). The the follo wing result w as the starting p oin t of this w ork. 1 Synesius of Cyrene, Falkra âgk¸ m i o n (kno wn as Calvitii en omium ), end of Chapter 8 (at 72A in the page n um b ering from Denis P étau's 1633 edition). The En omium w as written around 402; w e used the 1834 edition b y J. G. Krabinger (Ch. G. Löund, Stuttgart). The ab o v e exerpt translates roughly to: as muh pr aise as is given to the spher es is due to the b ald he ad to o . ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 5 Theorem 1.6. L et X b e a pr op er CA T(0) sp a e with nite-dimensional Tits b oundary. Assume that Is( X ) has no glob al xe d p oint in ∂ X . Then ther e is a anoni al lose d onvex Is( X ) -stable subset X ′ ⊆ X suh that G = Is( X ′ ) has a nite index op en har ateristi sub gr oup G ∗ ✁ G whih admits a anoni al de omp osition (1.i) G ∗ ∼ = S 1 × · · · × S p × R n ⋊ O ( n ) × D 1 × · · · × D q ( p, q , n ≥ 0) wher e S i ar e almost onne te d simple Lie gr oups with trivial entr e and D j ar e total ly dis- onne te d irr e duible gr oups with trivial amenable r adi al. A ny pr o dut de omp osition of G ∗ is a r e gr ouping of the fators in ( 1.i ) . Mor e over, al l non-trivial normal, subnormal or as ending sub gr oups N < D j ar e stil l irr e duible with trivial amenable r adi al and trivial entr aliser in D j . (These pr op erties also hold for latti es in N and their normal, subnormal or as ending sub gr oups, se e [ CM08b ℄ .) (A top ologial group is alled irreduible if no nite index op en subgroup splits non- trivially as a diret pro dut of losed subgroups. The amenable radial of a lo ally ompat group is the largest amenable normal subgroup; it is indeed a r adi al sine the lass of amenable lo ally ompat groups is stable under group extensions.) Remarks 1.7. (i) The nite-dimensionalit y assumption holds automatially when X has a o ompat group of isometries [ Kle99 , Theorem C℄. It is also automati for uniquely geo desi spaes, e.g. manifolds (Prop osition 7.11 ). (ii) The onlusion fails in v arious w a ys if G xes a p oin t in ∂ X . (iii) The quotien t G/G ∗ is just a group of p erm utations of p ossibly isomorphi fators in the deomp osition. In partiular, G = G ∗ ⋊ G/G ∗ . (iv) The anonial on tin uous homomorphism Is( X ) → Is( X ′ ) = G is prop er, but its image sometimes has innite o v olume. In Theorem 1.6 , the ondition that Is( X ) has no global xed p oin t at innit y ensures the existene of a losed on v ex Is( X ) -in v arian t subset Y ⊆ X on whi h Is( X ) ats minimally (see Prop osition 4.1 ). The set of these minimal on v ex subsets p ossesses a anonial elemen t, whi h is preisely the spae X ′ whi h app ears in Theorem 1.6 . Prop osition 1.5 explains wh y the distintion b et w een X and X ′ did not app ear in Theorem 1.1 . . De Rham deomp ositions. It is kno wn that pro dut deomp ositions of isometry groups ating minimally and without global xed p oin t at innit y indue a splitting of the spae (for o ompat Hadamard manifolds, this is the La wsonY au [ L Y72 ℄ and GromollW olf [ GW71 ℄ theorem; in general and for more referenes, see [ Mon06 ℄). It is therefore natural that Theorem 1.6 is supplemen ted b y a geometri statemen t. A ddendum 1.8. In The or em 1.6 , ther e is a anoni al isometri de omp osition (1.ii) X ′ ∼ = X 1 × · · · × X p × R n × Y 1 × · · · × Y q wher e G ∗ ats omp onentwise a or ding to ( 1.i ) and G/G ∗ p ermutes any isometri fators. A l l X i and Y j ar e irr e duible and minimal. As it turns out, a geometri deomp osition is the rst of t w o indep enden t steps in the pro of of Theorem 1.6 . In fat, w e b egin with an analogue of the de Rham deomp osition [ dR52 ℄ 6 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD whose pro of uses (a mo diation of ) argumen ts from the generalised de Rham theorem of F o erts hLyt hak [ FL06 ℄. In purely geometrial terms, w e ha v e the follo wing statemen t. Theorem 1.9. L et X b e a pr op er b oundary-minimal CA T(0) sp a e with ∂ X nite-dimensional. Then X admits a anoni al maximal isometri splitting X ∼ = R n × X 1 × · · · × X m ( n, m ≥ 0) with e ah X i irr e duible and 6 = R 0 , R 1 . Every isometry of X pr eserves this de omp osition up on p ermuting p ossibly isometri fators X i . Mor e over, if X is minimal, so is every X i . T o apply this theorem, it is desirable to kno w onditions ensuring b oundary-minimalit y . In addition to the onditions pro vided b y Prop osition 1.5 , w e sho w that a anonial b oundary- minimal subspae exists as so on as the b oundary has irumradius > π / 2 (Corollary 3.10 ). In the seond part of the pro of of Theorem 1.6 , w e analyse the irreduible ase where X admits no isometri splitting, resulting in Theorem 1.10 to whi h w e shall no w turn. Com bining these t w o steps, w e then pro v e the unique deomp osition of the gr oups , using also the splitting theorem from [ Mon06 ℄. . Geometry of normal subgroups. In É. Cartan's orresp ondene b et w een symmetri spaes and semi-simple Lie groups as w ell as in BruhatTits theory , irreduible fators of the spae orresp ond to simple groups. F or general CA T(0) spaes and groups, simpliit y fails of ourse v ery dramatially (free groups are p erhaps the simplest, and y et most non- simple, CA T(0) groups). Nonetheless, w e establish a geometri w eak ening of simpliit y . Theorem 1.10. L et X 6 = R b e an irr e duible pr op er CA T(0) sp a e with nite-dimensional Tits b oundary and G < I s ( X ) any sub gr oup whose ation is minimal and do es not have a glob al xe d p oint in ∂ X . Then every non-trivial normal sub gr oup N ✁ G stil l ats minimal ly and without xe d p oint in ∂ X . Mor e over, the amenable r adi al of N and the entr aliser Z Is( G ) ( N ) ar e b oth trivial; N do es not split as a pr o dut. This result an for instane b e om bined with the solution to Hilb ert's fth problem in order to understand the onneted omp onen t of the isometry group. Corollary 1.11. Is( X ) is either total ly dis onne te d or an almost onne te d simple Lie gr oup with trivial entr e. The same holds for any lose d sub gr oup ating minimal ly and without xe d p oint at in- nity. A more elemen tary appliation of Theorem 1.10 uses the fat that elemen ts with a disrete onjugay lass ha v e op en en traliser. Corollary 1.12. If G is non-disr ete, N annot b e a nitely gener ate d disr ete sub gr oup. A feature of Theorem 1.10 is that is an b e iterated and th us applies to subnormal subgroups. Reall that more generally a subgroup H < G is asending if there is a (p ossibly transnite) hain of normal subgroups starting with H and abutting to G . Using limiting argumen ts, w e b o otstrap Theorem 1.10 and sho w: Theorem 1.13. L et N < G b e any non-trivial as ending sub gr oup. Then al l onlusions of The or em 1.10 hold for N . ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 7 . A few ases of sup errigidit y . Com bining the preeding general struture results with some of Margulis' theorems, w e obtain the follo wing sup errigidit y statemen t. Theorem 1.14. L et X b e a pr op er CA T(0) sp a e whose isometry gr oup ats o omp atly and without glob al xe d p oint at innity. L et Γ = SL n ( Z ) with n ≥ 3 and G = SL n ( R ) . F or any isometri Γ -ation on X ther e is a non-empty Γ -invariant lose d onvex subset Y ⊆ X on whih the Γ -ation extends uniquely to a ontinuous isometri ation of G . ( The orr esp onding statement applies to al l those latti es in semi-simple Lie gr oups that have virtual ly b ounde d gener ation by unip otents. ) Observ e that the ab o v e theorem has no assumptions whatso ev er on the ation; o om- patness is an assumption on the giv en CA T(0) spae. It an happ en that Γ xes p oin ts in ∂ X , but its ation on Y is without xed p oin ts at innit y and minimal (as w e shall establish in the pro of ). The assumption on b ounded generation holds onjeturally for all non-uniform irreduible latties in higher rank semi-simple Lie groups (but alw a ys fails in rank one). It is kno wn to hold for arithmeti groups in split or quasi-split algebrai groups of a n um b er eld K of K - rank ≥ 2 b y [ T a v90 ℄, as w ell as in a few ases of isotropi but non-quasi-split groups [ ER06 ℄; see also [ WM07 ℄. More generally , Theorem 1.14 holds for (S-)arithmeti groups pro vided the arithmeti subgroup (giv en b y in tegers at innite plaes) satises the ab o v e b ounded generation prop- ert y . F or instane, the SL n example is as follo ws: Theorem 1.15. L et X b e a pr op er CA T(0) sp a e whose isometry gr oup ats o omp atly and without glob al xe d p oint at innity. L et m b e an inte ger with distint prime fators p 1 , . . . p k and set Γ = SL n ( Z [ 1 m ]) , G = SL n ( R ) × SL n ( Q p 1 ) × · · · × SL n ( Q p k ) , wher e n ≥ 3 . Then for any isometri Γ -ation on X ther e is a non-empty Γ -invariant lose d onvex subset Y ⊆ X on whih the Γ -ation extends uniquely to a ontinuous isometri ation of G . W e p oin t out that a xed p oin t prop ert y for similar groups ating on lo w-dimensional CA T(0) ell omplexes w as established b y B. F arb [ F ar08 ℄. Some of our general results also allo w us to impro v e on the generalit y of the CA T(0) su- p errigidit y theorem for irreduible latties in arbitrary pro duts of lo ally ompat groups pro v ed in [ Mon06 ℄. F or ations on prop er CA T(0) spaes, the results of lo . it. establish an unrestrited sup errigidit y on the b oundary but require, in order to dedue sup errigidit y on the spae itself, the assumption that the ation b e r e du e d (or alternativ ely indeom- p osable). W e pro v e that, as so on as the b oundary is nite-dimensional, an y ation without global xed p oin t at innit y is alw a ys redued after suitably passing to subspaes and diret fators. It follo ws that the sup errigidit y theorem for arbitrary pro duts holds in that generalit y , see Theorem 8.4 b elo w. 8 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Contents 1. In tro dution 1 1.A. Spaes with extensible geo desis 2 1.B. General ase 4 2. Notation and preliminaries 9 3. Con v ex subsets of the Tits b oundary 10 3.A. Boundary subsets of small radius 10 3.B. Subspaes with b oundary of large radius 12 3.C. Minimal ations and b oundary-minimal spaes 13 4. Minimal in v arian t subspaes for subgroups 14 4.A. Existene of a minimal in v arian t subspae 14 4.B. Di hotom y 15 4.C. Normal subgroups 15 5. Algebrai and geometri pro dut deomp ositions 16 5.A. Preliminary deomp osition of the spae 16 5.B. Pro of of Theorem 1.6 and A ddendum 1.8 18 5.C. CA T(0) spaes without Eulidean fator 19 6. T otally disonneted group ations 20 6.A. Smo othness 20 6.B. Lo ally nite equiv arian t partitions and ellular deomp ositions 21 6.C. Alexandro v angle rigidit y 22 6.D. Algebrai struture 23 7. Co ompat CA T(0) spaes 25 7.A. Fixed p oin ts at innit y 25 7.B. A tions of simple algebrai groups 26 7.C. No bran hing geo desis 31 7.D. No op en stabiliser at innit y 32 7.E. Co ompat stabilisers at innit y 33 8. A few ases of CA T(0) sup errigidit y 35 8.A. CA T(0) sup errigidit y for some lassial non-uniform latties 35 8.B. CA T(0) sup errigidit y for irreduible latties in pro duts 37 Referenes 37 ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 9 2. Not a tion and preliminaries A metri spae is prop er if ev ery losed ball is ompat. W e refer to Bridson and Haeiger [ BH99 ℄ for ba kground on CA T(0) spaes. W e reall that the omparison angle ∠ p ( x, y ) determined b y three p oin ts p, x, y in an y metri spae is dened purely in terms of the orresp onding three distanes b y lo oking at the orresp ond- ing Eulidean triangle. In other w ords, it is dened b y d 2 ( x, y ) = d 2 ( p, x ) + d 2 ( p, y ) − 2 d ( p, y ) d ( p, y ) cos ∠ p ( x, y ) . The Alexandro v angle ∠ p ( x, y ) in a CA T(0) spae X is the non-inreasing limit of the omparison angle near p along the geo desi segmen ts [ p, x ] and [ p, y ] , see [ BH99 , I I.3.1℄. In partiular, ∠ p ( x, y ) ≤ ∠ p ( x, y ) . Lik ewise, geo desi ra ys from p determine the Alexandro v angle ∠ p ( ξ , η ) for ξ , η ∈ ∂ X . The Tits angle ∠ T ( ξ , η ) is dened as the suprem um of ∠ p ( ξ , η ) o v er all p ∈ X and has sev eral useful haraterisations giv en in Prop osition I I.9.8 of [ BH99 ℄. Reall that to an y p oin t at innit y ξ ∈ ∂ X is asso iated the Busemann funtion B ξ : X × X → R : ( x, y ) 7→ B ξ ,x ( y ) dened b y B ξ ,x ( y ) = lim t →∞ ( d ( ( t ) , y ) − d ( ( t ) , x )) , where : [0 , ∞ ) → X is an y geo desi ra y p oin ting to w ards ξ . The Busemann funtion do es not dep end on the hoie of and satises the follo wing: B ξ ,x ( y ) = − B ξ ,y ( x ) B ξ ,x ( z ) = B ξ ,x ( y ) + B ξ ,y ( z ) (the o yle relation) B ξ ,x ( y ) ≤ d ( x, y ) . Com bining the denition of the Busemann funtion and of the omparison angle, w e nd that if r is the geo desi ra y p oin ting to w ards ξ with r (0) = x , then for an y y 6 = x w e ha v e lim t →∞ cos ∠ x ( r ( t ) , y ) = − B ξ ,x ( y ) d ( x, y ) (the asymptoti angle form ula) . By abuse of language, one refers to a Busemann funtion when it is more on v enien t to onsider the on v ex 1 -Lips hitz funtion b ξ : X → R dened b y B ξ ,x for some (usually impliit) hoie of base-p oin t x ∈ X . W e shall simply denote su h a funtion b y b ξ in lo w er ase; they all dier b y a onstan t only in view of the o yle relation. The b oundary at innit y ∂ X is endo w ed with the ne top ology [ BH99 , I I.8.6℄ as w ell as with the (m u h ner) top ology dened b y the Tits angle. The former is often impliitly understo o d, but when referring to dimension or radius, the top ology and distane dened b y the Tits angle are onsidered (this is sometimes emphasised b y referring to the Tits b oundary). The later distane is not to b e onfused with the asso iated length metri alled Tits distane in the literature; w e will not need this onept (exept in the disussions at the b eginning of Setion 7 ). Reall that an y omplete CA T(0) spae splits o a anonial maximal Hilb ertian fator (Eulidean in the prop er ase studied here) and an y isometry deomp oses aordingly , see Theorem I I.6.15(6) in [ BH99 ℄. Normalisers and en tralisers in a group G are resp etiv ely denoted b y N G and Z G . When some group G ats on a set and x is a mem b er of this set, the stabiliser of x in G is denoted b y Stab G ( x ) or b y the shorthand G x . F or the notation regarding algebrai groups, w e follo w the standard notation as in [ Mar91 ℄. 