At infinity of finite-dimensional CAT(0) spaces

A T INFINITY OF FINITE-DIMEN SIONAL CA T(0) SP A CES PIERRE-EMMA NUEL CAPRA CE* AND ALEXAND ER L YTCHAK † Abstract. W e show that an y ﬁlter ing family of closed con vex subsets of a ﬁnite-dimensional CA T(0) space X has a non-empty in tersection in the visual bo rdiﬁcation X = X ∪ ∂ X . Using this fa c t, several results known for pro per CA T(0) spaces ma y be extended to ﬁnit e-dimensio nal spaces, including the exis- tence of canonica l ﬁxed po in ts a t inﬁnity for parab olic is ometries, algebra ic a nd geometric restrictions on amenable group actions, and geometr ic superr igidit y for no n-element ar y actions of irreducible unifor m lattices in pro ducts of lo ca lly compact gr o ups. 1. Introduction Sev eral families of ﬁnite-dimensional CA T(0) spaces naturally include sp ecimens whic h are not lo cally compact; e.g. buildings of ﬁnite rank (Euclidean or not), ﬁnite-dimensional CA T(0) cub e complexes, or asymptotic cones of Hadamard manifolds or of CA T(0) groups. A ma jor diﬃcult y one encoun ters when dealing with non-prop er spaces is that the visual b o undary ma y hav e a v ery pathological b eha viour. F or example, an un b ounded CA T(0) space ma y w ell hav e an em pty visual b oundary . The purp o se of this pap er is to sho w that for ﬁnite-dimensional spaces, the visual b oundary nev ertheless enjoys similarly nice prop erties as in the case of prop er spaces. F ollow ing B. Kleiner [ Kle99 ], we deﬁne the (geom etric) d imension of a CA T(0) space X to b e the suprem um ov er all compact subsets K ⊂ X of the top ological dimension of K . W e refer to lo c. c i t. for more details and sev eral c haracterizations of this notion. A 0 -dime nsional CA T(0) space is reduced to a singleton, while 1 -dimensional CA T(0) spaces coincide with R - trees. W e empha- size that the no t io n of g eometric dimension is lo c al . It turns out that, for our purp oses, it will b e suﬃcien t to demand that the spaces ha v e ﬁnite dimension at lar ge sc ale . In order to deﬁne this condition precisely , w e shall say that a CA T(0) space X has telescopic dimension ≤ n if eve ry asymptotic cone lim ω ( ε n X , ⋆ n ) has geometric dimensi on ≤ n . A space has telescopic dimension 0 if and only if it is b ounded. It has telescopic dimension ≤ 1 if and only if it is Gro mo v h yp erb olic. A CA T(0) space o f ﬁnite geometric dimension has ﬁnite telescopic dimension. W e refer to § 2.1 b elow for more details and some examples. Theorem 1.1. L et X b e a c omplete CA T(0) sp ac e of ﬁnite telesc op i c di m ension and { X α } α ∈ A b e a ﬁltering family o f close d c on vex subsp ac es. Then either the interse ction T α ∈ A X α is non-empty, o r the interse ction of the visual b oundarie s T α ∈ A ∂ X α is a non-empty subset of ∂ X of intrinsic r ad ius at most π / 2 . Date : Oc to ber 9, 20 08. 1991 Mathematics Subje ct Classiﬁc ation. 53 C 2 0, 20F65. *F.N.R.S. Resear ch Asso ciate. † Suppor ted in part by SFB 61 1 and MPI für Mathematik in Bonn. 1 Recall that a family F o f subsets o f a giv en set is called ﬁltering if for all E , F ∈ F there exists D ∈ F suc h that D ⊆ E ∩ F . In particular the preceding a pplies to nested fa milies of closed conv ex subsets, and prov ides a criterion ensurin g that the visual b oundary ∂ X is non- empt y . In the course of the pro of, we shall establish a result similar to Theorem 1.1 fo r ﬁnite-dimens ional CA T (1) spaces (see Prop osition 5.3 b elo w). W e remark how ev er that Theorem 1.1 fails for complete CA T(0) spaces with ﬁnite-dimensional Tits b oundary , see Example 5.6 b elow . R emark 1.2 . Theorem 1.1 ma y b e reformulated using t he top ology T c in tro duced b y Nicolas Mono d [ Mon06 , §3.7] on the set X = X ∪ ∂ X . It is deﬁned as the coarsest top ology suc h tha t for any conv ex subset Y ⊆ X , the (usual) closure Y in X is T c -closed. It is kno wn that an y b ounded closed subset of X is T c -quasi- compact (see [ Mon06 , Theorem 14]) and that, if X is Gromov h yp erb olic, then X is T c -quasi-compact (see Prop osition 23 in lo c. cit. ). How ev er, if X is inﬁnite- dimensional then X is generally not T c -quasi-compact. Theorem 1.1 just means that, given a c omplete CA T(0) sp ac e of ﬁ nite telesc opic dim ension, the set X is quasi-c o mp act for the top olo gy T c . This compactness prop ert y is thus shared by prop er CA T(0) spaces, Gromov h yp erb olic CA T(0) spaces and ﬁnite-dimensional CA T(0) spaces. A k ey idea in the pro of of Theorem 1.1 is to obtain p oin ts at inﬁnity b y applying (a v ery sp ecial case o f ) a result of A. Ka rlsson and G. Margulis [ KM99 ] to the gradien t ﬂo w of a conv ex function that is asso ciated in a canonical w a y to the giv en ﬁltering family . This strategy requires to sho w that the v elo cit y of escape o f the g ra dien t ﬂow in question is strictly p ositiv e. This is where the assumption o n the telescopic dimension of the am bien t space is used; the main p oint in estimating that vel o cit y is the fo llowing natural generalisation to non-p o sitive ly curv ed spaces of H. Jung’s classic al theorem [ Jun01 ]. Another closely related generalisation was established in [ LS97 ]. Theorem 1.3. L et X b e a CA T(0) sp a c e and n b e a p ositive inte ger. Then X has ge om etric dimension ≤ n if and on l y if for e ach subset Y of X of ﬁnite diameter we have rad X ( Y ) ≤ r n 2( n + 1) diam( Y ) . Similarly X has telesc opic dimensi o n ≤ n if and o nly if