10 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Finally , w e presen t t w o remarks that will nev er b e used b elo w but giv e some on text on ertain frequen t assumptions. The rst remark is the follo wing CA T(0) v ersion of the HopfRino w theorem: Every ge o desi al ly omplete lo al ly omp at CA T(0) sp a e is pr op er . Surprisingly , w e ould not nd this statemen t in the literature (though a dieren t statemen t is often referred to as the HopfRino w theorem, see [ BH99 , I.3.7℄). As p oin ted out orally b y A. Lyt hak, the ab o v e result is readily established b y follo wing the strategy of pro of of [ BH99 , I.3.7℄ and extending geo desis. The seond fat is that if a prop er CA T(0) spae is nite-dimensional (in the sense of [ Kle99 ℄), then so is its Tits b oundary (generalising for instane Prop osition 7.11 b elo w). The argumen t is giv en in [ CL08 , Prop osition 2.1℄ and ma y b e outlined as follo ws. F or an y sphere S in the spae X , the visual map ∂ X → S is Tits-on tin uous; if it w ere injetiv e, the result w ould follo w. Ho w ev er, it b eomes injetiv e after replaing S with the ultrapro dut of spheres of un b ounded radius b y the v ery denition of the b oundary; the ultrapro dut onstrution preserv es the b ound on the dimension, nishing the pro of. 3. Convex subsets of the Tits bound ar y 3.A. Boundary subsets of small radius. Giv en a metri spae X and a subset Z ⊆ X , one denes the irumradius of Z in X as inf x ∈ X sup z ∈ Z d ( x, z ) . A p oin t x realising the inm um is alled a irumen tre of Z in X . The in trinsi ir- umradius of Z is its irumradius in Z itself; one denes similarly an in trinsi irum- en tre . It is alled anonial if it is xed b y ev ery isometry of X whi h stabilises Z . W e shall mak e frequen t use of the follo wing onstrution of irumen tres, due to A. Balser and A. Lyt hak [ BL05 , Prop osition 1.4℄: Prop osition 3.1. L et X b e a omplete CA T(1) sp a e and Y ⊆ X b e a nite-dimensional lose d onvex subset. If Y has intrinsi ir umr adius ≤ π / 2 , then the set C ( Y ) of intrinsi ir um entr es of Y has a unique ir um entr e, whih is ther efor e a anoni al (intrinsi) ir um entr e of Y . Let no w X b e a prop er CA T(0) spae. Prop osition 3.2. L et X 0 ⊃ X 1 ⊃ . . . b e a neste d se quen e of non-empty lose d onvex subsets of X suh that T n X n is empty. Then the interse tion T n ∂ X n is a non-empty lose d onvex subset of ∂ X of intrinsi ir umr adius at most π / 2 . In p artiular, if the Tits b oundary is nite-dimensional, then T n ∂ X n has a anoni al intrinsi ir um entr e. Pr o of. Pi k an y x ∈ X and let x n b e its pro jetion to X n . The assumption T n X n = ∅ implies that x n go es to innit y . Up on extrating, w e an assume that it on v erges to some p oin t ξ ∈ ∂ X ; observ e that ξ ∈ T n ∂ X n . W e laim that an y η ∈ T n ∂ X n satises ∠ T ( ξ , η ) ≤ π / 2 . The prop osition then follo ws b eause (i) the b oundary of an y losed on v ex set is losed and π -on v ex [ BH99 , I I.9.13℄ and (ii) ea h ∂ X n is non-empt y sine otherwise X n w ould b e b ounded, on traditing T n X n = ∅ . When ∂ X has nite dimension, there is a anonial in trinsi irumen tre b y Prop osition 3.1 . F or the laim, observ e that there exists a sequene of p oin ts y n ∈ X n on v erging to η . It sues to pro v e that the omparison angle ∠ x ( x n , y n ) is b ounded b y π / 2 for all n , ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 11 see [ BH99 , I I.9.16℄. This follo ws from ∠ x n ( x, y n ) ≥ ∠ x n ( x, y n ) ≥ π / 2 , where the seond inequalit y holds b y the prop erties of the pro jetion on a on v ex set [ BH99 , I I.2.4(3)℄. The om bination of the preeding t w o prop ositions has the follo wing onsequene, whi h impro v es the results established b y F ujiw ara, Nagano and Shio y a (Theorems 1.1 and 1.3 in [ FNS06 ℄). Corollary 3.3. L et g b e a p ar ab oli isometry of X . The fol lowing assertions hold: (i) The xe d p oint set of g in ∂ X has intrinsi ir umr adius at most π / 2 . (ii) If ∂ X nite-dimensional, then the entr aliser Z Is( X ) ( g ) has a anoni al glob al xe d p oint in ∂ X . (iii) F or any sub gr oup H < Is( X ) ontaining g , the (p ossibly empty) xe d p oint set of H in ∂ X has ir umr adius at most π / 2 . Here is another immediate onsequene. Corollary 3.4. L et G b e a top olo gi al gr oup with a ontinuous ation by isometries on X without glob al xe d p oint. Supp ose that G is the union of an inr e asing se quen e of omp at sub gr oups and that ∂ X is nite-dimensional. Then ther e is a anoni al G -xe d p oint in ∂ X , xe d by al l isometries normalising G . Pr o of. Consider the sequene of xed p oin t sets X K n of the ompat subgroups K n . Its in tersetion is empt y b y assumption and th us Prop osition 3.2 applies. Finally , w e reord the follo wing elemen tary fat, whi h ma y also b e dedued b y means of Prop osition 3.2 : Lemma 3.5. L et ξ ∈ ∂ X . Given any lose d hor ob al l B entr e d at ξ , the b oundary ∂ B oinides with the b al l of Tits r adius π / 2 entr e d at ξ in ∂ X . Pr o of. An y t w o horoballs en tred at the same p oin t at innit y lie at b ounded Hausdor dis- tane from one another. Therefore, they ha v e the same b oundary at innit y . In partiular, the b oundary ∂ B of the giv en horoball oinides with the in tersetion of the b oundaries of all horoballs en tred at ξ . By Prop osition 3.2 , this is of irumradius at most π / 2 ; in fat the pro of of that prop osition sho ws preisely that the set is on tained in the ball of radius at most π / 2 around ξ . Con v ersely , let η ∈ ∂ X b e a p oin t whi h do es not b elong to ∂ B . W e laim that ∠ T ( ξ , η ) ≥ π / 2 . This sho ws that ev ery p oin t of ∂ X at Tits distane less than π / 2 from ξ b elongs to ∂ B . Sine the latter is losed, it follo ws that ∂ B on tains the losed ball of Tits radius π / 2 W e turn to the laim. Let b ξ b e a Busemann funtion en tred at ξ . Sine ev ery geo desi ra y p oin ting to w ards η esap es ev ery horoball en tred at ξ , there exists a ra y : [0 , ∞ ) → X p oin ting to η su h that b ξ ( (0)) = 0 and b ξ ( ( t )) > 0 for all t > 0 (atually , this inreases to innit y b y on v exit y). Let c : [0 , ∞ ) → X b e the geo desi ra y emanating from (0) and p oin ting to ξ . W e ha v e ∠ T ( ξ , η ) = lim t,s →∞ ∠ (0) ( ( t ) , c ( s )) , see [ BH99 , I I.9.8℄. Therefore the laim follo ws from the asymptoti angle form ula (Setion 2 ) b y taking y = c ( s ) with s large enough. 12 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD 3.B. Subspaes with b oundary of large radius. As b efore, let X b e a prop er CA T(0) spae. The follo wing result impro v es Prop osition 2.2 in [ Lee00 ℄: Prop osition 3.6. L et Y ⊆ X b e a lose d onvex subset suh that ∂ Y has intrinsi ir- umr adius > π / 2 . Then ther e exists a lose d onvex subset Z ⊆ X with ∂ Z = ∂ Y whih is minimal for these pr op erties. Mor e over, the union Z 0 of al l suh minimal subsp a es is lose d, onvex and splits as a pr o dut Z 0 ∼ = Z × Z ′ . Pr o of. If no minimal su h Z existed, there w ould b e a hain of su h subsets with empt y in tersetion. The distane to a base-p oin t m ust then go to innit y and th us the hain on- tains a oun table sequene to whi h w e apply Prop osition 3.2 , on traditing the assumption on the irumradius. Let Z ′ denote the set of all su h minimal sets and Z 0 = S Z ′ b e its union. As in [ Lee00 , p. 10℄ one observ es that for an y Z 1 , Z 2 ∈ Z ′ , the distane z 7→ d ( z , Z 2 ) is onstan t on Z 1 and that the nearest p oin t pro jetion p Z 2 restrited to Z 1 denes an isometry Z 1 → Z 2 . By the Sandwi h Lemma [ BH99 , I I.2.12℄, this implies that Z 0 is on v ex and that the map Z ′ × Z ′ → R + : ( Z 1 , Z 2 ) 7→ d ( Z 1 , Z 2 ) is a geo desi metri on Z ′ . As in [ Mon06 , Setion 4.3℄, this yields a bijetion α : Z 0 → Z × Z ′ : x 7→ ( p Z ( x ) , Z x ) , where Z x is the unique elemen t of Z ′ on taining x . The pro dut of metri spaes Z × Z ′ is giv en the pro dut metri. In order to establish that α is an isometry , it remains as in [ Mon06 , Prop osition 38℄, to trivialise holonom y; it the urren t setting, this is a hiev ed b y Lemma 3.7 , whi h th us onludes the pro of of Prop osition 3.6 . (Notie that Z 0 is indeed losed sine otherwise w e ould extend α − 1 to the ompletion of Z × Z ′ .) Lemma 3.7. F or al l Z 1 , Z 2 , Z 3 ∈ Z ′ , we have p Z 1 ◦ p Z 3 ◦ p Z 2 | Z 1 = Id Z 1 . Pr o of of L emma 3.7 . Let ϑ : Z 1 → Z 1 b e the isometry dened b y p Z 1 ◦ p Z 3 ◦ p Z 2 | Z 1 and let f b e its displaemen t funtion. Then f : Z 1 → R is a non-negativ e on v ex funtion whi h is b ounded ab o v e b y d ( Z 1 , Z 2 ) + d ( Z 2 , Z 3 ) + d ( Z 3 , Z 1 ) . In partiular, the restrition of f to an y geo desi ra y in Z 1 is non-inreasing. Therefore, a sublev el set of f is a losed on v ex subset Z of Z 1 with full b oundary , namely ∂ Z = ∂ Z 1 . By denition, the subspae Z 1 is minimal with resp et to the prop ert y that ∂ Z 1 = ∂ Y and hene w e dedue Z = Z 1 . It follo ws that the on v ex funtion f is onstan t. In other w ords, the isometry ϑ is a Cliord translation. If it is not trivial, then Z 1 w ould on tain a ϑ -stable geo desi line on whi h ϑ ats b y translation. But b y [ BH99 , Lemma I I.2.15℄, the restrition of ϑ to an y geo desi line is the iden tit y . Therefore ϑ is trivial, as desired. Let Γ b e a group ating on X b y isometries. F ollo wing [ Mon06 , Denition 5℄, w e sa y that the Γ -ation is redued if there is no un b ounded losed on v ex subset Y ( X su h that g .Y is at nite Hausdor distane from Y for all g ∈ Γ . Corollary 3.8. L et X b e a pr op er irr e duible CA T(0) sp a e with nite-dimensional Tits b oundary, and Γ < Is( X ) b e a sub gr oup ating minimal ly without xe d p oint at innity. Then the Γ -ation is r e du e d. Pr o of. Supp ose for a on tradition that the Γ -ation on X is not redued. Then there exists an un b ounded losed on v ex subset Y ( X su h that g .Y is at nite Hausdor distane from Y for all g ∈ Γ . In partiular ∂ Y is Γ -in v arian t. By Prop osition 3.1 , it m ust ha v e in trinsi irumradius > π / 2 . Prop osition 3.6 therefore yields a anonial losed on v ex subset Z 0 = Z × Z ′ with ∂ ( Z × { z ′ } ) = ∂ Y for all z ′ ∈ Z ′ ; learly Z 0 is Γ -in v arian t and hene w e ha v e Z 0 = X b y minimalit y . Sine X is irreduible b y assumption, w e dedue X = Z and hene X = Y , as desired. ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 13 3.C. Minimal ations and b oundary-minimal spaes. Boundary-minimalit y and min- imalit y , as dened in the In tro dution, are t w o p ossible w a ys for a CA T(0) spae to b e non-degenerate, as illustrated b y the follo wing. Lemma 3.9. L et X b e a omplete CA T(0) sp a e. (i) A gr oup G < Is( X ) ats minimal ly if and only if any ontinuous onvex G -invariant funtion on X is onstant. (ii) If X is b oundary-minimal then any b ounde d onvex funtion on X is onstant. Pr o of. Neessit y in the rst assertion follo ws immediately b y onsidering sub-lev el sets (see [ Mon06 , Lemma 37℄). Suieny is due to the fat that the distane to a losed on v ex set is a on v ex on tin uous funtion [ BH99 , I I.2.5℄. The seond assertion w as established in the pro of of Lemma 3.7 . Prop osition 3.6 has the follo wing imp ortan t onsequene: Corollary 3.10. L et X b e a pr op er CA T(0) sp a e. If ∂ X has ir umr adius > π / 2 , then X p ossesses a anoni al lose d onvex subsp a e Y ⊆ X suh that Y is b oundary-minimal and ∂ Y = ∂ X . Pr o of. Let Z 0 = Z × Z ′ b e the pro dut deomp osition pro vided b y Prop osition 3.6 . The group Is( X ) p erm utes the elemen ts of Z ′ and hene ats b y isometries on Z ′ . Under the presen t h yp otheses, the spae Z ′ is b ounded sine ∂ Z = ∂ X . Therefore it has a irumen tre z ′ , and the bre Y = Z × { z ′ } is th us Is( X ) -in v arian t. Prop osition 3.11. L et X b e a pr op er CA T(0) sp a e whih is minimal. Assume that ∂ X has nite dimension. Then ∂ X has ir umr adius > π / 2 (unless X is r e du e d to a p oint). In p artiular, X is b oundary-minimal. The pro of of Prop osition 3.11 requires some preliminaries. Giv en a p oin t at innit y ξ , onsider the Busemann funtion B ξ ; the o yle prop ert y (realled in Setion 2 ) implies in partiular that for an y isometry g ∈ Is( X ) xing ξ and an y x ∈ X the real n um b er B ξ ,x ( g .x ) is indep enden t on the hoie of x and yields a anonial homomorphism β ξ : Is( X ) ξ − → R : g 7− → B ξ ,x ( g .x ) alled the Busemann harater en tred at ξ . Giv en an isometry g , it follo ws b y the CA T(0) prop ert y that inf n ≥ 0 d ( g n x, x ) /n oinides with the translation length of g indep enden tly of x . W e all an isometry ballisti when this n um b er is p ositiv e. An imp ortan t fat ab out a ballisti isometry g of an y omplete CA T(0) spae X is that for an y x ∈ X the sequene { g n .x } n ≥ 0 on v erges to a p oin t η g ∈ ∂ X indep enden t of x ; η g is alled the (anonial) attrating xed p oin t of g in ∂ X . Moreo v er, this on v ergene holds also in angle, whi h means that lim ∠ x ( g n x, r ( t )) v anishes as n, t → ∞ when r : R + → X is an y ra y p oin ting to η g . This is a (v ery) sp eial ase of the results in [ KM99 ℄. Lemma 3.12. L et ξ ∈ X and g ∈ Is( X ) ξ b e an isometry whih is not annihilate d by the Busemann har ater entr e d at ξ . Then g is b al listi. F urthermor e, if β ξ ( g ) > 0 then ∠ T ( ξ , η g ) > π / 2 . Pr o of. W e ha v e β ξ ( g ) = B ξ ,x ( g .x ) ≤ d ( x, g .x ) for all x ∈ X . Th us g is ballisti as so on as β ξ ( g ) is non-zero. Assume β ξ ( g ) > 0 and supp ose for a on tradition that ∠ T ( ξ , η g ) ≤ π / 2 . Cho ose x ∈ X and let , σ b e the ra ys issuing from x and p oin ting to w ards ξ and η g resp etiv ely . Reall 14 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD from [ BH99 , I I.9.8℄ that ∠ T ( ξ , η g ) = lim t,s →∞ ∠ x ( ( t ) , σ ( s )) . The on v ergene in diretion of g n x implies that this angle is also giv en b y lim t,n →∞ ∠ x ( ( t ) , g n x ) . Sine β ξ ( g ) > 0 w e an x n large enough to ha v e cos lim in f t →∞ ∠ x ( ( t ) , g n x ) > − β ξ ( g ) d ( g x, x ) . W e no w apply the asymptoti angle form ula from Setion 2 with y = g n x and dedue that the left hand side is − β ξ ( g n x ) /d ( g n x, x ) . Sine β ξ ( g n x ) = nβ ξ ( g x ) and d ( g n x, x ) ≤ nd ( g x, x ) , w e ha v e a on tradition. Pr o of of Pr op osition 3.11 . W e an assume that ∂ X is non-empt y sine otherwise X is a p oin t b y minimalit y . Supp ose for a on tradition that its irumradius is ≤ π / 2 . Then Is( X ) p ossesses a global xed p oin t ξ ∈ ∂ X and ξ is a irumen tre of ∂ X , see Prop osition 3.1 . Lemma 3.12 implies that Is( X ) = I s ( X ) ξ is annihilated b y the Busemann harater en tred at ξ . Th us Is( X ) stabilises ev ery horoball, on traditing minimalit y . W e shall use rep eatedly the follo wing elemen tary fat. Lemma 3.13. L et G b e a gr oup with an isometri ation on a pr op er ge o desi al ly omplete CA T(0) sp a e X . If G ats o omp atly or mor e gener al ly has ful l limit set, then the ation is minimal. (This holds mor e gener al ly when ∆ G = ∂ X in the sense of Se tion 4.B b elow.) Pr o of. Let Y ⊆ X b e a a non-empt y losed on v ex in v arian t subset, ho ose y ∈ Y and supp ose for a on tradition that there is x / ∈ Y . Let r : R + → X b e a geo desi ra y starting at y and going through x . By on v exit y [ BH99 , I I.2.5(1)℄, the funtion d ( r ( t ) , Y ) tends to innit y and th us r ( ∞ ) / ∈ ∂ Y . This is absurd sine ∆ G ⊆ ∂ Y . Pr o of of Pr op osition 1.5 . (i) See Prop osition 3.11 . (ii) Sine Is( X ) has full limit set, an y Is( X ) -in v arian t subspae has full b oundary . Minimalit y follo ws, sine b oundary-minimalit y ensures that X p ossesses no prop er subspae with full b oundary . (iii) X is minimal b y Lemma 3.13 , hene b oundary-minimal b y (i), sine an y o ompat spae has nite-dimensional b oundary b y [ Kle99 , Theorem C℄. 4. Minimal inv ariant subsp a es f or subgr oups 4.A. Existene of a minimal in v arian t subspae. F or the reord, w e reall the follo wing elemen tary di hotom y; a renemen t will b e giv en in Theorem 4.3 b elo w: Prop osition 4.1. L et G b e a gr oup ating by isometries on a pr op er CA T(0) . Then either G has a glob al xe d p oint at innity, or any ltering family of non-empty lose d onvex G -invariant subsets has non-empty interse tion. (Reall that a family of sets is ltering if it is direted b y on tainmen t ⊇ .) Pr o of. (Remark 36 in [ Mon06 ℄.) Supp ose Y is su h a family , ho ose x ∈ X and let x Y b e its pro jetion on ea h Y ∈ Y . If the net { x Y } Y ∈ Y is b ounded, then T Y ∈ Y Y is non- empt y . Otherwise it go es to innit y and an y aum ulation p oin t in ∂ X is G -xed in view of d ( g x Y , x Y ) ≤ d ( g x, x ) . ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 15 4.B. Di hotom y. Let G b e a group ating b y isometries on a omplete CA T(0) spae X . Lemma 4.2. Given any two x, y ∈ X , the onvex losur es of the r esp e tive G -orbits of x and y in X have the same b oundary in ∂ X . Pr o of. Let Y b e the on v ex losure of the G -orbit of x . In partiular Y is the minimal losed on v ex G -in v arian t subset on taining x . Giv en an y losed on v ex G -in v arian t subset Z , let r = d ( x, Z ) . Reall that the tubular losed neigh b ourho o d N r ( Z ) is on v ex [ BH99 , I I.2.5(1)℄. Sine it is also G -in v arian t and on tains x , the minimalit y of Y implies Y ⊆ N r ( Z ) . This yields a anonial losed on v ex G -in v arian t subset of the b oundary ∂ X , whi h w e denote b y ∆ G . It on tains the limit set Λ G but is sometimes larger. Com bining what w e established th us far with the splitting argumen ts from [ Mon06 ℄, w e obtain a di hotom y: Theorem 4.3. L et G b e a gr oup ating by isometries on a omplete CA T(0) sp a e X and H < G any sub gr oup. If H admits no minimal non-empty lose d onvex invariant subset and X is pr op er, then: (A.i) ∆ H is a non-empty lose d onvex subset of ∂ X of intrinsi ir umr adius at most π / 2 . (A.ii) If ∂ X is nite-dimensional, then the normaliser N G ( H ) of H in G has a glob al xe d p oint in ∂ X . If H admits a minimal non-empty lose d onvex invariant subset Y ⊆ X , then: (B.i) The union Z of al l suh subsets is a lose d onvex N G ( H ) -invariant subset. (B.ii) Z splits H -e quivariantly and isometri al ly as a pr o dut Z ≃ Y × C , wher e C is a omplete CA T(0) sp a e whih admits a anoni al N G ( H ) /H -ation by isometries. (B.iii) If the H -ation on X is non-evanes ent, then C is b ounde d and ther e is a anoni al minimal non-empty lose d onvex H -invariant subset whih is N G ( H ) -stable. (When X is prop er, the non-evanes en e ondition of (iii) simply means that H has no xed p oin t in ∂ X ; see [ Mon06 ℄.) Pr o of. In view of Lemma 4.2 , the set ∆ H is on tained in the b oundary of an y non-empt y losed on v ex H -in v arian t set and is N G ( H ) -in v arian t. Th us the assertions (A.i) and (A.ii) follo w from Prop osition 3.2 , notiing that in a prop er spae ∆ H is non-empt y unless H has b ounded orbits, in whi h ase it xes a p oin t, pro viding a minimal subspae. F or (B.i), (B.ii) and (B.iii), see Remarks 39 in [ Mon06 ℄. 4.C. Normal subgroups. Pr o of of The or em 1.10 . W e adopt the notation and assumptions of the theorem. By (A.ii), N admits a minimal non-empt y losed on v ex in v arian t subset Y ⊆ X . This set is un- b ounded, sine otherwise N xes a p oin t and th us b y G -minimalit y X N = X , hene N = 1 . Sine X is irreduible, p oin ts (B.i) and (B.ii) sho w Y = X and th us N ats indeed minimally . Sine the displaemen t funtion of an y g ∈ Z G ( N ) is a on v ex N -in v arian t funtion, it is onstan t b y minimalit y . Hene g is a Cliord translation and m ust b e trivial sine otherwise X splits o a Eulidean fator, see [ BH99 , I I.6.15℄. The deriv ed subgroup N ′ = [ N , N ] is also normal in G and therefore ats minimally b y the previous disussion, notiing that N ′ is non-trivial sine otherwise N ⊆ Z G ( N ) . If N xed a p oin t at innit y , N ′ w ould preserv e all orresp onding horoballs, on traditing minimalit y . 16 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Ha ving established that N ats minimally and without xed p oin t at innit y , w e an apply the splitting theorem (Corollary 10 in [ Mon06 ℄) and dedue from the irreduibilit y of X that N do es not split. Finally , let R ✁ N b e the amenable radial and observ e that it is normal in G . The theorem of A damsBallmann [ AB98a ℄ states that R either (i) xes a p oin t at innit y or (ii) preserv es a Eulidean at in X . (Although their result is stated for amenable groups without men tioning an y top ology , the pro of applies indeed to ev ery top ologial group that preserv es a probabilit y measure whenev er it ats on tin uously on a ompat metrisable spae.) If R is non-trivial, w e kno w already from the ab o v e disussion that (i) is imp ossible and that R ats minimally; it follo ws that X is a at. By irreduibilit y and sine X 6 = R , this fores X to b e a p oin t, on traditing R 6 = 1 . Corollary 1.11 will b e pro v ed in Setion 5.B . F or Corollary 1.12 , it sues to observ e that the en traliser of an y elemen t of a disrete normal subgroup is op en. Next, w e reall the follo wing denition. A subgroup N of a group G is asending if there is a family of subgroups N α < G indexed b y the ordinals and su h that N 0 = N , N α ✁ N α +1 , N α = S β <α N β if α is a limit ordinal and N α = G for α large enough. The smallest su h ordinal is the order . Prop osition 4.4. Consider a gr oup ating minimal ly by isometries on a pr op er CA T(0) sp a e. Then any as ending sub gr oup without glob al xe d p oint at innity stil l ats minimal ly. Pr o of. W e argue b y transnite indution on the order ϑ of asending subgroups N < G , the ase ϑ = 0 b eing trivial. Let X b e a spae as in the statemen t. By Prop osition 4.1 , ea h N α has a minimal set. If ϑ = ϑ ′ + 1 , it follo ws from (B.iii) that N ϑ ′ ats minimally and w e are done b y indution h yp othesis. Assume no w that ϑ is a limit ordinal. F or all α , w e denote as in (B.i) b y Z α ⊆ X the union of all N α -minimal sets. The indution h yp othesis implies that for all α ≤ β < ϑ , an y N β -minimal set is N α -minimal. Th us, if Z 0 = Y 0 × C 0 is a splitting as in (B.ii) with a N -minimal set Y 0 , w e ha v e a nested family of deomp ositions Z α = Y 0 × C α for a nested family of losed on v ex subspaes C α of the ompat CA T(0) spae C 0 , indexed b y α < ϑ . Th us, for an y c ∈ T α<ϑ C α , the spae Y 0 × { c } is G -in v arian t and hene Y 0 = X indeed. Remark 4.5. Prop osition 4.4 holds more generally for omplete CA T(0) spaes if N is non-ev anesen t. Indeed Prop osition 4.1 hold in that generalit y (Remark 36 in [ Mon06 ℄) and C remains ompat in a w eak er top ology (Theorem 14 in [ Mon06 ℄). Pr o of of The or em 1.13 . In view of Theorem 1.10 , it sues to pro v e that an y non-trivial asending subgroup N < G as in that statemen t still ats minimally and without global xed p oin t at innit y . W e argue b y indution on the order ϑ and w e an assume that ϑ is a limit ordinal b y Theorem 1.10 . Then T α<ϑ ( ∂ X ) N α is empt y and th us b y ompatness there is some α < ϑ su h that ( ∂ X ) N α is empt y . No w N α ats minimally on X b y Prop osition 4.4 and th us w e onlude using the indution h yp othesis. 5. Algebrai and geometri pr odut deompositions 5.A. Preliminary deomp osition of the spae. W e shall prepare our spaes b y means of a geometri deomp osition. F or an y geo desi metri spae with nite ane r ank , F o erts h Lyt hak [ FL06 ℄ established a anonial deomp osition generalising the lassial theorem of de Rham [ dR52 ℄. Ho w ev er, su h a statemen t fails to b e true for CA T(0) spaes that ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 17 are merely prop er, due notably to ompat fators that an b e innite pro duts. Nev er- theless, using asymptoti CA T(0) geometry and Setion 3.A , w e an adapt the argumen ts from [ FL06 ℄ and obtain: Theorem 5.1. L et X b e a pr op er CA T(0) sp a e with ∂ X nite-dimensional and of ir- umr adius > π / 2 . Then ther e is a anoni al lose d onvex subset Z ⊆ X with ∂ Z = ∂ X , invariant under al l isometries, and admitting a anoni al maximal isometri splitting (5.i) Z ∼ = R n × Z 1 × · · · × Z m ( n, m ≥ 0) with e ah Z i irr e duible and 6 = R 0 , R 1 . Every isometry of Z pr eserves this de omp osition up on p ermuting p ossibly isometri fators Z i . Remark 5.2. It is w ell kno wn that in the ab o v e situation the splitting ( 5.i ) indues a deomp osition Is( Z ) = Is( R n ) × Is( Z 1 ) × · · · × Is( Z m ) ⋊ F , where F is the p erm utation group of { 1 , . . . , d } p erm uting p ossible isometri fators amongst the Y j . Indeed, this follo ws from the statemen t that isometries preserv e the splitting up on p erm utation of fators, see e.g. Prop osition I.5.3(4) in [ BH99 ℄. Of ourse, this do es not a priori mean that w e ha v e a unique, nor ev en anonial, splitting in the ate gory of gr oups ; this shall ho w ev er b e established for Theorem 1.6 . The h yp otheses of Theorem 5.1 are satised in some naturally o urring situations: Corollary 5.3. L et X b e a pr op er CA T(0) sp a e with nite-dimensional b oundary. (i) If Is( X ) has no xe d p oint at innity, then X p ossesses a subsp a e Z satisfying al l the onlusions of The or em 5.1 . (ii) If Is( X ) ats minimal ly, then X admits a anoni al splitting as in 5.i . Pr o of of Cor ol lary 5.3 . By Prop osition 3.1 , if Is( X ) has no xed p oin t at innit y , then ∂ X has irumradius > π / 2 . By Prop osition 3.11 , the same onlusion holds is Is( X ) ats minimally . Pr o of of The or ems 1.9 and 5.1 . F or Theorem 5.1 , w e let Z ⊆ X b e the anonial b oundary- minimal subset with ∂ Z = ∂ X pro vided b y Corollary 3.10 ; w e shall not use the irumradius assumption an y more. F or Theorem 1.9 , w e let Z = X . The remainder of the argumen t is ommon for b oth statemen ts. Realling that in omplete generalit y all isometries preserving the Eulidean fator de- omp osition [ BH99 , I I.6.15℄, w e an assume that Z has no Eulidean fator and shall obtain the deomp osition ( 5.i ) with n = 0 . Sine Z is minimal amongst losed on v ex subsets with ∂ Z = ∂ X , it has no non-trivial ompat fator. On the other hand, an y prop er geo desi metri spae admits some maximal pro dut deomp osition in to non-ompat fators. In onlusion, Z admits some maximal splitting Z = Z 1 × · · · × Z m with ea h Z i irreduible and 6 = R 0 , R 1 . (This an fail in presene of ompat fators). It remains to pro v e that an y other su h deomp osition Z = Z ′ 1 × · · · × Z ′ m ′ oinides with the rst one after p ossibly p erm uting the fators (in partiular, m ′ = m ). W e no w b orro w from the argumen tation in [ FL06 ℄, indiating the steps and the neessary hanges. It is assumed that the reader has a op y of [ FL06 ℄ at hand but k eeps in mind that our spaes migh t la k the nite ane rank ondition assumed in that pap er. W e shall replae the notion of ane subspaes with a large-sale partiular ase: a ne shall b e an y subspae 18 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD isometri to a losed on v ex ne in some Eulidean spae. This inludes the partiular ases of a p oin t, a ra y or a full Eulidean spae. Whenev er a spae Y has some pro dut deomp osition and Y ′ is a fator, write Y ′ y ⊆ Y for the orresp onding bre Y ′ y ∼ = Y ′ through y ∈ Y . The follo wing is an analogue of Corollary 1.2 in [ FL06 ℄. Lemma 5.4. L et Y b e a pr op er CA T(0) sp a e with nite-dimensional b oundary and without omp at fators. Supp ose given two de omp ositions Y = Y 1 × Y 2 = S 1 × S 2 with al l four ( Y i ) y ∩ ( S j ) y r e du e d to { y } for some y ∈ Y . Then Y is a Eulide an sp a e. Pr o of of L emma 5.4 . An y y ∈ Y is on tained in a maximal ne based at y sine ∂ Y has nite dimension; b y abuse of language w e all su h nes maximal. The argumen ts of Setions 3 and 4 in [ FL06 ℄ sho w that an y maximal ne is r e tangular , whi h means that it inherit a pro dut struture from an y pro dut deomp osition of the am bien t CA T(0) spae. Sp eially , it sues to observ e that the pro dut of t w o nes is a ne and that the pro jetion of a ne along a pro dut deomp osition of CA T(0) spaes remains a ne. (In fat, the equalit y of slop es of Setion 4.2 in [ FL06 ℄, namely the fat that parallel geo desi segmen ts in a CA T(0) spae ha v e iden tial slop es in pro dut deomp ositions, is a general fat for CA T(0) spaes. It follo ws from the on v exit y of the metri, see for instane [ Mon06 , Prop osition 49℄ for a more general statemen t.) The dedution of the statemen t of Lemma 5.4 from the retangularit y of maximal nes follo wing [ FL06 ℄ is partiularly short sine all prop er CA T(0) Bana h spaes are Eulidean. Lemma 5.5. F or a given z ∈ Z and any pr o dut de omp osition Z = S × S ′ , the interse tion ( Z i ) z ∩ S z is either { z } or ( Z i ) z . Pr o of of L emma 5.5 . W rite P S : Z → S and P Z i : Z → Z i for the pro jetions and set F z = S z ∩ ( Z i ) z . F ollo wing [ FL06 ℄, dene T ⊆ Z b y T = P S ( F z ) × S ′ . W e on tend that P Z i ( T ) has full b oundary in Z i . Indeed, giv en an y p oin t in ∂ ( Z i ) z , w e represen t is b y a ra y r originating from z . W e an ho ose a maximal ne in Z based at z and on taining r . W e kno w already that this ne is retangular, and therefore the pro of of Lemma 5.2 in [ FL06 ℄ sho ws that P Z i ( r ) lies in P Z i ( T ) , justifying our on ten tion. W e observ e that Z i inherits from Z the prop ert y that it has no losed on v ex prop er subset of full b oundary . In onlusion, sine P Z i ( T ) is a on v ex set, it is dense in Z . Ho w ev er, aording to Lemma 5.1 in [ FL06 ℄, it splits as P Z i ( T ) = P Z i ( F z ) × P Z i ( S ′ ) . Up on p ossibly replaing P Z i ( S ′ ) b y its ompletion (whilst P Z i ( F z ) is already losed in Z i sine P Z i is isometri on ( Z i ) z ), w e obtain a splitting of the losure of P Z i ( T ) , and hene of Z i . This ompletes the pro of of the lemma sine Z i is irreduible. No w the main argumen t runs b y indution o v er m ≥ 2 . Lemma 5.5 iden ties b y indution Z i with some Z ′ j . Indeed, Lemma 5.4 exludes that all pairwise in tersetions redue to a p oin t sine Z has no Eulidean fator. 