for any δ > 0 ther e exists some c onstant D > 0 such that for any b ounde d subset Y ⊂ X of diameter > D , we have rad X ( Y ) ≤  δ + r n 2( n + 1)  diam( Y ) . Recall t hat the circumradius rad X ( Y ) of a subset Y ⊆ X is deﬁned as the inﬁm um of all p ositive real nu mbers r suc h that Y is con tained in some closed ball of radius r of X . R emark 1.4 . In the case of an n - dimensional regular Euclidean simplex one has equalit y in the theorem ab ov e. F or a short discussion of the case of equalit y as w ell as analogous statemen ts in other curv a ture b ounds we refer to Section 3 . It turns out that Theorem 1.1 provide s a k ey prop erty that allo ws one to extend to ﬁnite-dimensional CA T(0) spaces sev eral results whic h are kno wn to hold for prop er spaces. W e now pro ceed to describe a few of these applications. 2 P arab olic isometries. A ﬁrst elemen tary consequence of Theorem 1.1 is the existenc e of canonical ﬁxed p oints at inﬁnit y for parab o lic isometries. This extends the results obta ined in [ FNS06 , Theorem 1.1] a nd [ CM09 , Corollary 2.3] in the lo cally compact setting. Corollary 1.5. L et g b e a p ar ab o lic isometry of a CA T(0) sp ac e X of ﬁnite telesc op i c dimension. T hen the c entr aliser Z Is( X ) ( g ) p ossesse s a c anonic al ﬁxe d p oint in ∂ X . Amenable group actions. The next application provide s obstructions to iso- metric actions of amenable groups; in the lo cally compact case the corresp onding statemen t is due to S. Adams a nd W. Ba llmann [ AB98 ], and generalizes earlier results b y M. Burger and V. Sc hro eder [ BS87 ]. Theorem 1.6. L et X b e a c omp lete CA T(0) sp ac e of ﬁ n ite telesc opic dime n sion. L et G b e an ame n able lo c al ly c omp act gr oup acting c ontinuously on X b y isome- tries. T h en either G stabili s e s a ﬂat subs p ac e (p ossibly r e duc e d to a p oint) or G ﬁxes a p oint in the ide al b ounda ry ∂ X . Com bining this with the argumen ts of [ Cap09 ], one obtains the follo wing de- scription of the algebraic structure of amenable g roups acting on CA T(0) cell complexes . Theorem 1.7. L et X b e a CA T(0) c el l c o mplex with ﬁnitely ma ny typ e s of c el l s and G b e a lo c al l y c omp act gr oup admitting an iso metric action on X which is c ontinuous, c el lular and metric al ly pr op er. Then a close d sub gr oup H < G is amenable if and only if it i s (top olo gic al ly lo c al ly ﬁnite)-by-(virtual ly A b elian). By deﬁnition, a subgroup H of a top ological group G is top ologically lo cally ﬁnite if t he closure of ev ery ﬁnitely g enerated subgroup of H is compact. W e refer to [ Cap09 ] for more details. The pro of of Theorem 1.7 pro ceeds as in lo c. cit. One in tro duces the reﬁned b oundary ∂ ﬁne X of the CA T(0) space a nd shows , using Theorem 1.6 , that an y amenable subgroup o f G virtually ﬁxes a p oint in X ∪ ∂ ﬁne X ; con ve rsely any p oin t of X ∪ ∂ ﬁne X has an amenable stabilizer in G . Minimal and reduced actions. A basic prop ert y of CA T(0) spaces with ﬁnite telescopic dimension is that their Tits b o undary has ﬁnite geometric dimension (see Prop osition 2.1 b elo w). G iv en this observ ation, Theorem 1.1 ma y b e used to extend sev eral results of [ CM09 , P art I] to the ﬁnite-dimensional case. The follo wing collects a few of these statemen ts. Prop osition 1.8. L et X b e a c omplete CA T(0) sp ac e of ﬁnite telesc opic dime n - sion and let G < Is( X ) b e any gr oup of isometries. (i) If the G -action is evanes c ent, then G ﬁxes a p oin t in X ∪ ∂ X . (ii) If G do es not ﬁx a p oint in the ide al b oundary, then ther e is a non-empty close d c onvex G -invariant subset Y ⊆ X on which G acts m inimal ly. (iii) Supp ose that X is irr e d ucib le. If G a c ts minimal ly without ﬁxe d p oin t at inﬁ nity on X , then so do es every non-trivial normal sub gr o up of G ; furthermor e, the G -a ction is r e duc e d. (iv) If Is( X ) acts minimal ly o n X , then for e ach close d c onvex subset Y ( X we have ∂ Y ( ∂ X . 3 F ollow ing Nicolas Mono d [ Mon06 ], we say that the action o f a g roup G on a CA T(0) space X is ev anescen t if there is an un b ounded subset T ⊆ X suc h that for eve ry compact set Q ⊂ G the set { d ( g x, x ) : g ∈ Q, x ∈ T } is b ounded. Recall further that the G -action is said to b e minimal if there is no non-empt y closed con v ex G -in v ariant subset Y ( X . Finally , it is called reduced if there is no non-empty closed con v ex subset Y ( X suc h that for each g ∈ G , the sets Y and g .Y are at b ounded Hausdorﬀ distance from one another. The relev ance of the notions of ev anescen t and reduced a ctions w as ﬁrst highligh ted by Nico- las Mono d [ Mon06 ] in the con text of geometric sup errigidity . In particular, the com bination of [ Mon06 , Theorem 6] with Prop o sition 1.8 (iii) yields the following (see [ CM09 , Theorem 9 .4] for the correspo nding statemen t in the lo cally compact case). Corollary 1.9. L et Γ b e an irr e ducible uniform (or squar e-inte g r able we akly c o- c omp act) lattic e in a pr o d uct G = G 1 × · · · × G n of n ≥ 2 lo c al ly c omp act σ -c omp act gr oups. L e t X b e a c om plete CA T(0) sp ac e of ﬁnite telesc opic dimensio n without Euclide an factor. The n any minimal isometric Γ -action on X without ﬁxe d p oin t at inﬁnity extends to a c ontinuous G -action by iso m etries. On the other hand, combin ing Prop osition 1.8 with Theorem 1.6 yields the follo wing extens ion of [ CM09 , Corollary 4.8]. Corollary 1.10. L et