5.B. Pro of of Theorem 1.6 and A ddendum 1.8 . The follo wing onsequene of the solution to Hilb ert's fth problem b elongs to the mathematial lore. Theorem 5.6. L et G b e a lo al ly omp at gr oup with trivial amenable r adi al. Then G p ossesses a anoni al nite index op en normal sub gr oup G † suh that G † = L × D , wher e L is a onne te d semi-simple Lie gr oup with trivial entr e and no omp at fators, and D is total ly dis onne te d. ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 19 Pr o of. This follo ws from the GleasonMon tgomeryZippin solution to Hilb ert's fth prob- lem and the fat that onneted semi-simple Lie groups ha v e nite outer automorphism groups. More details ma y b e found for example in [ Mon01 , 11.3℄. Com bining Theorem 5.6 with Theorem 1.10 , w e nd the statemen t giv en as Corollary 1.11 in the In tro dution. Theorem 5.7. L et X 6 = R b e an irr e duible pr op er CA T(0) sp a e with nite-dimensional Tits b oundary and G < Is( X ) any lose d sub gr oup whose ation is minimal and do es not have a glob al xe d p oint in ∂ X . Then G is either total ly dis onne te d or an almost onne te d simple Lie gr oup with trivial entr e. Pr o of. By Theorem 1.10 , G has trivial amenable radial. Let G † b e as in Theorem 5.6 . Applying Theorem 1.10 to this normal subgroup of G , dedue that w e ha v e either G † = L with L simple or G † = D . W e an no w omplete the pro of of Theorem 1.6 and A ddendum 1.8 and w e adopt their notation. Sine Is( X ) has no global xed p oin t at innit y , there is a anonial minimal non-empt y losed on v ex Is( X ) -in v arian t subset X ′ ⊆ X (Remarks 39 in [ Mon06 ℄). W e apply Corollary 5.3 to Z = X ′ and Remark 5.2 to G = Is( Z ) , setting G ∗ = Is( R n ) × Is( Z 1 ) × · · · × Is( Z m ) . All the laimed prop erties of the resulting fator groups are established in Theorem 1.10 , Theorem 1.13 and Theorem 1.1 in [ CM08b ℄ (the pro of of whi h is ompletely indep enden t from the presen t onsiderations). Finally , the laim that an y pro dut deomp osition of G ∗ is a regrouping of the fators in ( 1.i ) is established as follo ws. Notie that the G ∗ -ation on Z is still minimal and without xed p oin t at innit y (this is almost b y denition but alternativ ely also follo ws from Theorem 1.10 ). Therefore, giv en an y pro dut deomp osition of G ∗ , w e an apply the splitting theorem (Corollary 10 in [ Mon06 ℄) and obtain a orresp onding splitting of Z . No w the uniqueness of the deomp osition of the spae Z (a w a y from the Eulidean fator) implies that the giv en deomp osition of G ∗ is a regrouping of the fators o urring in Remark 5.2 . 5.C. CA T(0) spaes without Eulidean fator. F or the sak e of future referenes, w e reord the follo wing onsequene of the results obtained th us far: Corollary 5.8. L et X b e a pr op er CA T(0) sp a e with nite-dimensional b oundary and no Eulide an fator, suh that G = Is ( X ) ats minimal ly without xe d p oint at innity. Then G has trivial amenable r adi al and any sub gr oup of G ating minimal ly on X has triv- ial entr aliser. F urthermor e, given a non-trivial normal sub gr oup N ✁ G , any N -minimal N -invariant lose d subsp a e of X is a r e gr ouping of fators in the de omp osition of A d- dendum 1.8 . In p artiular, if e ah irr e duible fator of G is non-disr ete, then G has no non-trivial nitely gener ate d lose d normal sub gr oup. Pr o of. The trivialit y of the amenable radial omes from the orresp onding statemen t in irreduible fators of X , see Theorem 1.10 . By the seond paragraph of the pro of of Theo- rem 1.10 , an y subgroup of G ating minimally has trivial en traliser. The fat that minimal in v arian t subspaes for normal subgroups are bres in the pro dut deomp osition ( 1.ii ) fol- lo ws sine an y pro dut deomp osition of X is a regrouping of fators in ( 1.ii ) and sine an y normal subgroup of G yields su h a pro dut deomp osition b y Theorem 4.3 (B.i) and (B.ii). Assume nally that ea h irreduible fator in ( 1.i ) is non-disrete and let N < G b e a nitely 20 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD generated losed normal subgroup. Then N is disrete b y Baire's ategory theorem, and N ats minimally on a bre, sa y Y , of the spae deomp osition ( 1.ii ). Therefore, the pro jetion of N to Is( Y ) has trivial en traliser, unless N is trivial. Sine N is disrete, normal and nitely generated, its en traliser is op en. Sine Is( Y ) is non-disrete b y assumption, w e dedue that N is trivial, as desired. 6. Tot all y disonneted gr oup a tions 6.A. Smo othness. When onsidering ations of totally disonneted groups, a desirable prop ert y is smo othness , namely that p oin ts ha v e op en stabilisers. This ondition is imp or- tan t in represen tation theory , but also in our geometri on text, see p oin t ( ii ) of Corollary 6.3 b elo w and [ Cap07 ℄. In general, this ondition do es not hold, ev en for ations that are o ompat, minimal and without xed p oin t at innit y . An example will b e onstruted in Setion 6.C in [ CM08b ℄. Ho w ev er, w e establish it under a rather ommon additional h yp othesis. Reall that a metri spae X is alled geo desially omplete (or said to ha v e extensible geo desis ) if ev ery geo desi segmen t of p ositiv e length ma y b e extended to a lo ally isometri em b edding of the whole real line. The follo wing on tains Theorem 1.2 from the In tro dution. Theorem 6.1. L et G b e a total ly dis onne te d lo al ly omp at gr oup with a minimal, on- tinuous and pr op er ation by isometries on a pr op er CA T(0) sp a e X . If X is ge o desi al ly omplete, then the ation is smo oth. In fat, the p ointwise stabiliser of every b ounde d set is op en. Remark 6.2. In partiular, the stabiliser of a p oin t ats as a nite group of isometries on an y giv en ball around this p oin t in the setting of Theorem 6.1 . Corollary 6.3. L et X b e a pr op er CA T(0) sp a e and G b e a total ly dis onne te d lo al ly omp at gr oup ating ontinuously pr op erly on X by isometries. Then: (i) If the G -ation is o omp at, then every element of zer o tr anslation length is el lipti. (ii) If the G -ation is o omp at and every p oint x ∈ X has an op en stabiliser, then the G -ation is semi-simple. (iii) If the G -ation is o omp at and X is ge o desi al ly omplete, then the G -ation is semi-simple. Pr o of of Cor ol lary 6.3 . P oin ts ( i ) and ( ii ) follo w readily from Theorem 6.1 , see [ Cap07 , Corollary 3.3℄. ( iii ) In view of Lemma 3.13 , this follo ws from Theorem 6.1 and ( ii ) . The follo wing is a k ey fat for Theorem 6.1 : Lemma 6.4. L et X b e a ge o desi al ly omplete pr op er CA T(0) sp a e. L et ( C n ) n ≥ 0 b e an inr e asing se quen e of lose d onvex subsets whose union C = S n C n is dense in X . Then every b ounde d subset of X is ontaine d in some C n ; in p artiular, C = X . Pr o of. Supp ose for a on tradition that for some r > 0 and x ∈ X the r -ball around x on- tains an elemen t x n not in C n for ea h n . W e shall onstrut indutiv ely a sequene { c k } k ≥ 1 of pairwise r -disjoin t elemen ts in C with d ( x, c k ) ≤ 2 r + 2 , on traditing the prop erness of X . If c 1 , . . . , c k − 1 ha v e b een onstruted, ho ose n large enough to that C n on tains them all and d ( x, C n ) ≤ 1 . Consider the (non-trivial) geo desi segmen t from x n to its nearest p oin t pro jetion x n on C n ; b y geo desi ompleteness, it is on tained in a geo desi line and ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 21 w e ho ose y at distane r + 1 from C n on this line. Notie that x n ∈ [ x n , y ] and hene d ( y , x ) ≤ 2 r + 1 . Moreo v er, d ( y , c i ) ≥ r + 1 for all i < k . Sine C is dense, w e an ho ose c k lose enough to y to ensure d ( c k , x ) ≤ 2 r + 2 and d ( c k , c i ) ≥ r for all i < k , ompleting the indution step. End of pr o of of The or em 6.1 . The subset C ⊆ X onsisting of those p oin ts x ∈ X su h that the stabiliser G x is op en is learly on v ex and G -stable. By [ Bou71 , I I I 4 No 6℄, the group G on tains a ompat op en subgroup and hene C is non-empt y . Th us C is dense b y minimalit y of the ation. Sine Is( X ) is seond oun table, w e an ho ose a desending hain Q n < G of ompat op en subgroups whose in tersetion ats trivially on X . Therefore, C ma y b e written as the union of an asending family of losed on v ex subsets C n ⊆ X , where C n is the xed p oin t set of Q n . No w the statemen t of the theorem follo ws from Lemma 6.4 . 6.B. Lo ally nite equiv arian t partitions and ellular deomp ositions. Let X b e a lo ally nite ell omplex and G b e its group of ellular automorphisms, endo w ed with the top ology of p oin t wise on v ergene on b ounded subsets. Then G is a totally disonneted lo ally ompat group and ev ery b ounded subset of X has an op en p oin t wise stabiliser in G . One of the in terest of Theorem 6.1 is that it allo ws for a partial on v erse to the latter statemen t: Prop osition 6.5. L et X b e a pr op er CA T(0) sp a e and G b e a total ly dis onne te d lo al ly omp at gr oup ating ontinuously pr op erly on X by isometries. Assume that the p ointwise stabiliser of every b ounde d subset of X is op en in G . Then we have the fol lowing: (i) X admits a anoni al lo al ly nite G -e quivariant p artition. (ii) Denoting by σ ( x ) the pie e supp orting the p oint x ∈ X in that p artition, we have Stab G ( σ ( x )) = N G ( G x ) and N G ( G x ) /G x ats fr e ely on σ ( x ) . (iii) If G \ X is omp at, then so is Stab G ( σ ( x )) \ σ ( x ) for al l x ∈ X . Pr o of. Consider the equiv alene relation on X dened b y x ∼ y ⇔ G x = G y . This yields a anonial G -in v arian t partition of X . W e need to sho w that it is lo ally nite. Assume for a on tradition that there exists a on v erging sequene { x n } n ≥ 0 su h that the subgroups G x n are pairwise distint. Let x = lim n x n . W e laim that G x n < G x for all suien tly large n . Indeed, up on extrating there w ould otherwise exist a sequene g n ∈ G x n su h that g n .x 6 = x for all n . Up on a further extration, w e ma y assume that g n on v erges to some g ∈ G . By onstrution g xes x . Sine G x is op en b y h yp othesis, this implies that g n xes x for suien tly large n , a on tradition. This pro v es the laim. By h yp othesis the p oin t wise stabiliser of an y ball en tred at x is op en. Th us G x p ossesses a ompat op en subgroup U whi h xes ev ery x n . This implies that w e ha v e the inlusion U < G x n < G x for all n . Sine the index of U in G x is nite, there are only nitely man y subgroups of G x on taining U . This nal on tradition nishes the pro of of (i). (ii) Straigh tforw ard in view of the denitions. (iii) Supp ose for a on tradition that H \ σ ( x ) is not ompat, where H = Stab G ( σ ( x )) . Let then y n ∈ σ ( x ) b e a sequene su h that d ( y n , H .x ) > n . Let no w g n ∈ G b e su h that { g n .y n } is b ounded, sa y of diameter C . By (i), the set { g n G y n g − 1 n } is th us nite. Up on extrating, w e shall assume that it is onstan t. No w, for all n < k , the elemen t g − 1 n g k 22 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD normalises G y k = G x and maps y k to a p oin t at distane ≤ C from y n . In view of (ii), this is absurd. Remark 6.6. The partition of X onstruted ab o v e is non-trivial whenev er G do es not at freely . This is for example the ase whenev er G is non-disrete and ats faithfully . The piees in the ab o v e partition are generally neither b ounded (ev en if G \ X is ompat), nor on v ex, nor ev en onneted. Ho w ev er, if one assumes that the spae admits a suien tly large amoun t of symmetry , then one obtains a partition whi h deserv es to b e view ed as an equiv arian t ellular deomp osition. Corollary 6.7. L et X b e a pr op er CA T(0) sp a e and G b e a total ly dis onne te d lo al ly omp at gr oup ating ontinuously pr op erly on X by isometries. Assume that the p ointwise stabiliser of every b ounde d subset of X is op en in G , and that no op en sub gr oup of G xes a p oint at innity. Then X admits admits a anoni al lo al ly nite G -e quivariant de omp osition into omp at onvex pie es. Pr o of. F or ea h x ∈ X , let τ ( x ) b e the xed-p oin t-set of G x . Then τ ( x ) is learly on v ex; it is ompat b y h yp othesis. F urthermore the map x 7→ τ ( x ) is G -equiv arian t. The fat that the olletion { τ ( x ) | x ∈ X } is lo ally nite follo ws from Prop osition 6.5 . 6.C. Alexandro v angle rigidit y. A further onsequene of Theorem 6.1 is a phenomenon of angle rigidit y . Giv en an ellipti isometry g of omplete a CA T(0) spae X and a p oin t x ∈ X , w e denote b y c g ,x the pro jetion of x on the losed on v ex set of g -xed p oin ts. Prop osition 6.8. L et G b e a total ly dis onne te d lo al ly omp at gr oup with a ontinuous and pr op er o omp at ation by isometries on a ge o desi al ly omplete pr op er CA T(0) sp a e X . Then ther e is ε > 0 suh that for any el lipti g ∈ G and any x ∈ X with g x 6 = x we have ∠ c g,x ( g x, x ) ≥ ε . (W e will later also pro v e an angle rigidit y for the Tits angle, see Prop osition 7.15 .) Pr o of. First w e observ e that this b ound on the Alexandro v angle is really a lo al prop ert y at c g ,x of the germ of the geo desi [ c g ,x , x ] sine for an y y ∈ [ c g ,x , x ] w e ha v e c g ,y = c g ,x . Next, w e laim that for an y n ∈ N , an y isometry of order ≤ n of an y omplete CA T(0) spae B satises ∠ c g,x ( g x, x ) ≥ 1 /n for all x ∈ B that are not g -xed. Indeed, it follo ws from the denition of Alexandro v angles (see [ BH99 , I I.3.1℄) that for an y y ∈ [ c g ,x , x ] w e ha v e d ( g y, y ) ≤ d ( c g ,x , y ) ∠ c g,x ( g x, x ) . Therefore, if ∠ c g,x ( g x, x ) < 1 /n , the en tire g -orbit of y w ould b e on tained in a ball around y not on taining c g ,x = c g ,y . This is absurd sine the irumen tre of this orbit is a g -xed p oin t. In order to pro v e the prop osition, w e no w supp ose for a on tradition that there are sequenes { g n } of ellipti elemen ts in G and { x n } in X with g n x n 6 = x n and ∠ c n ( g n x n , x n ) → 0 , where c n = c g n ,x n . Sine the G -ation is o ompat, there is (up on extrating) a sequene { h n } in G su h that h n c n on v erges to some c ∈ X . Up on onjugating g n b y h n , replaing x n b y h n x n and c n b y h n c n , w e an assume c n → c without lo osing an y of the onditions on g x , x n and c n , inluding the relation c n = c g n ,x n . Sine d ( g n c, c ) ≤ 2 d ( c n , c ) , w e an further extrat and assume that { g n } on v erges to some limit g ∈ G ; notie also that g xes c . By Lemma 3.13 , the ation is minimal and hene Theorem 6.1 applies. Therefore, w e an assume that all g n oinide with g on some ball B around c and in partiular preserv e B . Using Remark 6.2 , this pro vides a on tradition. ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 23 A rst onsequene is an analogue of a result that E. Sw enson pro v ed for disrete groups (Theorem 11 in [ Sw e99 ℄). Corollary 6.9. L et G b e a total ly dis onne te d lo al ly omp at gr oup with a ontinuous and pr op er o omp at ation by isometries on a ge o desi al ly omplete pr op er CA T(0) sp a e X not r e du e d to a p oint. Then G ontains hyp erb oli elements (thus in p artiular elements of innite or der). Bey ond the totally disonneted ase, w e an app eal to Theorem 1.6 and A ddendum 1.8 and state the follo wing. Corollary 6.10. L et G b e any lo al ly omp at gr oup with a ontinuous and pr op er o omp at ation by isometries on a ge o desi al ly omplete pr op er CA T(0) sp a e X not r e du e d to a p oint. Then G ontains elements of innite or der; if mor e over ( ∂ X ) G = ∅ , then G ontains hyp erb oli elements. Pr o of of Cor ol lary 6.9 . Prop osition 6.8 allo ws us use the argumen t form [ Sw e99 ℄: W e an ho ose a geo desi ra y r : R + → X , an inreasing sequene { t i } going to innit y in R + and { g i } in G su h that the funtion t 7→ g i r ( t + t i ) on v erges uniformly on b ounded in terv als (to a geo desi line). F or i < j large enough, the angle ∠ h ( r ( t i )) ( r ( t i ) , h 2 ( r ( t i ))) dened with h = g − 1 i g j is arbitrarily lose to π . In order to pro v e that h is h yp erb oli, it sues to sho w that this angle will ev en tually equal π . Supp ose this do es not happ en; b y Corollary 6.3 ( iii ), w e an assume that h is ellipti. W e set x = r ( t i ) and c = c h,x . Considering the ongruen t triangles ( c, x, hx ) and ( c, hx, h 2 x ) , w e nd that ∠ c ( x, hx ) is arbitrarily small. This is in on tradition with Prop osition 6.8 . Pr o of of Cor ol lary 6.10 . If the onneted omp onen t G ◦ is non-trivial, then it on tains ele- men ts of innite order; if it is trivial, w e an apply Corollary 6.9 . Assume no w ( ∂ X ) G = ∅ . Then Theorem 1.6 and A ddendum 1.8 apply . Therefore, w e obtain h yp erb oli elemen ts either from Corollary 6.9 or from the fat that an y non-ompat semi-simple group on tains elemen ts that are algebraially h yp erb oli, om bined with the fat that the latter at as h yp erb oli isometries. That fat is established in Theorem 7.4 ( i ) b elo w, the pro of of whi h is indep enden t of Corollary 6.10 . 6.D. Algebrai struture. Giv en a top ologial group G , w e dene its so le so c( G ) as the subgroup generated b y all minimal non-trivial losed normal subgroups of G . Notie that G migh t ha v e no minimal non-trivial losed normal subgroup, in whi h ase its so le is trivial. W e also reall that the quasi-en tre of a lo ally ompat group G is the subset Q Z ( G ) onsisting of all those elemen ts p ossessing an op en en traliser. Clearly Q Z ( G ) is a (top o- logially) harateristi subgroup of G . Sine an y elemen t with a disrete onjugay lass p ossesses an op en en traliser, it follo ws that the quasi-en tre on tains all disrete normal subgroups of G . Prop osition 6.11. L et X b e a pr op er CA T(0) sp a e without Eulide an fator and G < Is( X ) b e a lose d sub gr oup ating minimal ly o omp atly without xe d p oint at innity. If G has trivial quasi- entr e, then so c( G ∗ ) is dir e t pr o dut of r non-trivial har ateristi al ly simple gr oups, wher e r is the numb er of irr e duible fators of X and G ∗ is the anoni al nite index op en normal sub gr oup ating trivial ly on the set of fators of X . The pro of will use the follo wing general fat inspired b y a statemen t for tree automor- phisms, Lemma 1.4.1 in [ BM00 ℄. 24 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Prop osition 6.12. L et G b e a omp atly gener ate d total ly dis onne te d lo al ly omp at gr oup without non-trivial omp at normal sub gr oups. Then any ltering family of non- disr ete lose d normal sub gr oups has non-trivial (thus non- omp at) interse tion. A v arian t of this prop osition is pro v ed in [ CM08a ℄; sine the pro of is short, w e giv e it for the sak e of ompleteness. Pr o of of Pr op osition 6.12 . Let g b e a S hreier graph for G . W e reall that it onsists in ho osing an y op en ompat subgroup U < G (whi h exists b y [ Bou71 , I I I 4 No 6℄), dening the v ertex set of g as G/U and dra wing edges aording to a ompat generating set whi h is a union of double osets mo dulo U ; see [ Mon01 , 11.3℄. Sine G has no non-trivial ompat normal subgroup, the on tin uous G -ation on g is faithful. Let v 0 b e a v ertex of g and denote b y v ⊥ 0 the set of neigh b ouring v erties. Sine G is v ertex-transitiv e on g , it follo ws that for an y normal subgroup N ✁ G , the N v 0 -ation on v ⊥ 0 denes a nite p erm utation group F N < Sym( v ⊥ 0 ) whi h, as an abstrat p erm utation group, is indep enden t of the hoie of v 0 . Therefore, if N is non-disrete, this p erm utation group F N has to b e non-trivial sine U is op en and g onneted. No w a ltering family F of non-disrete normal subgroups yields a ltering family of non-trivial nite subgroups of Sym( v ⊥ 0 ) . Th us the in tersetion of these nite groups is non-trivial. Let g b e a non-trivial elemen t in this in tersetion. F or an y N ∈ F , let N g b e the in v erse image of { g } in N v 0 . Th us N g is a non-empt y ompat subset of N for ea h N ∈ F . Sine the family F is ltering, so are { N v 0 | N ∈ F } and { N g | N ∈ F } . The result follo ws, sine a ltering family of non-empt y losed subsets of the ompat set G v 0 has a non-empt y in tersetion. Eviden tly op en normal subgroups form a ltering family; w e an th us dedue: Corollary 6.13. L et G b e a omp atly gener ate d lo al ly omp at gr oup without any non- trivial omp at normal sub gr oup. If G is r esidual ly disr ete, then it is disr ete. Pr o of of Pr op osition 6.11 . W e rst observ e that G ∗ has no non-trivial disrete normal sub- group. Indeed, su h a subgroup has nitely man y G -onjugates, whi h implies that ea h of its elemen ts has disrete G -onjugay lass and hene b elongs to Q Z ( G ) , whi h w as assumed trivial. Let no w { N i } b e a hain of non-trivial losed normal subgroups of G ∗ . If N i is totally disonneted for some i , then the in tersetion T i N i is non-trivial b y Prop osition 6.12 . Otherwise N ◦ i is non-trivial and normal in ( G ∗ ) ◦ for ea h i , and the in tersetion T i N i is non-trivial b y Theorem 1.6 (sine the latter desrib es in partiular the p ossible normal onneted subgroups of G ∗ ). In all ases, Zorn's lemma implies that the ordered set of non-trivial losed normal subgroups of G ∗ p ossesses minimal elemen ts. Giv en t w o minimal losed normal subgroups M , M ′ , the in tersetion M ∩ M ′ is th us trivial and, hene, so is [ M , M ′ ] . Th us minimal losed normal subgroups of G ∗ en tralise one another. W e dedue from Corollary 5.8 that the n um b er of minimal losed normal subgroups is at most r . Consider no w an irreduible totally disonneted fator H of G ∗ . W e laim that the olletion of non-trivial losed normal subgroups of H forms a ltering family . Indeed, giv en t w o su h normal subgroup N 1 , N 2 , then N 1 ∩ N 2 is again a losed normal subgroup of H . It is is trivial, then the omm utator [ N 1 , N 2 ] is trivial and, hene, the en traliser of N 1 in H is non-trivial, on traditing Theorem 1.10 . This onrms the laim. Th us the in tersetion of all non-trivial losed normal subgroups of H is non-trivial b y Prop osition 6.12 . Clearly this in tersetion is the so le of H ; it is lear w e ha v e just established that it is on tained in ev ery non-trivial losed normal subgroup of H . In partiular so c( H ) is harateristially ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 25 simple. The desired result follo ws, sine so c( H ) is learly a minimal losed normal subgroup of G ∗ . Theorem 6.14. L et X b e a pr op er irr e duible ge o desi al ly omplete CA T(0) sp a e. L et G < Is( X ) b e a lose d total ly dis onne te d sub gr oup ating o omp atly, in suh a way that no op en sub gr oup xes a p oint at innity. Then we have the fol lowing: (i) Every omp at sub gr oup of G is ontaine d in a maximal one; the maximal omp at sub gr oups fal l into nitely many onjugay lasses. (ii) Q Z ( G ) = 1 . (iii) so c( G ) is a non-disr ete har ateristi al ly simple gr oup. Pr o of. (i) By Lemma 3.13 , the ation is minimal and hene Theorem 6.1 applies. In par- tiular, w e an apply Corollary 6.7 and onsider the resulting equiv arian t deomp osition. Let Q < G b e a ompat subgroup and x b e a Q -xed p oin t. If G x is not on tained in a maximal ompat subgroup of G , then there is an innite sequene ( x n ) n ≥ 0 su h that x 0 = x and G x n ⊆ G x n +1 . By Corollary 6.7 , the sequene x n lea v es ev ery b ounded subset. Sine the xed p oin ts X G x n form a nested sequene, it follo ws that X G x is un b ounded. In partiular its visual b oundary ∂ ( X G x ) is non-empt y and the op en subgroup G x has a xed p oin t at innit y . This on tradits the h yp otheses, and the laim is pro v ed. Notie that a similar argumen t sho ws that for ea h x ∈ X , there are nitely man y maximal ompat subgroups Q i < G on taining G x . The fat that G p ossesses nitely man y onjugay lasses of maximal ompat subgroups no w follo ws from the ompatness of G \ X . (ii) W e laim that Q Z ( G ) is top ologially lo ally nite , whi h means that ev ery nite subset of it is on tained in a ompat subgroup. The desired result follo ws sine it is then amenable but G has trivial amenable radial b y Theorem 1.10 . Let S ⊆ Q Z ( G ) b e a nite subset. Then G p ossesses a ompat op en subgroup U en tralising S . By h yp othesis the xed p oin t set of U is ompat. Sine h S i stabilises X U , it follo ws that h S i is ompat, whene the laim. (iii) F ollo ws from (ii) and Prop osition 6.11 . 7. Coomp a t CA T(0) sp a es 7.A. Fixed p oin ts at innit y. W e b egin with a simple observ ation. W e reall that t w o p oin ts at innit y are opp osite if they are the t w o endp oin ts of a geo desi line. W e denote b y ξ op the set of p oin ts opp osite to ξ . Reall from [ Bal95 , Theorem 4.11(i)℄ that, if X is prop er, then t w o p oin ts ξ , η ∈ ∂ X at Tits distane > π are neessarily opp osite. (Reall that Tits distane is b y denition the length metri asso iated to the Tits angle.) Ho w ev er, it is not true in general that t w o p oin ts at Tits distane π are opp osite. Prop osition 7.1. L et X b e a pr op er CA T(0) sp a e and H < Is( X ) a lose d sub gr oup ating o omp atly. If H xes a p oint ξ at innity, then ξ op 6 = ∅ and H ats tr ansitively on ξ op . Pr o of. First w e laim that there is a geo desi line σ : R → X with σ ( ∞ ) = ξ . Indeed, let r : R + → X ′ b e a ra y p oin ting to ξ and { g n } a sequene in H su h that g n r ( n ) remains b ounded. The ArzelàAsoli theorem implies that g n r ( R + ) sub on v erges to a geo desi line in X . Sine ξ is xed b y all g n , this line has an endp oin t at ξ . Let no w σ ′ : R → X b e an y other geo desi with σ ′ ( ∞ ) = ξ and ho ose a sequene { h n } n ∈ N in H su h that d ( h n σ ( − n ) , σ ′ ( − n )) remains b ounded. By on v exit y and sine all h n x ξ , d ( h n σ ( t ) , σ ′ ( t )) is b ounded for all t and th us sub on v erges (uniformly for t in 26 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD b ounded in terv als). On the one hand, it implies that { h n } has an aum ulation p oin t h . On the other hand, it follo ws that hσ ( −∞ ) = σ ′ ( −∞ ) . Reall that an y omplete CA T(0) spae X admits a anonial splitting X = X ′ × V preserv ed b y all isometries, where V is a (maximal) Hilb ert spae alled the Eulidean fator of X , see [ BH99 , I I.6.15(6)℄. F urthermore, there is a anonial em b edding X ′ ⊆ X ′′ × V ′ , where V ′ is a Hilb ert spae generated b y all diretions in X ′ p oin ting to at p oin ts at innit y , namely p oin ts for whi h the Busemann funtions are ane on X ′ ; moreo v er, ev ery isometry of X ′ extends uniquely to an isometry of X ′′ × V ′ whi h preserv es that splitting. This is a result of A damsBallmann [ AB98a , Theorem 1.6℄, who all V ′ the pseudo-Eulidean fator (one ould also prop ose Eulidean pseudo-fator). Corollary 7.2. L et X b e a pr op er CA T(0) sp a e with a o omp at gr oup of isometries. Then the pseudo-Eulide an fator of X is trivial. Pr o of. In view of the ab o v e disussion, X ′ is also a prop er CA T(0) spae with a o ompat group of isometries. The set of at p oin ts in ∂ X ′ admits a anonial (in trinsi) irumen tre ξ b y Lemma 1.7 in [ AB98a ℄. In partiular, ξ is xed b y all isometries and therefore, b y Prop osition 7.1 , it has an opp osite p oin t, whi h is imp ossible for a at p oin t unless it lies already in the Eulidean fator (see [ AB98a ℄). Prop osition 7.3. L et G b e a gr oup ating o omp atly by isometries on a pr op er CA T(0) sp a e X without Eulide an fator and assume that the stabiliser of every p oint at innity ats minimal ly on X . Then G has no xe d p oint at innity. Pr o of. If G has a global xed p oin t ξ , then the stabiliser G η of an opp osite p oin t η ∈ ξ op (whi h exists b y Prop osition 7.1 ) preserv es the union Y ⊆ X of all geo desi lines onneting ξ to η . By [ BH99 , I I.2.14℄, this spae is on v ex and splits as Y = R × Y 0 . Sine G η ats minimally , w e dedue Y = X whi h pro vides a Eulidean fator. 