G b e a lo c al l y c omp act gr oup ac ting c ontinuously and mini- mal ly on a CA T(0) sp ac e X of ﬁnite teles c opic dimension, without ﬁxi n g any p oint at inﬁnity. Then the am enable r adic al R of G stabil i z es the maximal Euclide an factor of X . In p a rticular, if X has no non-trivial Euclide an fa ctor, then R acts trivial ly. A ckn owledge men ts. W e w ould lik e to thank Viktor Sc hro eder for fruitful con- v ersations on a ﬃne functions. W e are gra teful to An ton Pe trunin for providin g Example 4.4 . W e thank Nicolas Mono d for n umerous illuminating con v ersations and for p oin ting out that no separabilit y assumption on G is needed for Theo- rem 1.6 to hold. Finally , thanks are due to the referee for his/her commen ts. 2. Preliminaries 2.1. Geometric and telescopic dimension. W e r ecall some facts ab out di- mensions of spaces with upp er curv ature b o unds. The geometric dimension (sometimes simply called dimension ) of suc h spaces w as deﬁned inductiv ely in [ Kle99 ], by setting the dimension of a discrete space to b e 0 and b e deﬁning dim( X ) = sup { dim( S x X ) + 1 | x ∈ X } , where S x X denotes the space of directions at the p oin t x . It turns out that this notio n of dimension is closely related to more top ological notions. Namely dim ( X ) ≤ n if and only if for all op en subsets V ⊂ U of X the relativ e singular homology H k ( U, V ) v anishes f or all k > n . Moreov er, this is equiv alen t to the fact that the top ological dimension of all compact subsets of X is b ounded ab o v e b y n , see lo c. cit. By deﬁnition, a CA T(0) space X is said to ha v e telescopic dimension ≤ n if ev ery asymptotic cone lim ω ( ε n X , ⋆ n ) has geometric dimension ≤ n . Although this will not pla y any role in the sequel, we remark that the telescopic dimension is a quasi-isometry in v a rian t. Moreo v er, it fo llo ws from [ Kle99 , Th. C] that a lo cally compact CA T(0) space with a co compact isometry gro up has ﬁnite telescopic dimension. 4 Con v ex subsets inherit the geometric dimension b ound f r o m the am bien t space. Moreo v er, if ( X i , x i ) is a sequence of p oin ted CA T ( κ ) spaces of geometric dimen- sion ≤ n , then their ultralimit lim ω ( X i , x i ) with respect to some ultraﬁlter is a CA T( κ ) space of dimension a t most n , see [ Lyt05b , Lemma 11.1]. In particu- lar, it f ollo ws that a CA T(0) space of geometric dimension ≤ n has telescopic dimension ≤ n . F urthermore, w e ha v e the follow ing. Prop osition 2.1. L et X b e a CA T(0) sp ac e. If X has telesc opic dime nsion ≤ n , then the visual b oundary ∂ X endowe d with the Tits metric has ge ometric dime n - sion at most n − 1 . Pr o of. Let o ∈ X b e a base p oin t and C ω X b e the asymptotic cone lim ω ( 1 n X , o ) . The Euclidean cone C ( ∂ X ) em b eds isometrically in to C ω X , see [ Kle99 , Lemma 10.6]. Th us dim( ∂ X ) = dim( C ( ∂ X )) − 1 ≤ dim( C ω X ) ≤ n .  W e emphasize t hat a CA T(0) space X ma y ha ve ﬁnite-dimensional Tits b ound- ary without b eing of ﬁnite telescopic dimension, ev en if X is prop er. Indeed, consider for instance the p ositiv e real half-line and glue a t eac h p oint n ∈ N an n -dimensional Euclidean ball of radius n . The resulting space is prop er and CA T(0), its ideal b oundary consists of a single p oin t, but eac h of its asymptotic cones contains a n inﬁnite-dimen sional Hilb ert space. W e shall use a top o logical v ersion Helly’s classical theorem that holds in mu ch greater generalit y (see [ Dug67 ] as wel l as [ F ar 0 9 , §3] fo r a r elated discussion). The follo wing statemen t is an immediate consequenc e of [ Kle99 , Prop osition 5.3] since in tersections o f con v ex sets are either empt y o r con tractible. Lemma 2.2. L et X b e a CA T(0) s p ac e of ge ometric dimensi o n ≤ n . L et { U α } α ∈ A b e a ﬁnite fami l y of op en c onvex subsets of X . I f for e ach subset B ⊂ A with a t most n + 1 elements the interse ction T α ∈ B U α is non-empty, then T α ∈ A U α is non-empty.  2.2. Inner p oints. F ollo wing [ LS07 ], w e shall sa y tha t a p oin t o of a CA T(0) space X is a top ologically inner p oint if X \ { o } is not contractible. F o r each top ologically inner p oint there is some ε > 0 and a compact subset K of X with d ( o, K ) ≥ ε with the follo wing prop erty : F or eac h x ∈ X there is some ¯ x ∈ K suc h that xo ¯ x is a geo desic. Thus ev ery geo desic segmen t which terminates at o ma y b e lo cally prolonged b ey ond o ; in lo ose terms, the space X is g e o desic al ly c omp lete at the p oint o . In a CA T(0) space whic h is lo cally o f ﬁnite geometric dimension, the set of top olog ically inner p oin ts is dense, see [ LS07 , Theorem 1.5]. In particular it is non-empt y . 3. Jung’s t heorem 3.1. CA T(0) case. Throughout the pa p er, w e shall adopt the follo wing nota- tional conv en tion. Giv en a subset Y ⊆ X we denote the distance to Y b y d Y , namely d Y : X → R : x 7→ inf y ∈ Y d ( x, y ) . W e further recall that the intrinsic radius of a subset Z of a metric space X is deﬁned as rad( Z ) = inf z ∈ Z { r ∈ R > 0 | Z ⊆ B ( z , r ) } . This notion should not b e confused with the circumradius (or relativ e radius ), deﬁned as rad X ( Z ) = inf x ∈ X { r ∈ R > 0 | Z ⊆ B ( x, r ) } . 