7.B. A tions of simple algebrai groups. Let X b e a CA T(0) spae and G b e an alge- brai group dened o v er the eld k . An isometri ation of G ( k ) on X is alled algebrai if ev ery (algebraially) semi-simple elemen t g ∈ G ( k ) ats as a semi-simple isometry . When G is semi-simple, w e denote b y X model the Riemannian symmetri spae or Bruhat Tits building asso iated with G ( k ) . Theorem 7.4. L et k b e a lo al eld and G b e an absolutely almost simple simply onne te d k -gr oup. L et X b e a non- omp at pr op er CA T(0) sp a e on whih G = G ( k ) ats ontinuously by isometries. Assume either: (a) the ation is o omp at; or: (b) it has ful l limit set, is minimal and ∂ X is nite-dimensional. Then: (i) The G -ation is algebr ai. (ii) Ther e is a G -e quivariant bije tion ∂ X ∼ = ∂ X model whih is an isometry with r e- sp e t to Tits' metri and a home omorphism with r esp e t to the ne top olo gy. This bije tion extends to a G -e quivariant r ough isometry β : X model → X . (iii) If X is ge o desi al ly omplete, then X is isometri to X model . (iv) F or any semi-simple k -sub gr oup L < G , ther e non-empty lose d onvex subsp a e Y ⊆ X minimal for L = L ( k ) ; mor e over, ther e is no L -xe d p oint in ∂ Y . In the ab o v e p oin t ( ii ), a rough isometry refers to a map β : X model → X su h that there is a onstan t C with d X mod el ( x, y ) − C ≤ d X ( β ( x ) , β ( y )) ≤ d X mod el ( x, y ) + C ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 27 for all x, y ∈ X model and su h that β ( X model ) has nite o diameter in X . Su h a map is also alled a (1 , C ) -quasi-isometry . Remarks 7.5. (i) Notie that there is no assumption on the k -rank of G in this result. (ii) W e reall for (b) that minimalit y follo ws from full limit set in the geo desially omplete ase (Lemma 3.13 ). (iii) In the on text of ( ii ), w e reall that in general t w o CA T(0) spaes with the same o ompat isometry group need not ha v e homeomorphi b oundaries [ CK00 ℄. (iv) A p osteriori , p oin t ( ii ) sho ws in partiular that the ation is also o ompat under the assumption (b). Before pro eeding to the pro of, w e giv e t w o examples sho wing that the assumptions made in Theorem 7.4 are neessary . Example 7.6 . Without the assumption of geo desi ompleteness, it is not true in general that, in the setting of the theorem, the spae X on tains a losed on v ex G -in v arian t subspae whi h is isometri to X model . A simple example of this situation ma y obtained as follo ws. Consider the ase where k is non-Ar himedean and G has k -rank one. Let 0 < r < 1 / 2 and let X b e the spae obtained b y replaing the r -ball en tred at ea h v ertex in the tree X model b y an isometri op y of a giv en Eulidean n -simplex, where n + 1 is v alene of the v ertex. In this w a y , one obtains a CA T(0) spae whi h is still endo w ed with an isometri G -ation that is o ompat and minimal, but learly X is not isometri to X model . W e do not kno w whether su h a onstrution ma y also b e p erformed in the Ar himedean ase (see Problem 7.2 in [ CM08b ℄). Example 7.7 . Under the assumptions (b), minimalit y is needed. Indeed, w e laim that for an y CA T(0) spae X 0 there is a anonial CA T( − 1 ) spae X (in partiular X is a CA T(0) spae) together with a anonial map i : Is( X 0 ) ֒ → Is( X ) with the follo wing prop erties: The b oundary ∂ X is redued to a single p oin t; X non-ompat; X is prop er if and only if X 0 is so; the map i is an isomorphism of top ologial groups on to its image. This laim justies that minimalit y is needed sine w e an apply it to the ase where X 0 is the symmetri spae or BruhatTits building asso iated to G ( k ) . (In that ase the ation has indeed full limit set, a heap feat as the isometry group is non-ompat and the b oundary rather inapaious.) T o pro v e the laim, onsider the parab oli ne Y asso iated to X 0 . This is the metri spae with underlying set X 0 × R ∗ + where the distane is dened as follo ws: giv en t w o p oin ts ( x, t ) and ( x ′ , t ′ ) of Y , iden tify the in terv al [ x, x ′ ] ⊆ X 0 with an in terv al of orresp onding length in R and measure the length from the resulting p oin ts ( x, t ) and ( x ′ , t ′ ) in the upp er half-plane mo del for the h yp erb oli plane. This is a partiular ase of the syn theti v ersion ([ Che99 ℄, [ AB98b ℄) of the BishopO'Neill w arp ed pro duts [ BO69 ℄ and its prop erties are desrib ed in [ BGP92 ℄, [ AB04 , 1.2(2A)℄ and [ HLS00 , 2℄. In partiular, Y is CA T( − 1 ). W e no w let ξ ∈ ∂ Y b e the p oin t at innit y orresp onding to t → ∞ and dene X ⊆ Y to b e an asso iated horoball; for deniteness, set X = X 0 × [1 , ∞ ) . W e no w ha v e ∂ X = { ξ } b y the CA T( − 1 ) prop ert y or alternativ ely b y the expliit desription of geo desi ra ys ( e.g. 2(iv) in [ HLS00 ℄). The remaining prop erties follo w readily . Pr o of of The or em 7.4 . W e start with a few preliminary observ ations. Finite-dimensionalit y of the b oundary alw a ys holds sine it is automati in the o ompat ase. Sine X is non- ompat, the ation is non-trivial, b eause it has full limit set. It is w ell kno wn that ev ery non-trivial on tin uous homomorphism of G to a lo ally ompat seond oun table group is prop er [ BM96 , Lemma 5.3℄. Th us the G -ation on X is prop er. 28 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD W e laim that the stabiliser of an y p oin t ξ ∈ ∂ X on tains the unip oten t radial of some prop er parab oli subgroup of G . Indeed, x a p olar deomp osition G = K T K . Let x 0 ∈ X b e a K -xed p oin t. Cho ose a sequene { g n } n ≥ 0 of elemen ts of G su h that g n .x 0 on v erges to ξ . W rite g n = k n .a n .k ′ n with k n , k ′ n ∈ K and a n ∈ T . W e ma y furthermore assume, up on replaing { g n } b y a subsequene, that { k n } on v erges to some k ∈ K , that { a n .x 0 } on v erges in X ∪ ∂ X and that { a n .p } on v erges in X model ∪ ∂ X model , where p ∈ X model is some base p oin t. Let η = lim n →∞ a n .x 0 and observ e η = k − 1 ξ . F urthermore, the stabiliser of η on tains the group U = { g ∈ G | li m n →∞ a − 1 n g a n = 1 } . The on v ergene in diretion of { a n } in T implies that U on tains the unip oten t radial U Q of the parab oli subgroup Q < G orresp onding to lim n →∞ a n .p ∈ ∂ X model . (In fat, the argumen ts for Lemma 2.4 in [ Pra77 ℄ probably sho w U = U Q ; this follo ws a p osteriori from (ii) b elo w.) Therefore, the stabiliser of ξ = k .η in G on tains the unip oten t radial of k Qk − 1 , pro ving the laim. Notie that w e ha v e seen in passing that an y p oin t at innit y lies in the limit set of some torus; in the ab o v e notation, ξ is in the limit set of k T k − 1 . ( i ) Ev ery elemen t of G whi h is algebraially ellipti ats with a xed p oin t in X , sine it generates a relativ ely ompat subgroup. W e need to sho w that ev ery non-trivial elemen t of a maximal split torus T < G ats as a semi-simple isometry . Assume for a on tradition that some elemen t t ∈ T ats as a parab oli isometry . Sine X has nite-dimensional b oundary and w e an apply Corollary 3.3 ( ii ) . It follo ws that the Ab elian group T has a anonial xed p oin t at innit y ξ xed b y the normaliser N G ( T ) . By the preeding paragraph, w e kno w furthermore that the stabiliser of ξ in G also on tains the unip oten t radial of some parab oli subgroup of G . Reall that G is generated b y N G ( T ) together with an y su h unip oten t radial: this follo ws from the fat that N G ( T ) has no xed p oin t at innit y in X model and that G is generated b y the unip oten t radials of an y t w o distint parab oli subgroups. Therefore ξ is xed b y the en tire group G . Sine G has trivial Ab elianisation, its image under the Busemann harater en tred at ξ v anishes, thereb y sho wing that G m ust stabilise ev ery horoball en tred at ξ . This is absurd b oth in the minimal and the o ompat ase. ( ii ) Let T < G b e a maximal split torus. Let F model ⊆ X model b e the (maximal) at stabilised b y T . In view of ( i ) and the prop erness of the T -ation, w e kno w that T also stabilises a at F ⊆ X with dim F = dim T , see [ BH99 , I I.7.1℄. Cho ose a base p oin t p 0 ∈ F model in su h a w a y that its stabiliser K := G p 0 is a maximal ompat subgroup of G . The union of all T -in v arian t ats whi h are parallel to F is N G ( T ) -in v arian t. Therefore, up on replaing F b y a parallel at, w e ma y and shall assume that F on tains a p oin t x 0 whi h is stabilised b y N K := N G ( T ) ∩ K . Note that, sine N G ( T ) = h N K ∪ T i , the at F is N G ( T ) -in v arian t. Therefore, there is a w ell dened N G ( T ) -equiv arian t map α of the N G ( T ) -orbit of p 0 to F , dened b y α ( g .p 0 ) = g .x 0 for all g ∈ N G ( T ) . W e laim that, up to a saling fator, the map α is isometri and indues an N G ( T ) - equiv arian t isometry α : F model → F . In order to establish this, remark that the W eyl group W := N G ( T ) / Z G ( T ) ats on F , sine W = N K /T K , where T K := Z G ( T ) ∩ K ats trivially on F . The group N K normalises the oro ot lattie Λ < T . F urthermore N K . Λ ats on F model as an ane W eyl group sine N K . Λ /T K ∼ = W ⋉ Λ . Moreo v er, sine an y reetion in W en tralises an Ab elian subgroup of orank 1 in Λ , it follo ws that N K . Λ ats on F as a disrete reetion group. But a giv en ane W eyl group has a unique (up to saling fator) disrete o ompat ation as a reetion group on Eulidean spaes, as follo ws from [ Bou68 , ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 29 Ch. VI, 2, Prop osition 8℄. Therefore the restrition of α to Λ .x 0 is a homothet y . Sine Λ is a uniform lattie in T , the laim follo ws. A t this p oin t, it follo ws that α indues an N G ( T ) -equiv arian t map ∂ α : ∂ F model → ∂ F , whi h is isometri with resp et to Tits' distane. W e reall that N G ( T ) is the stabiliser of ∂ F model in G . Moro v er, for an y η ∈ ∂ F model , the stabilisers in G of η is on tained in that of α ( η ) b eause of the geometri desription of parab oli subgroups alluded to in the preliminary observ ation: see the argumen t for Lemma 2.4 in [ Pra77 ℄. Therefore, ∂ α extends to a w ell dened G -equiv arian t map ∂ X model → ∂ X , whi h w e denote again b y ∂ α . Sine an y t w o p oin ts of ∂ X model are on tained in a ommon maximal sphere (i.e. an apartmen t), and sine G ats transitiv ely on these spheres, the map ∂ α is isometri, b eause so is its restrition to the sphere ∂ F model . Note that ∂ α is surjetiv e: indeed, this follo ws from the last preliminary observ ation, whi h, om bined with ( i ), sho ws in partiular that ∂ X = K.∂ F . W e no w sho w that ∂ α is a homeomorphism with resp et to the ne top ology . Sine ∂ X model is ompat, it is enough to sho w that ∂ α is on tin uous. No w an y on v ergen t sequene in ∂ X model ma y b e written as { k n .ξ n } n ≥ 0 , where { k n } n ≥ 0 (resp. { ξ n } n ≥ 0 ) is a on v ergen t sequene of elemen ts of K (resp. ∂ F model ). On the sphere ∂ F model , the ne top ology oinides with the one indued b y Tits' metri. Therefore, the equiv ariane of the Tits' isometry ∂ α sho ws that { ∂ α ( k n .ξ n ) } n ≥ 0 is a on v ergen t sequene in ∂ X , as w as to b e pro v ed. W e next laim that that G -ation on X is o ompat ev en under the assumption (b). T o w ards a on tradition, assume otherwise. Cho ose a sequene { y n } in X with y 0 a K - xed p oin t and su h that d ( y n , g .y 0 ) ≥ n for all g ∈ G . Up on replaing y n ( n ≥ 1 ) b y an appropriate G -translate, w e an and shall assume that moreo v er (7.i) d ( y n , y 0 ) ≤ d ( y n , g .y 0 ) + c ∀ g ∈ G, n ≥ 1 , where c is some onstan t. Up on extrating a subsequene, the sequene { y n } on v erges to some p oin t η ∈ ∂ X . It w as established ab o v e that ∂ X = K.∂ F ; in partiular there exists k ∈ K su h that k .