5 Bounded closed conv ex subsets of non-p ositiv ely curv ed spaces ha v e the ﬁ- nite in tersection prop ert y (see [ LS97 , Pro of of Theorem B] or [ Mon06 , The- orem 14]). This means that for an y family { X α } α ∈ A of b ound e d closed con v ex subsets of a CA T(0) space X the in tersection T α ∈ A X α is non- empty whenev er the in tersection of eac h ﬁnite sub-family is non-empt y . Lemma 3.1. L et X b e a CA T(0) sp ac e of ge ometric dimension ≤ n and Y ⊆ X b e a subset of ﬁ nite diameter. If for al l subsets Y ′ ⊆ Y of c ar dinality | Y ′ | ≤ n + 1 we have rad X ( Y ′ ) ≤ r then ra d X ( Y ) ≤ r . Pr o of. Fix an arbitrary r ′ > r . F or y ∈ Y , denote b y O y the op en ball of radius r ′ around y . These balls are conv ex and, b y assumption, the in tersection of an y collection of at most ( n + 1) suc h balls is non-empt y . By Lemma 2.2 the in tersection of a n y ﬁnite collection of suc h balls is no n- empt y . Since r ′ > r is arbitrary , this implies t hat eac h ﬁnite subset Y ′ of Y has radius at most r . F or y ∈ Y , denote no w b y B y the closed ball of radius r ar o und y . Then the inters ection of each ﬁnite collection of B y is non- empty , hence the in tersection o f all B y is non- empty . F or an y p oin t x in this interse ction, w e get d ( x, y ) ≤ r fo r all y ∈ Y . Hence rad X ( Y ) ≤ r .  Pr o of of The or em 1.3 . Theorem A f r o m [ LS97 ] ensures that for an y CA T(0) space X and each subset Y ⊂ X of cardinalit y at most n + 1 , the inequalit y rad X ( Y ) ≤ q n 2( n +1) diam( Y ) holds. In view of this, it follow s from Lemma 3.1 that the in- equalit y ra d X ( Y ) ≤ q n 2( n +1) diam( Y ) holds for any subset Y of a CA T(0) space X of geometric dimension ≤ n . Assume con v ersely that X has geometric dimension > n . By [ Kle99 , Theo- rem 7.1], there exist a sequence ( λ k ) of p ositiv e real n umbers suc h that lim k λ k = ∞ , and a sequen ce ( Y k , ⋆ k ) k ≥ 0 of p ointe d subsets of X suc h that lim ω ( λ k Y k , ⋆ k ) = R n +1 for an y non- principal ultraﬁlter ω . W e ma y then ﬁnd n + 2 sequences ( y i k ) k ≥ 0 of p oin ts o f Y k indexed b y i ∈ { 0 , 1 , . . . , n + 1 } suc h that the set ∆ = { lim ω ( y i k ) | i = 0 , . . . , n + 1 } coincides with the v ertex set o f a regular simplex o f diameter 1 in R n +1 . Since the equalit y case of the ( n + 1 ) - dimensional Jung inequalit y is ac hiev ed in the case of ∆ , w e deduce that there exists some k ≥ 0 suc h that the n - dimensional Jung inequalit y fails for the subset ∆ k = { y i k | i = 0 , . . . , n + 1 } ⊂ X . Assume no w t ha t X has telescopic dimension ≤ n and supp ose fo r a con tra- diction that for some ﬁxed δ > 0 and for eac h in teger k > 0 there is a subset Y k ⊂ X suc h that diam( Y k ) > k and r a d X ( Y k ) ≥ ( q n 2( n +1) + δ )diam( Y k ) . Let ⋆ k b e the circumcen tre of Y k . Setting λ k = rad X ( Y k ) , it then follo ws that the asymptotic cone lim ω ( 1 λ k X , ⋆ k ) p ossesses a subset lim ω ( Y k ) whic h fails to satisfy the n -dimensional Jung inequalit y . The contradicts the ﬁrst part of the statemen t whic h has already b een established. Assume con v ersely that X has telescopic dimension > n . Then, b y [ Kle99 , Theorem 7.1] there exists a sequence ( Y k , ⋆ k ) k ≥ 0 of p ointed subsets of X suc h that lim ω ( Y k , ⋆ k ) = R n +1 . In pa rt icular diam( Y k ) tends to ∞ with k and w e conclude b y the same argumen t as b efore.  6 3.2. Rigidit y and other curv ature b ounds. In this subsection, w e brieﬂy sk etc h the analogues o f Theorem 1.3 in the case of non-zero curv ature b ounds and address the equalit y case. Since the results are not used in the sequel, w e do not pro vide complete pro ofs. F ollow ing word b y w ord t he pro of of Theorem 1.3 and using the results of [ LS97 ] for other curv a ture b ounds, one obtains the f ollo wing. Prop osition 3.2. L et X b e a CA T( − 1) sp ac e o f ge ometric dimen s ion at m o st n . L et Y b e a subset of X o f diam e ter at m o st D . Then the r adius of Y in X is b ounde d ab ov e by r n ( D ) , wher e r n ( D ) denote the r adius of the r e gular n - dimensional sim p lex ∆ D in the n -dimens ional r e a l hyp erb oli c sp ac e H n of diameter D .  In the p ositiv ely curv ed case one needs to assume a b ound on the radius in order for t he balls in question to b e con v ex. An additional tech nical diﬃcult y arises from the fact the the whole space ma y b e non-con tractible in this case, and the statemen t of Lemma 2.2 has therefore to b e sligh tly mo diﬁed in that case. The resulting ra dius–diameter estimate is the follo wing. Prop osition 3.3. L et X b e a CA T(1) sp ac e o f dim ension ≤ n . L et Y b e a subset of X of cir cumr adius r < π 2 . Then the diameter of Y is at l e ast s n ( r ) , wher e s n ( r ) is the diameter of the r e gular simplex of r adius r in the r ound S n . R emark 3.4 . In a similar wa y it can b e sho wn, that the assumption r = rad X ( Y ) < π 2 is fulﬁlled as so on as diam( Y ) < k n = arccos( − 1 / ( n + 1)) . It is sho wn in [ LS97 ] that for a subset Y of cardinalit y ≤ n + 1 , the equalit y in Theorem 1.3 holds if and o nly if the con v ex hull of these p oints is isometric to a regular Euclidean simplex. Arguing as in the pro of of Theorem 1.3 one o btains that if X is lo cally compact, the inequality b ecomes an equalit y if and only if the con v ex h ull of Y con tains a regular n -dimensional Euclidean simplex of diameter equal to t he diameter of Y . If X is not lo cally compact the same statemen t holds for the con v ex hu ll of the ultrapro duct Y ω ⊂ X ω . Similarly , the ana lo g ous rigidity statemen t s hold for spaces with o ther curv ature b ounds for the same reasons. 4. Convex funct ions and their gradient flo w 4.1. Gradien t ﬂo w. W e recall some basics ab out gradien t ﬂo ws asso ciated to con v ex f unctions. W e refer to [ Ma y98 ] fo r the general case and to [ Lyt05a ] for the simpler case of Lipsc hitz con tin uous functions; only the la t t er is relev a n t to our purp oses. Giv en a CA T(0) space X , a map f : X → R is called c on v ex if its restriction f ◦ γ to each geo desic γ is con v ex. Basic examples o f con v ex functions o n CA T(0) spaces are distance functions to p oints or t o con v ex subsets, and Busemann func- tions, see [ BH99 , I I.2 and I I.8]. Let f b e a con tin uous conv ex function on a CA T(0) space X . F or a p oin t p ∈ X , the absolute gradien t of the concav e function ( − f ) at p is deﬁned b y the formula |∇ p ( − f ) | = max n 0 , lim sup x → p f ( p ) − f ( x ) d ( p, x ) o . 7 The absolute gradient is a non-negative , p ossibly inﬁnite function. It is b ounded ab o ve b y the Lipsc hitz constan t if f is Lipsc hitz contin uous. A f undamental ob ject attached to the function f is the gradien t ﬂo w whic h consists of a map φ : [0 , ∞ ) × X → X whic h, lo osely sp eaking, has the pro p ert y that φ 0 = Id and φ t ( x ) follows for eac h x the path of steep est descen t o f f from x . The gradien t ﬂo w is indeed a ﬂo w in the sense that it satisﬁes φ s + t ( x ) = φ s ◦ φ t ( x ) for all x ∈ X . The most imp ortant prop ert y of gradien t ﬂow s, originally observ ed b y Vladimir Sharafutdino v [ Šar77 ] in the Riemannian con text, is that the ﬂow φ t is semi- con tracting . In other w ords, for each t ≥ 0 , the map φ t : X → X is 1 - Lipsc hitz (see [ Lyt05a , Theorem 1.7]). W e refer to [ Ma y98 ] or [ Lyt05a , §9] for more details and historical commen ts. R emark 4.1 . Originally , the gradien t lines and ﬂo ws w ere deﬁned for conca v e func- tions by Sharaf utdino v [ Šar77 ] in the case o f manifolds; they a re also commonly used fo r semi-conca v e (but not semi-con v ex) functions. Moreov er t he g ra dien t usually represen ts the direction of the maximal gr owth of the function ra ther than its maximal de c ay . This explains the sligh tly cum b ersome notation |∇ x ( − f ) | that w e use here. F or eac h x ∈ X the gradien t curv e t 7→ φ t ( x ) of f has the followi ng pro p erties (and is unique ly c haracterised by them). (1) The curv e t 7→ φ t ( x ) has ve lo cit y | φ t ( x ) ′ | = |∇ φ t ( x ) ( − f ) | for almost all t ≥ 0 . (2) The restriction t 7→ f ( φ t ( x )) of f to the gr a dien t curv e is conv ex. F ur- thermore it satisﬁes ( f ◦ φ t ( x )) ′ = −|∇ φ t ( x ) ( − f ) | 2 . W e deﬁne the v elo city of escap e of the ﬂo w φ t at the p o in t x ∈ X b y lim sup t →∞ d ( x, φ t ( x )) t . Since the ﬂo w φ t is semi-con tracting, the lim sup in the ab ov e deﬁnition ma y b e replaced b y a usual limit. Moreo v er, it do es not dep end on the starting p oin t x . The follow ing statemen t is an application of the main result o f [ KM99 ] (to a deterministic setting). Prop osition 4.2. L et f b e a c o n vex Lipschitz function on a CA T(0) sp ac e X . If ε = inf x ∈ X |∇ x ( − f ) | > 0 then ther e is a unique p o int ξ f ∈ ∂ X such that for a l l x ∈ X the gr ad ient curve φ t ( x ) deﬁne d by f c onver ges to ξ f for t → ∞ . R emark 4.3 . In particular, the existenc e of a function f as in Prop osition 4.2 implies that the ideal b oundary of X is non-empt y . The follo wing construction due to Anton Pe trunin sho ws that the conclusi on of Prop osition 4.2 fails without a uniform low er b ound on the absolute g radien t. Example 4.4 . Cho ose an acute angle in R 2 enclosed by tw o ra ys γ ± ( t ) = t · v ± emanating fro m the origin. Let f n ( w ) = h w , x n i b e linear maps on R 2 suc h that the followi ng conditions hold. First, for all n , w e require that h x n , v ± i b e p ositiv e. F or o dd (r esp. eve n) n , the direction v + (resp. v − ) is b etw een x n and v − (resp. x n and v + ). Moreo v er, the sequence ( x n ) satisﬁes the recursiv e condition h x n , v − i = h x n − 1 , v + i fo r ev en n and h x n , v + i = h x n − 1 , v − i for o dd n . Finally , w e require that the length k x n k tends to 0 as n tends to inﬁnit y . It is easy to see that suc h a sequence ( x n ) exists. 8 No w let p 1 = v − and deﬁne inductiv ely p n on γ + (resp. γ − ) for n ev en (resp. o dd) to b e the p oint suc h that p n − p n − 1 is para llel to x n . This just means that p n arises from p n − 1 b y following the gradien t ﬂow o f the aﬃne (and hence concav e) function f n . Deﬁne the num b ers C n b y C 0 = 0 and f n ( p n +1 ) − f n +1 ( p n +1 ) = C n +1 − C n . Consider the conca v e function f ( x ) = inf ( f n ( x ) + C n ) . One veriﬁe s that on the geo desic segmen t ( p n , p n +1 ) the function f coincides with f n (in fact on a neigh b ourho o d o f all p oints except p n +1 ). Hence the segmen t joining p n to p n +1 is part of a gradien t curv e of f . Therefore the appropriately parametrised piecew ise inﬁnite geo desic γ running through all p i is a gradient curv e of f . It is clear that b oth v − and v + (and all unit v ectors b etw een them) considered as p oints in the ideal b oundary are accum ulation p oin ts of γ at inﬁnit y . Pr o of of Pr o p osition 4.2 . F rom the assumption that ε = inf x ∈ X |∇ x ( − f ) | > 0 , we deduce that f ( φ t ( x )) − f ( x ) ≤ − ε 2 t for all x ∈ X . In view of Prop ert y (2) of the gradien t curv e recalled ab o ve and the fact that f is Lipsc hitz, w e deduce that the v elo cit y