η ∈ ∂ F . No w, up on replaing y n b y k .y n , w e obtain a sequene { y n } whi h still satises all ab o v e onditions but whi h on v erges to a b oundary p oin t η ′ of the at F . Let r : R + → F b e a geo desi ra y p oin ting to w ards η ′ . Sine N G ( T ) ats o ompatly on the at F , it follo ws from ( 7.i ) that for some onstan t c ′ , w e ha v e (7.ii) d ( y n , y 0 ) ≤ d ( y n , r ( t )) + c ′ ∀ t ≥ 0 , n ≥ 1 . Fix no w s > d ( y 0 , r (0)) + c ′ . F or n suien tly large, let z n b e the p oin t on [ r (0) , y n ] at distane s of r (0) . W e ha v e d ( y n , r ( s )) ≤ d ( y n , z n ) + d ( z n , r ( s )) = d ( y n , r (0)) − s + d ( z n , r ( s )) < d ( y n , r (0)) − d ( y 0 , r (0)) − c ′ + d ( z n , r ( s )) ≤ d ( y n , y 0 ) − c ′ + d ( z n , r ( s )) . As n go es to innit y , this pro vides a on tradition to ( 7.ii ) sine z n on v erges to r ( s ) ; th us o ompatness is established. It remains for ( ii ) to pro v e that ∂ α extends to a G -equiv arian t rough isometry β : X model → X . The orbital map g 7→ g .y 0 asso iated to y 0 yields a map β : G/K → X ; when k is Ar himedean, X model = G/K whereas w e extend β linearly to ea h ham b er of the building X model in the non-Ar himedean ase. It is a w ell-kno wn onsequene of o om- patness that the G -equiv arian t map β : X model → X is a quasi-isometry (see e.g. the pro of 30 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD of the Milnorv ar lemma giv en in [ BH99 , I.8.19℄). F or our stronger statemen t, it sues, in view of the K T K deomp osition and of equiv ariane, to pro v e that there is a onstan t C ′ su h that d X mod el ( a.p 0 , p 0 ) − C ′ ≤ d X ( a.y 0 , y 0 ) ≤ d X mod el ( a.p 0 , p 0 ) + C ′ for all a ∈ T . This follo ws from the fat that β and α are at b ounded distane from ea h other on F model (indeed, at distane d ( y 0 , x 0 ) ) and that β is isometri on F model . ( iii ) In the higher rank ase, assertion ( iii ) follo ws from ( ii ) and the main result of [ Lee00 ℄. Ho w ev er, the full strength of lo . it. is really not needed here, sine the main diult y there is preisely the absene of an y group ation, whi h is part of the h yp otheses in our setting. F or example, when the ground eld k is the eld of real n um b ers, the argumen ts ma y b e dramatially shortened as follo ws; they are v alid without an y rank assumption. Giv en an y ξ ∈ ∂ X , the unip oten t radial of the parab oli subgroup G ξ ats sharply transitiv ely on the b oundary p oin ts opp osite to ξ . In view of this and of the prop erness of the G -ation, the argumen ts of [ Lee00 , Prop osition 4.27℄ sho w that geo desi lines in X do not bran h; in other w ords X has uniquely extensible geo desis. F rom this, it follo ws that the group N K = N G ( T ) ∩ K onsidered in the pro of of ( ii ) has a unique xed p oin t in X , sine otherwise it w ould x p oin t wise a geo desi line, and hene, b y ( ii ), opp osite p oin ts in ∂ X model . The fat that this is imp ossible is purely a statemen t on the lassial symmetri spae X model ; w e giv e a pro of for the reader's on v eniene: Let F model b e the at orresp onding to T and p 0 ∈ F model b e the K -xed p oin t. If N K xed a p oin t ξ ∈ ∂ X model , then the ra y [ p 0 , ξ ) w ould b e p oin t wise xed and, hene, the group N K w ould x a non-zero v etor in the tangen t spae of X model at p 0 . A Cartan deomp osition g = k ⊕ p of the Lie algebra g of G yields an isomorphism b et w een the isotrop y represen tation of N K on T p 0 X model and the represen tation of the W eyl group W on p . An easy expliit omputation sho ws that the latter represen tation has no non-zero xed v etor. Sine N K has a unique xed p oin t, the latter is stabilised b y the en tire group K . Hene K xes a p oin t lying on a at F stabilised b y T . F rom the K T K -deomp osition, it follo ws that the G -orbit of this xed p oin t is on v ex. Sine the G -ation on X is minimal b y geo desi ompleteness (Lemma 3.13 ), w e dedue that G is transitiv e on X . In partiular X is o v ered b y ats whi h are G -onjugate to F , and the existene of a G -equiv arian t homothet y X model → X follo ws from the existene of a N G ( T ) -equiv arian t homothet y F model → F , whi h has b een established ab o v e. It remains only to ho ose the righ t sale on X model to mak e it an isometry . In the non-Ar himedean ase, w e onsider only the rank one ase, referring to [ Lee00 ℄ for higher rank. Let K b e a maximal ompat subgroup of G and x 0 ∈ X b e a K -xed p oin t. By ( ii ), the group K ats transitiv ely on ∂ X . Sine X is geo desially omplete, it follo ws that the K -translates of an y ra y emanating from x 0 o v er X en tirely . On the other hand ev ery p oin t in X has an op en stabiliser b y Theorem 6.1 , an y p oin t in X has a nite K -orbit. This implies that the spae of diretions at ea h p oin t p ∈ X is nite. In other w ords X is 1 -dimensional. Sine X is CA T(0) and lo ally ompat, it follo ws that X is a lo ally nite metri tree. As w e ha v e just seen, the group K ats transitiv ely on the geo desi segmen ts of a giv en length emanating from x 0 . One dedues that G is transitiv e on the edges of X . In partiular all edges of X ha v e the same length, whi h w e an assume to b e as in X model , G has at most t w o orbits of v erties, and X is either regular or bi-regular. The v alene of an y v ertex p equals min g 6∈ N G ( G p ) G p : G p ∩ g G p g − 1 , ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 31 and oinides therefore with the v alene of X model . It nally follo ws that X and X model are isometri, as w as to b e pro v ed. ( iv ) Let P b e a k -parab oli subgroup of G that is minimal amongst those on taining L . W e ma y assume P 6 = G sine otherwise L has no xed p oin t in ∂ X and the onlusion holds in view of Prop osition 4.1 . It follo ws that L en tralises a k -split torus T of p ositiv e dimension d . It follo ws from ( i ) that T = T ( k ) ats b y h yp erb oli isometries, and th us there is a T -in v arian t losed on v ex subset Z ⊆ X of the form Z = Z 1 × R d su h that the T -ation is trivial on the Z 1 fator; this follo ws from Theorem I I.6.8 in [ BH99 ℄ and the prop erness of the ation. Moreo v er, L preserv es Z and its deomp osition Z = Z 1 × R d , ating b y translations on the R d fator ( lo . it. ). Sine L is semi-simple, this translation ation is trivial and th us L preserv es an y Z 1 bre, sa y for instane Z 0 := Z 1 × { 0 } ⊆ Z . F or b oth the existene of a minimal set Y and the ondition ( ∂ Y ) L = ∅ , it sues to sho w that L has no xed p oin t in ∂ Z 0 (Prop osition 4.1 ). W e laim that ∂ Z 0 is Tits-isometri to the spherial building of the Lévi subgroup Z G ( T ) . Indeed, w e kno w from ( ii ) that ∂ X is equiv arian tly isometri to ∂ X model and the building of Z G ( T ) is haraterised as the p oin ts at distane π / 2 from the b oundary of the T -in v arian t at in ∂ X model . On the other hand, L has maximal semi-simple rank in Z G ( T ) b y the hoie of P and therefore annot b e on tained in a prop er parab oli subgroup of Z G ( T ) . This sho ws that L has no xed p oin t in ∂ Z 0 and ompletes the pro of. 7.C. No bran hing geo desis. Reall that in a geo desi metri spae X , the spae of diretions Σ x at a p oin t x is the ompletion of the spae e Σ x of geo desi germs equipp ed with the Alexandro v angle metri at x . If X has uniquely extensible geo desis, then e Σ x = Σ x . The follo wing is a result of V. Beresto vskii [ Ber02 ℄ (w e read it in [ Ber , 3℄; it also follo ws from A. Lyt hak's argumen ts in [ Lyt05 , 4℄). Theorem 7.8. L et X b e a pr op er CA T(0) sp a e with uniquely extensible ge o desis and x ∈ X . Then e Σ x = Σ x is isometri to a Eulide an spher e. W e use this result to establish the follo wing. Prop osition 7.9. L et X b e a pr op er CA T(0) sp a e with uniquely extensible ge o desis. Then any total ly dis onne te d lose d sub gr oup D < Is( X ) is disr ete. Pr o of. There is some ompat op en subgroup Q < D , see [ Bou71 , I I I 4 No 6℄. Let x b e a Q -xed p oin t. The isometry group of Σ x is a ompat Lie group b y Theorem 7.8 and th us the image of the pronite group Q in it is nite. Let th us K < Q b e the k ernel of this represen tation, whi h is op en. Denote b y S ( x, r ) the r -sphere around x . The Q - equiv arian t visual map S ( x, r ) → Σ x is a bijetion b y unique extensibilit y . It follo ws that K is trivial. W e are no w ready for: End of pr o of of The or em 1.1 . Sine the ation is o ompat, it is minimal b y Lemma 3.13 . The fat that extensibilit y of geo desis is inherited b y diret fators of the spae follo ws from the haraterisation of geo desis in pro duts, see [ BH99 , I.5.3(3)℄. Ea h fator X i is th us a symmetri spae in view of Theorem 7.4 ( iii ). By virtue of Corollary 6.3 ( iii ), the totally disonneted fators D j at b y semi-simple isometries. 32 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Assume no w that X has uniquely extensible geo desis. F or the same reason as b efore, this prop ert y is inherited b y ea h diret fator of the spae. Th us ea h D j is disrete b y Prop osition 7.9 . Theorem 7.10. L et X b e a pr op er irr e duible CA T(0) sp a e with uniquely extensible ge o desis. If X admits a non-disr ete gr oup of isometries with ful l limit set but no glob al xe d p oint at innity, then X is a symmetri sp a e. The ondition on xed p oin ts at innit y is neessary in view of E. Hein tze's examples [ Hei74 ℄ of negativ ely urv ed homogeneous manifolds whi h are not symmetri spaes. In fat these spaes onsist of ertain simply onneted soluble Lie groups endo w ed with a left-in v arian t negativ ely urv ed Riemannian metri. Prop osition 7.11. L et X b e a pr op er CA T(0) sp a e with uniquely extensible ge o desis. Then ∂ X has nite dimension. Pr o of. Let x ∈ X and reall that b y Beresto vskii's result quoted in Theorem 7.8 ab o v e, Σ x is isometri to a Eulidean sphere. By denition of the Tits angle, the visual map ∂ X → Σ x asso iating to a geo desi ra y its germ at x is Tits-on tin uous (in fat, 1 -Lips hitz). It is furthermore injetiv e (atually , bijetiv e) b y unique extensibilit y . Therefore, the top ologial dimension of an y omp at subset of ∂ X is b ounded b y the dimension of the sphere Σ x . The laim follo ws no w from Kleiner's haraterisation of the dimension of spaes with urv a- ture b ounded ab o v e in terms of the top ologial dimension of ompat subsets (Theorem A in [ Kle99 ℄). Pr o of of The or em 7.10 . By Lemma 3.13 , the ation of G := Is( X ) is minimal. In view of Prop osition 7.11 , the b oundary ∂ X is nite-dimensional. Th us w e an apply Theorem 1.6 and A ddendum 1.8 . Sine X is irreduible and non-disrete, Prop osition 7.9 implies that G is an almost onneted simple Lie group (unless X = R , in whi h ase X is indeed a symmetri spae). W e onlude b y Theorem 7.4 . 7.D. No op en stabiliser at innit y . The follo wing statemen t sums up some of the pre- eding onsiderations: Corollary 7.12. L et X b e a pr op er ge o desi al ly omplete CA T(0) sp a e without Eulide an fator suh that some lose d sub gr oup G < Is( X ) ats o omp atly. Supp ose that no op en sub gr oup of G xes a p oint at innity. Then we have the fol lowing: (i) X admits a anoni al e quivariant splitting X ∼ = X 1 × · · · × X p × Y 1 × · · · × Y q wher e e ah X i is a symmetri sp a e and e ah Y j p ossesses a G -e quivariant lo al ly nite de omp osition into omp at onvex el ls. (ii) G p ossesses hyp erb oli elements. (iii) Every omp at sub gr oup of G is ontaine d in a maximal one; the maximal omp at sub gr oups fal l into nitely many onjugay lasses. (iv) Q Z ( G ) = 1 ; in p artiular G has no non-trivial disr ete normal sub gr oup. (v) so c( G ∗ ) is a dir e t pr o dut of p + q non-disr ete har ateristi al ly simple gr oups. Pr o of. (i) F ollo ws from Theorem 1.1 and Corollary 6.7 . (ii) Clear from Corollary 6.10 . (iii) and (iv) Immediate from (i) and Theorem 6.14 (i) and (ii). (v) F ollo ws from (i), (iv) and Prop osition 6.11 . ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 33 7.E. Co ompat stabilisers at innit y . W e undertak e the pro of of Theorem 1.3 whi h desrib es isometrially an y geo desially omplete prop er CA T(0) spae su h that the sta- biliser of ev ery p oin t at innit y ats o ompatly . Remark 7.13. (i) The form ulation of Theorem 1.3 allo ws for symmetri spaes of Eulidean t yp e. (ii) A BassSerre tree is a tree admitting an edge-transitiv e automorphism group; in partiular, it is regular or bi-regular (the regular ase b eing a sp eial ase of Eulidean buildings). Lemma 7.14. L et X b e a pr op er CA T(0) sp a e suh that the stabiliser of every p oint at innity ats o omp atly on X . F or any ξ ∈ ∂ X , the set of η ∈ ∂ X with ∠ T ( ξ , η ) = π is ontaine d in a single orbit under Is( X ) . Pr o of. W rite G = Is( X ) . In view of Prop osition 7.1 applied to G ξ , it sues to pro v e that the G -orbit of an y su h η on tains a p oin t opp osite to ξ . By denition of the Tits angle, there is a sequene { x n } in X su h that ∠ x n ( ξ , η ) tends to π . Sine G ξ ats o ompatly , it on tains a sequene { g n } su h that, up on extrating, g n x n on v erges to some x ∈ X and g n η to some η ′ ∈ ∂ X . The angle semi-on tin uit y argumen ts giv en in the pro of of Prop osition I I.9.5(3) in [ BH99 ℄ sho w that ∠ x ( ξ , η ′ ) = π , realling that all g n x ξ . This means that there is a geo desi σ : R → X through x with σ ( −∞ ) = ξ and σ ( ∞ ) = η ′ . On the other hand, sine G η is o ompat in G , the G -orbit of η is losed in the ne top ology . This means that there is g ∈ G with η ′ = g η , as w as to b e sho wn. W e shall need another form of angle rigidit y (ompare Prop osition 6.8 ), this time for Tits angles. Prop osition 7.15. L et X b e a ge o desi al ly omplete pr op er CA T(0) sp a e, G < Is( X ) a lose d total ly dis onne te d sub gr oup and ξ ∈ ∂ X . If the stabiliser G ξ ats o omp atly on X , then the G -orbit of ξ is disr ete in the Tits top olo gy. Pr o of. Supp ose for a on tradition that there is a sequene { g n } su h that g n ξ 6 = ξ for all n but ∠ T ( g n ξ , ξ ) tends to zero. Sine G ξ is o ompat, w e an assume that g n on v erges in G ; sine the Tits top ology is ner than the ne top ology for whi h the G -ation is on tin uous, the limit of g n m ust x ξ and w e an therefore assume g n → 1 . Let B ⊆ X b e an op en ball large enough so that G ξ .B = X . Sine b y Lemma 3.13 w e an apply Theorem 6.1 , there is no loss of generalit y in assuming that ea h g n xes B p oin t wise. Let c : R + → X b e a geo desi ra y p oin ting to w ards ξ with c (0) ∈ B . F or ea h n there is r n > 0 su h that c and g n c bran h at the p oin t c ( r n ) . In partiular, g n xes c ( r n ) but not c ( r n + ε ) no matter ho w small ε > 0 . W e no w ho ose h n ∈ G ξ su h that x n := h n c ( r n ) ∈ B and notie that the sequene k n := h n g n h − 1 n is b ounded sine k n xes x n . W e an therefore assume up on extrating that it on v erges to some k ∈ G ; in view of Theorem 6.1 , w e an further assume that all k n oinide with k on B and in partiular k xes all x n . Sine ∠ T ( k n ξ , ξ ) = ∠ T ( g n ξ , ξ ) , w e also ha v e k ∈ G ξ . Considering an y giv en n , it follo ws no w that k xes the ra y from x n to ξ . Th us k n xes an initial segmen t of this ra y at x n . This is equiv alen t to g n xing an initial segmen t at c ( r n ) of the ra y from c ( r n ) to ξ , on trary to our onstrution. Here is a rst indiation that our spaes migh t resem ble symmetri spaes or Eulidean buildings: Prop osition 7.16. L et X b e a pr op er CA T(0) sp a e suh that that the stabiliser of every p oint at innity ats o omp atly on X . Then any p oint at innity is ontaine d in an isometri al ly emb e dde d standar d n -spher e in ∂ X , wher e n = dim ∂ X . 