of escap e of the gradient curv e is strictly p ositiv e. An imp ortant conseque nce of [ KM99 , Theorem 2 .1 ] is that an y semi-con tr a cting map F : X → X of a complete CA T(0) space X with strictly p ositiv e v elo cit y of escap e lim sup n →∞ d ( p,F n ( p )) n has t he follo wing conv ergence prop ert y: There is a unique p oin t ξ F in the ideal b oundary ∂ X of X , suc h that f o r all p ∈ X the sequenc e p n = F n ( p ) con v erges to ξ F in the cone top o logy . In view of the ab ov e discuss ion, w e a r e in a p o sition to apply this result to F = φ 1 , from whic h the desired conclusion fo llo ws.  4.2. Asymptotic slop e and a radius estimate. Finally w e recall an observ a- tion of Eb erlein ([ Eb e96 ], Section 4.1) ab out the size of the set of p oin ts in the ideal b oundary with negativ e asymptotic slop es. Let f : X → R b e a contin uous conv ex function. F or eac h geo desic ray γ : [0 , ∞ ) → X one deﬁnes the asymptotic slop e of f on γ b y lim t →∞ ( f ◦ γ ′ ( t )) . This deﬁnes a n um b er in ( −∞ , + ∞ ] whic h dep ends only on the p oint at inﬁnit y γ ( ∞ ) ∈ ∂ X . Thus one obtains a function sl ope f : ∂ X → ( −∞ , + ∞ ] . One sa ys that a p oint ξ ∈ ∂ X is f -monotone if sl ope f ( ξ ) ≤ 0 . This is equiv alen t to say ing that the restriction of f to any ray asymptotic to ξ is non-increasing. One denotes the set of all f -monotone p oints b y X f ( ∞ ) . Lemma 4.5. L et f b e a c onvex Lipschitz function on a c omplete CA T(0) sp ac e X such that inf x ∈ X |∇ x ( − f ) | > 0 . Then for e ach p oin t ξ ∈ X f ( ∞ ) , we have d Tits ( ξ , ξ f ) ≤ π 2 , wher e ξ f is the c a n onic al p oint pr ovi d e d by Pr op osi tion 4.2 . Pr o of. Eb erlein’s argumen t for the pro of o f [ Eb e96 , Prop osition 4.1 .1 ] (whic h is also repro duced in the pro of of [ FNS06 , Theorem 1 .1 ]) shows , that for any p ∈ X and an y sequenc e t i , suc h that φ t i ( p ) con v erges to some p oin t ξ ∈ ∂ X , the Tits- distance b etw een ξ and a ny other p oint ψ ∈ X is at most π 2 .  4.3. The space of conv ex functions. Pic k a base p oin t o ∈ X . Denote b y C 0 the set of all 1 -L ipsch itz con v ex functions f on X with f ( o ) = 0 . W e view it as subset of the lo cally conv ex to p ological ve ctor space B of all functions f on X with f ( o ) = 0 , where the latt er is considered with the top olog y of p oin twis e con v ergence. The subset C 0 ma y thus b e considered as a closed subset of the inﬁnite pro duct Q x ∈ X I x , where I x is the in terv al I x = [ − d ( o, x ) , d ( o, x )] . Since a 9 con v ex com bination of con ve x 1 -Lipsc hitz functions is con v ex and 1 -L ipsch itz, the set C 0 is a conv ex compact subset of B . The isometry g r o up G = Is( X ) of X a cts contin uously on B b y g · f : x 7→ f ( g x ) − f ( g o ) and preserv es the subset C 0 . Consider the map i : X → C 0 giv en b y i ( x ) := ¯ d x , where ¯ d x is the normalized distance function ¯ d x ( y ) = d ( x, y ) − d ( x, o ) . Note that the map i is G -equiv arian t. In particular, the subse t C = i ( X ) ⊂ C 0 as w ell as its closure and closed conv ex h ull are G -inv aria nt. If X is lo cally compact, then C consists precisely of normalized distance and Busemann functions on X , and is thu s nothing but the visual compactiﬁcation of X . Ho w ev er, if X is not lo cally compact, then ¯ C ma y b e muc h larger, and the con v ergence in C 0 ma y b e rather strange. Example 4.6 . Let X ′ b e a separable Hilb ert space with origin o and a n orthonormal base { e n } n ≥ 0 . Then the sequence ¯ d ne n con v erges in C 0 to the constant function. Example 4.7 . Let X ′′ b e a metric tree consisting of a single v ertex o from whic h emanate coun tably man y inﬁnite rays η n . In other w ords X ′′ is the Euclidean cone ov er a discrete coun tably inﬁnite set. Let b n denote the Busemann function asso ciated with η n . Then b n con v erge in C 0 to the distance function d o . W e emphasize that the choice of the base p oint o do es not pla y an y role: a ny c hange of base p oint amounts to adding an additiv e constan t. In some sense, the set C ma y serv e in the non-lo cally compact case a s a general- ized ideal b oundary . It is therefore imp ortan t to understand ho w “large” it really is. This will b e the purp ose of the next subsection. 4.4. Aﬃne functions on spaces of ﬁnite telescopic dimension. Recall that a function f : X → R is called aﬃne if its restriction to any geo desic is aﬃne. Equiv a len tly , for all pairs x + , x − ∈ X with midp oin t m w e ha v e f ( x + ) + f ( x − ) = 2 f ( m ) . A simple-minded but notew orth y observ ation is that aﬃne functions a re precisely those conv ex f unctions f whose opp osite ( − f ) is also con v ex. Clearly , constan t functions are aﬃne; thus an y CA T(0) space admit aﬃne functions. How- ev er, the very existence of non-c onstant aﬃne functions imp oses very strong re- strictions on the underlying space, see [ LS07 ]. The follo wing result also pro vides an illustration of this phenomenon, whic h will b e relev an t to the pro of of Theo- rem 1.6 . Prop osition 4.8. L et X b e a CA T(0) sp ac e of ﬁnite telesc opic dimension whic h is no t r e duc e d to a single p oint and such that Is( X ) acts mini mal ly. If C c ontains an aﬃne function, then ther e is a splitting X = R × X ′ . Recall from [ BH99 , I I.6.15 ( 6 )] that an y complete CA T(0) space X admits a canonical splitting X = E × X ′ preserv ed b y all isometries, where E is a (maxi- mal) Hilb ert space called the Euclidean factor of X . It is shown in [ LS07 , Corol- lary 4.8] that if X is lo cally ﬁnite-dimensional and if Is( X ) acts minimally , then X ′ do es not admit an y non-constan t aﬃne function. The main tec hnical p oint in the pro of of the latter fa ct is the existence of inner p oin ts (see § 2.2 ). In order to deal with the case o f asymptotic dimension b ounds, w e need to sub- stitute this b y some coarse equiv alen t. This substitute is pro vided b y Lemma 4.9 , whic h is of tec hnical nature. In the sp ecial case o f spaces of ﬁnite ge ometric di- mension, it follow s quite easily from the existence of inner p oin ts ; therefore, the reader who is only in terested in those spaces may wish to skip it. 10 Lemma 4.9. L et X b e an unb ounde d sp ac e of ﬁnite telesc opic dimen s ion. Then ther e a r e se quenc es of p os i tive numb ers D j → ∞ , δ j → 0 and se quenc es of p oin ts p j ∈ X and of ﬁn i te subsets Q j ⊂ X with the fo l lowing two pr op erties. (1) Q j is c ontaine d in the b al l of r adius D j (1 + δ j ) ar ound p j . (2) F or al l s ∈ X , ther e is some q j ∈ Q j with d ( s, q j ) − d ( s, p j ) ≥ D j . Pr o of. Consider ˜ X = lim ω ( 1 n X , o ) and let ˜ p = ( p n ) b e a n inner p oint of ˜ X . Let ε > 0 and the compact subset K ⊂ ˜ X b e chos en as in § 2.2 . Mov ing p oin ts of K to w ards ˜ p w e ma y assume that a ll p oint of K ha v e distance ε to ˜ p . F urthermore, there is no loss of generality in assuming ε < 1 . Since K is compact, there exist ﬁnite subsets Q n ∈ 1 n X with lim ω Q n = K and d ( p n , q ) ≤ εn for all q ∈ Q n . In view of the deﬁning prop ert y of K , w e deduce that for a ll δ ∈ (0 , ε ) a nd all n 0 , there is some n = n ( δ, n 0 ) > n 0 suc h that for any s ∈ X with d ( s, p n ) ≤ n , there is some q ∈ Q n with d ( s, q ) ≥ d ( s, p n ) + n ( ε − δ ) . Assume now δ ∈ (0 , ε 2 ) . Give n ˜ s ∈ X with d ( ˜ s, p n ) ≥ n and c ho ose the p oin t s b et w een p n and ˜ s with d ( p n , s ) = n . Let q ∈ Q n b e suc h that d ( s, q ) ≥ d ( s, p n ) + n ( ε − δ ) . Using the law of c o s i n es in a comparison triang le for ∆( ˜ s, p n , q ) , w e deduce from that CA T(0) inequalit y that d ( ˜ s, q ) 2 − d ( ˜ s, p n ) 2 − d ( p n , q ) 2 d ( ˜ s, p n ) ≥ n 2 ( ε − δ )(2 + ε − δ ) − d ( p n , q ) 2 n . Since d ( p n , q ) ≤ εn and d ( ˜ s, p n ) = n + d ( s, ˜ s ) , w e deduce d ( ˜ s, q ) 2 − d ( ˜ s, p n ) 2 ≥ n 2 ( ε − δ )(2 + ε − δ ) + 2 n ( ε − δ (1 + ε ) + δ 2 2 ) d ( s, ˜ s ) ≥ n 2 ( ε − 2 δ )(2 + ε − 2 δ ) + 2 n ( ε − 2 δ ) d ( s, ˜ s ) = n 2 ( ε − 2 δ ) 2 + 2 n ( ε − 2 δ ) d ( ˜ s, p n ) . This implies that d ( ˜ s, q ) ≥ d ( ˜ s, p n ) + n ( ε − 2 δ ) . It remains to deﬁne the desired sequences by making the appropriate ch oices of indices. This ma y b e done as follows. F or each in teger j > 0 , we no w set δ j = ε 2 j and n j = n ( εδ j 4 , j ) , where n : ( 0 , ε ) × N → N : ( δ, n 0 ) 7→ n ( δ, n 0 ) is the map considered ab ov e. Finally w e set p j = p n j , Q j = Q n j and D j = εn j (1 − δ j 2 ) .  Pr o of of Pr o p osition 4.8 . Assume that the set A of aﬃne functions con tained in C is nonempt y . F or eac h in teger j > 0 w e set C j = { p ∈ X | ∀ f ∈ A ∃ z ∈ X , f ( z ) − f ( p ) = D j and d ( p, z ) ≤ (1 + δ j ) D j } , where ( D j ) and ( δ j ) are the sequenc es prov ided b y Lemma 4.9 . W e claim that C j is non-empty. Let us ﬁx the index j and consider the p oin t p j ∈ X pro vided by Lemma 4.9 . W e will sho w that p j b elongs to C j . T o this end, let f ∈ A . By deﬁnition, there is a sequence ( s n ) of p o ints of X suc h that the sequence ( ¯ d s n ) con v erges p o in t wise t o f , where ¯ d s denotes the normalised distance function ¯ d s ( · ) = d ( s, · ) − d ( o, s ) . Now for each n , Lemma 4.9 provides a p o in t q n ∈ Q j suc h that d ( p j , q n ) ≤ D j (1 + δ j ) and ¯ d s n ( q n ) − ¯ d s n ( p j ) = d s n ( q n ) − d s n ( p j ) ≥ D j . Up o n extracting, w e ma y assume that ( q n ) is constan t a nd is equal to q ∈ Q j , since Q j is ﬁnite. Now, passing to the limit as n → ∞ , w e obtain f ( q ) − f ( p j ) ≥ D j and d ( p, q ) ≤ (1 + δ j ) D j . F inally , since f is a ﬃne, there exists a unique p oin t z ∈ [ p j , q ] suc h that f ( z ) − f ( p j ) = D j . This conﬁrms that p j ∈ C j and the claim stands pro ve n. 11 W e claim that C j is c onve x . Indeed, let p 1 , p 2 ∈ C j , f ∈ A and z 1 , z 2 suc h that for i = 1 , 2 we hav e f ( z i ) − f ( p i ) = D j and d ( p i , z i ) ≤ (1 + δ j ) D j . Giv en p ∈ [ p 1 , p 2 ] at distance λd ( p 1 , p 2 ) from p 1 , where λ ∈ ( 0 , 1) , let z ∈ [ z 1 , z 2 ] b e the unique p oin t at distance λd ( z 1 , z 2 ) from z 1 . Since the distance function is con v ex, w e hav e d ( p, z ) ≤ (1 + δ j ) D j . F urthermore, since f is aﬃne w e ha v e f ( z ) − f ( p ) = D j . Hence p ∈ C j . W e claim that C j = X . Since A is Is( X ) -inv a r ia nt, so is C j . In view of the assumption of minimalit y on the Is( X ) -action, it follow s that C j is dense in X for all D j , δ j > 0 . Now the claim follo ws fro m a routine contin uity argumen t using the fact that f is 1 -Lipsc hitz. F or each f ∈ A and p ∈ C j , w e hav e |∇ p ( f ) | ≥ 1 1+ δ j since f is aﬃne. By the previous claim, the latter inequalit y holds for all p ∈ X and all δ j > 0 . Since f is 1 -Lipsc hitz it follows that |∇ p ( f ) | = 1 for all p ∈ X . Now Prop osition 4.2 yields a p o int ξ ∈ ∂ X to whic h the g radien t curv e t 7→ φ t ( p ) conv erges as t tends to inﬁnit y . Since the gradien t curv e as velocity 1 (see § 4.1 ) we deduce that it is a geo desic ray p oin ting to ξ . It follows that − f is a Busemann function asso ciated with ξ . In particular − f = lim n ¯ d ( φ n ( p ) , · ) b elongs to C , hence to A since f is aﬃne. This yields another p oint ξ ′ ∈ ∂ X and a geo desic ra y φ ′ t ( p ) p o in ting to ξ ′ . The concatenation of b oth rays is a geo desic line γ joining ξ ′ to ξ suc h that ( f ◦ γ ) ′ = 1 . A t this p oin t, [ LS07 , Lemma 4.1] yields the desire d splitting.  