34 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Pr o of. Let η ∈ ∂ X . There is some standard n -sphere S isometrially em b edded in ∂ X b eause X is o ompat (Theorem C in [ Kle99 ℄). By Lemma 3.1 in [ BL05 ℄, there is ξ ∈ S with ∠ T ( ξ , η ) = π . Let ϑ ∈ S b e the an tip o de in S of ξ . In view of Lemma 7.14 , there is an isometry sending ϑ to η . The image of S on tains η . W e need one more fat for Theorem 1.3 . The b oundary of a CA T(0) spae need not b e omplete, regardless of the geo desi ompleteness of the spae itself; ho w ev er, this is the ase in our situation in view of Prop osition 7.16 : Corollary 7.17. L et X b e a pr op er CA T(0) sp a e suh that that the stabiliser of every p oint at innity ats o omp atly on X . Then ∂ X is ge o desi al ly omplete. Pr o of. Supp ose for a on tradition that some Tits-geo desi ends at ξ ∈ ∂ X and let B ⊆ ∂ X b e a small on v ex Tits-neigh b ourho o d of ξ ; in partiular, B is on tratible. Sine b y Prop osition 7.16 there is an n -sphere through ξ for n = dim ∂ X , the relativ e homology H n ( B , B \ { ξ } ) is non-trivial. Our assumption implies that B \ { ξ } is on tratible b y using the geo desi on tration to some p oin t η ∈ B \ { ξ } on the giv en geo desi ending at ξ . This implies H n ( B , B \ { ξ } ) = 0 , a on tradition. (This argumen t is adapted from [ BH99 , I I.5.12℄.) End of pr o of of The or em 1.3 . W e shall use b elo w that pro dut deomp ositions preserv e ge- o desi ompleteness (this follo ws e.g. from [ BH99 , I.5.3(3)℄). W e an redue to the ase where X has no Eulidean fator. By Lemma 3.13 , the group G = Is( X ) as w ell as all sta- bilisers of p oin ts at innit y at minimally . In partiular, Prop osition 7.3 ensures that G has no xed p oin t at innit y and w e an apply Theorem 1.6 and A ddendum 1.8 . Therefore, w e an from no w on assume that X is irreduible. If the iden tit y omp onen t G ◦ is non-trivial, then Theorem 1.1 (see also Theorem 7.4 (iii)) ensures that X is a symmetri spae, and w e are done. W e assume heneforth that G is totally disonneted. F or an y ξ ∈ ∂ X , the olletion An t( ξ ) = { η : ∠ T ( ξ , η ) = π } of an tip o des is on tained in a G -orbit b y Lemma 7.14 and hene is Tits-disrete b y Prop osition 7.15 . This disreteness and the geo desi ompleteness of the b oundary (Corollary 7.17 ) are the assumptions needed for Prop osition 4.5 in [ Lyt05 ℄, whi h states that ∂ X is a building. Sine X is irreduible, ∂ X is not a (non-trivial) spherial join, see Theorem I I.9.24 in [ BH99 ℄. Th us, if this building has non-zero dimension, w e onlude from the main result of [ Lee00 ℄ that X is a Eulidean building of higher rank. If on the other hand ∂ X is zero-dimensional, then w e laim that it is homogeneous under G . Indeed, w e kno w already that for an y giv en ξ ∈ ∂ X , the set An t( ξ ) lies in a single orbit. Sine in the presen t ase An t( ξ ) is simply ∂ X \ { ξ } , the laim follo ws from the fat that G has no xed p oin t at innit y . W e ha v e to sho w that X is an edge-transitiv e tree. T o this end, onsider an y p oin t x ∈ X . The isotrop y group G x is op en b y Theorem 6.1 . In partiular, sine G ats transitiv ely on ∂ X and sine G ξ is o ompat, it follo ws that G x has nitely man y orbits in ∂ X . Let ρ 1 , . . . , ρ k b e geo desi ra ys emanating from x and p oin ting to w ards b oundary p oin ts whi h pro vide a omplete set of represen tativ es for the G x -orbits. F or r > 0 suien tly large, the v arious in tersetions of the ra ys ρ 1 , . . . , ρ k with the r -sphere S r ( x ) en tred at x forms a set of k distint p oin ts. This set is a fundamen tal domain for the G x -ation on S r ( x ) . Sine G x has disrete orbits on S r ( x ) , w e dedue from Theorem 6.1 that the sphere S r ( x ) is nite. Sine this holds for an y r > 0 suien tly large and an y x ∈ X , it follo ws that ev ery sphere in X is nite. This implies that X is 1 -dimensional (see [ Kle99 ℄). In other w ords X is a ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 35 metri tree. W e denote b y V the set of bran h p oin ts whi h w e shall all the v erties. It remains to sho w that G has at most t w o orbits on V . Giv en ξ ′ ∈ ∂ X , let β ξ ′ : G ξ ′ → R denote the Busemann harater en tred at ξ ′ (see 2 ). Sine X is a o ompat tree, it follo ws that β ξ ′ has disrete image. Let g ∈ G ξ ′ b e an elemen t su h that β ξ ′ ( g ) is p ositiv e and minimal. Then g is h yp erb oli and translates a geo desi line L . Let ξ ′′ denote the endp oin t of L distint from ξ . Let v ∈ L b e an y v ertex. W e denote b y e ′ and e ′′ the edges of L on taining v and p oin ting resp etiv ely to ξ ′ and ξ ′′ . Giv e an y edge e on taining v with e ′ 6 = e 6 = e ′′ , w e prolong e to a geo desi ra y ρ whose in tersetion with L is redued to { v } . Sine G ξ ′ is transitiv e on ∂ X \ { ξ ′ } there exists g ′ ∈ G ξ ′ su h that g ′ .ξ ′′ = ρ ( ∞ ) . Up on pre-omp osing g ′ with a suitable p o w er of g , w e ma y assume that β ξ ′ ( g ′ ) = 0 . In other w ords g ′ xes v . This sho ws that G ξ ′ ,v is transitiv e on the edges on taining v and dieren t from e ′ . The same argumen t with ξ ′ and ξ ′′ in ter hanged sho ws that G ξ ′′ ,v is transitiv e on the edges on taining v and dieren t from e ′′ . In partiular G v is transitiv e on the edges on taining v . A straigh tforw ard indution on the distane to v no w sho ws that for an y v ertex w ∈ V , the isotrop y group G w is transitiv e on the edges on taining w . This implies that G is indeed edge-transitiv e. 8. A few ases of CA T(0) superrigidity This Setion demonstrates that ertain forms of sup errigidit y an b e obtained b y om- bining the struture results of this pap er with kno wn sup errigidit y te hniques. Mu h more general results will b e established in the ompanion pap er [ CM08b ℄. 8.A. CA T(0) sup errigidit y for some lassial non-uniform latties. Let Γ b e a non- uniform lattie in a simple (real) Lie group G of rank at least 2 . By [ LMR00 , Theorem 2.15℄, unip oten t elemen ts of Γ are exp onen tially distorted. This means that, with resp et to an y nitely generating set of Γ , the w ord length of | u n | is an O (lo g n ) when u is a unip oten t. More generally an elemen t u is alled distorte d if | u n | is sublinear. If Γ is virtually b oundedly generated b y unip oten t elemen ts, one an therefore apply the follo wing xed p oin t priniple: Lemma 8.1. L et Γ b e a gr oup whih is virtual ly b ounde d ly gener ate d by distorte d elements. Then any isometri Γ -ation on a omplete CA T(0) sp a e suh that elements of zer o tr ans- lation length ar e el lipti has a glob al xe d p oint. Pr o of. F or an y Γ -ation on a CA T(0) spae, the translation length of a distorted elemen t is zero. Th us ev ery su h elemen t has a xed p oin t; the assumption on Γ no w implies that all orbits are b ounded, th us pro viding a xed p oin t [ BH99 , I I.2.8(1)℄. Bounded generation is a strong prop ert y , whi h onjeturally holds for all (non-uniform) latties of a higher rank semi-simple Lie group. It is kno wn to hold for arithmeti groups in split or quasi-split algebrai groups of a n um b er eld K of K -rank ≥ 2 b y [ T a v90 ℄, as w ell as in a few ases of isotropi but non-quasi-split groups [ ER06 ℄. As notied in a on v ersation with Sh. Mozes, Lemma 8.1 yields the follo wing elemen tary sup errigidit y statemen t. Prop osition 8.2. L et Λ = SL n ( Z [ 1 p 1 ··· p k ]) with n ≥ 3 and p i distint primes and set H = SL n ( Q p 1 ) × · · · × SL n ( Q p k ) . Given any isometri Λ -ation on any omplete CA T(0) sp a e suh that every element of zer o tr anslation length is el lipti, ther e exists a Λ -invariant lose d onvex subsp a e on whih the given ation extends uniquely to a ontinuous H -ation by isometries. 36 PIERRE-EMMANUEL CAPRA CE AND NICOLAS MONOD Pr o of. Let X b e a omplete CA T(0) spae endo w ed with a Λ -ation as in the statemen t. The subgroup Γ = SL n ( Z ) < Λ xes a p oin t b y Lemma 8.1 . The statemen t no w follo ws b eause Γ is the in tersetion of Λ with the op en subgroup SL n ( Z p 1 ) × · · · × SL n ( Z p k ) of H ; for later use, w e isolate this elemen tary fat as Lemma 8.3 b elo w. Lemma 8.3. L et H b e a top olo gi al gr oup, U < H an op en sub gr oup, Λ < H a dense sub gr oup and Γ = Λ ∩ U . A ny Λ -ation by isometries on a omplete CA T(0) sp a e with a Γ -xe d p oint admits a Λ -invariant lose d onvex subsp a e on whih the ation extends ontinuously to H . Pr o of. Let X b e the CA T(0) spae and x 0 ∈ X a Γ -xed p oin t. F or an y nite subset F ⊆ Λ , let Y F ⊆ X b e the losed on v ex h ull of F x 0 . The losed on v ex h ull Y of Λ x 0 is the losure of the union Y ∞ of the direted family { Y F } . Therefore, sine the ation is isometri and Y is omplete, it sues to sho w that the Λ -ation on Y ∞ is on tin uous for the top ology indued on Λ b y H . Equiv alen tly , it sues to pro v e that all orbital maps Λ → Y ∞ are on tin uous at 1 ∈ Λ . This is the ase ev en for the disrete top ology on Y ∞ b eause the p oin t wise xator of ea h Y F is an in tersetion of nitely man y onjugates of Γ , the latter b eing op en b y denition. The same argumen ts as b elo w sho w that Theorem 1.14 holds for an y lattie of a higher- rank semi-simple Lie group whi h is b oundedly generated b y distorted elemen ts (and a- ordingly Theorem 1.15 generalises to suitable (S-)arithmeti groups). Pr o of of The or ems 1.14 and 1.15 . W e start with the ase Γ = SL n ( Z ) . By Theorem 1.1 , w e obtain a losed on v ex subspae X ′ whi h splits as a diret pro dut X ′ ∼ = X 1 × · · · × X p × Y 0 × Y 1 × · · · × Y q in an Is( X ′ ) -equiv arian t w a y , where Y 0 ∼ = R n is the Eulidean fator. Ea h totally dison- neted fator D i of Is( X ′ ) ∗ ats b y semi-simple isometries on the orresp onding fator Y i of X ′ b y Corollary 6.3 . Therefore, b y Lemma 8.1 for ea h i = 0 , . . . , q , the indued Γ -ation on Y i has a global xed p oin t, sa y y i . In other w ords Γ stabilises the losed on v ex subset Z := X 1 × · · · × X p × { y 0 } × · · · × { y q } ⊆ X . Note that the isometry group of Z is an almost onneted semi-simple real Lie group L . Com bining Lemma VI I.5.1 and Theorems VI I.5.15 and VI I.6.16 from [ Mar91 ℄, it follo ws that the Zariski losure of the image of Γ in L is a omm uting pro dut L 1 .L 2 , where L 1 is ompat, su h that the orresp onding homomorphism Γ → L 2 extends to a on tin uous homomorphism G → L 2 . W e dene Y ⊆ Z as the xed p oin t set of L 1 . No w L 2 , and hene Γ , stabilises Y . Therefore the on tin uous homomorphism G → L 2 yields a G -ation on Y whi h extends the giv en Γ -ation, as desired. Applying Theorem 7.4 p oin t ( iv ) to the pair L 2 < L ating on Z , w e nd in partiular that L 2 has no xed p oin t at innit y in Y . Th us, up on replaing Y b y a subspae, it is L 2 -minimal. No w Theorem 2.4 in [ CM08b ℄ (whi h is ompletely indep enden t of the presen t onsiderations) implies that the Γ - and G -ations on Y are minimal and without xed p oin t in ∂ Y (although there migh t b e xed p oin ts in ∂ X ). T urning to Theorem 1.15 , the only hange is that one replaes Lemma 8.1 b y Prop osi- tion 8.2 . ISOMETR Y GR OUPS OF NON-POSITIVEL Y CUR VED SP A CES: STR UCTURE THEOR Y 37 8.B. CA T(0) sup errigidit y for irreduible latties in pro duts. The aim of this se- tion is to state a v ersion of the sup errigidit y theorem [ Mon06 , Theorem 6℄ with CA T(0) targets. The original statemen t from lo . it. onerns ations of latties on arbitr ary CA T(0) spaes, with redued un b ounded image. The follo wing statemen t sho ws that, when the underlying CA T(0) spae is nie enough, the assumption on the ation an b e onsid- erably w eak ened. W e reall for the statemen t that an y isometri ation on a prop er CA T(0) spae without glob el xed p oin t at innit y admits a anoni al minimal non-empt y losed on v ex in v arian t subspae, see Remarks 39 in [ Mon06 ℄. Theorem 8.4. L et Γ b e an irr e duible uniform (or squar e-inte gr able we akly o omp at) latti e in a pr o dut G = G 1 × · · · × G n of n ≥ 2 lo al ly omp at σ - omp at gr oups. L et X b e a pr op er CA T(0) sp a e with nite-dimensional b oundary. Given any Γ -ation on X without xe d p oint at innity, if the anoni al Γ -minimal subset Y ⊆ X has no Eulide an fator, then the Γ -ation on Y extends to a ontinuous G -ation by isometries. Remark 8.5. Although the ab o v e ondition on the Eulidean fator in the Γ -minimal sub- spae Y migh t seem a wkw ard, it annot b e a v oided, as illustrated b y Example 64 in [ Mon06 ℄. Notie ho w ev er that if Γ has the prop ert y that an y isometri ation on a nite-dimensional Eulidean spae has a global xed (for example if Γ has Kazhdan's prop ert y (T)), then an y minimal Γ -in v arian t subspae has no Eulidean fator. Pr o of of The or em 8.4 . Let Y ⊆ X b e the anonial subspae realled ab o v e. Then Is( Y ) ats minimally on Y , without xed p oin t at innit y . In partiular w e ma y apply Theorem 1.6 and A ddendum 1.8 . In order to sho w that the Γ -ation on Y extends to a on tin uous G - ation, it is suien t to sho w that the indued Γ -ation on ea h irreduible fator of Y extends to a on tin uous G -ation, fatoring through some G i . But the indued Γ -ation on ea h irreduible fator of Y is redued b y Corollary 3.8 . Th us the result follo ws from [ Mon06 , Theorem 6℄. Referenes [AB98a℄ Sot A dams and W erner Ballmann, A menable isometry gr oups of Hadamar d sp a es , Math. Ann. 312 (1998), no. 1, 183195. [AB98b℄ Stephanie B. Alexander and Ri hard L. 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