W e conclude this section with a tec hnical prop ert y of the space of functions C 0 v alid for arbitrary CA T(0) spaces. Lemma 4.10. L et X b e an y CA T(0) sp ac e and C 0 b e as ab ov e. Given a c om p act subset A ⊂ C 0 , if A do es not c ontain any aﬃne function, then the close d c onvex hul l C onv ( A ) do es not c ontain any aﬃn e function either. Pr o of. F or a n y f ∈ A w e ﬁnd some pair of p oin ts x + f , x − f ∈ X with midp oin t m f , suc h that ε f = f ( x + f ) + f ( x − f ) − 2 f ( m f ) > 0 . F or eac h f ∈ A , let U f b e the op en subset of A consisting of all h with h ( x + f ) + h ( x − f ) − 2 h ( m f ) > ε f / 2 . Since A is compact, ﬁnitely many U f i co v er A . Therefore, using the con v exit y of h , we deduce that r ( h ) := X i  h ( x + f i ) + h ( x − f i ) − 2 h ( m f i )  ≥ inf { ε f i } > 0 for a ll h ∈ A . Thus the con tin uous functional h 7→ r ( h ) is strictly p ositiv e on A , hence it is p ositiv e on the compact con v ex h ull of A . Therefore each f ∈ C onv ( A ) is non-aﬃne o n at least one of the geo desics [ x + f i , x − f i ] .  5. Fil tering f amilies of convex sets The purp ose of this section is to prov e Theorem 1.1 . W e start b y considering an analogous prop ert y for ﬁnite-dimens ional CA T(1) spaces. 5.1. CA T(1) case. W e start with the following analogue of the ﬁnite in tersection prop ert y fo r b ounded conv ex sets in CA T(0) spaces. Lemma 5.1. L et X b e a c omplete CA T(1) sp ac e of r adius < π 2 . Then any ﬁltering family { X α } α ∈ A of close d c onvex subsp ac es has a no n -empty interse ction. 12 Pr o of. Giv en [ BH99 , II.2.6 ( 1 ) and I I.2.7], the pro of is iden tical to that in [ Mon06 , Theorem 14].  Lemma 5.2. L et X b e a ﬁnite-dim e n sional CA T(1) sp ac e and { X i } i ≥ 0 b e a de- cr e asing se quenc e of close d c onvex subsets such that rad( X i ) ≤ π 2 . Then the inter- se ction T i X i is a non-empty subset of intrinsic r ad i us ≤ π / 2 . Pr o of. Let z i b e a cen tre of X i and Z = { z i | i ≥ 0 } . By assumption d ( z i , z j ) ≤ π 2 for all i, j . Since any ball of radius ≤ π 2 is con v ex, it follow s that the closed con v ex h ull C of Z ha s in trinsic ra dius ≤ π 2 . W e claim that r a d( C ) < π 2 . Otherwise w e hav e r a d( C ) = π 2 and ev ery z ∈ Z is a cen tre of C . Since the set of all centres is c onv e x , it f o llo ws that ev ery p oin t of C is a cen tre. This implies diam( C ) ≤ rad( C ) , whic h contradicts [ BL05 , Prop osition 1.2] and thereb y establishes the claim. Let C i b e the con v ex h ull of { z j | j ≥ i } . Then ( C i ) i ≥ 0 is a decreasin g sequence of closed conv ex subsets in a CA T(1) space of ra dius < π / 2 . By Lemma 5.2 , the in tersection Q = T i C i is non-empt y . Notice t ha t C i ⊆ X i whence Q ⊆ T i X i . The latter interse ction is thus non-empt y . F or eac h x ∈ T i X i w e ha v e d ( x, z j ) ≤ π 2 for a ll j . Th us C j is contained in the ball of radius π 2 around x . Therefore d ( x, q ) ≤ π / 2 for all x ∈ T i X i and q ∈ Q . This sho ws that T i X i has radius at most π 2 .  Prop osition 5.3. L et X b e a ﬁnite-dime n sional CA T(1) sp a c e an d { X α } α ∈ A b e a ﬁltering family o f close d c onvex subsets such that rad( X α ) ≤ π 2 for e ach α ∈ A . Then the interse ction T α ∈ A X α is a non-empty subset of intrinsic r ad i us ≤ π / 2 . Pr o of. W e pro ceed by induction on n = dim X . There is nothing to pro v e in dimension 0 , hence the induction can start. If dim( X 0 ) < n for some index 0 ∈ A , then the induction h yp othesis applied to the ﬁltering fa mily { X 0 ∩ X α } α ∈ A yields the desired conclusion. W e assume henceforth that dim( X α ) = n for eac h α ∈ A . F or β ∈ A , let z β b e a cen tre of X β . If d X α ( z β ) = π 2 for some β ∈ A , then the closed con ve x h ull o f z β and X α coincides with the spherical suspension of z β and X α (see [ Lyt05b , Lemma 4.1]) and hence has dimension 1 + dim( X α ) . This is absurd since dim( X α ) = dim( X ) . W e deduce d X α ( z β ) < π 2 for all α , β ∈ A . Assume no w that sup α ∈ A d X α ( z β ) = π 2 . Then there is a coun table sequenc e ( X α i ) i ≥ 0 with α i ∈ A suc h that lim i d X α i ( z β ) = π 2 . Up on replacing X α j b y T j i =0 X α i w e ma y and shall a ssume that the sequenc e ( X α i ) i ≥ 0 is decreasing. By Lemma 5.2 the in tersection Y = T i ≥ 0 X α i is a non-empty closed con v ex subset o f X . F ur- thermore b y deﬁnition w e ha v e d Y ( z β ) = π 2 . In particular, w e deduce b y the same argumen t as ab o ve t hat dim( Y ) < n . No w for each α ∈ A , w e apply Lemma 5.2 to the nested fa mily ( X α ∩ X α i ) i ≥ 0 , whic h sho ws that Y α = T i ≥ 0 ( X α ∩ X α i ) is a closed conv ex non-empt y subset of Y with in trinsic radius at most π 2 . Moreov er, the family { Y α } α ∈ A is ﬁltering and we ha v e T α Y α = T α ( X α ) . It follo ws b y induction that T α X α is non-empt y and of in trinsic radius at most π 2 , as desired. It remains to consider the case when r = sup α d X α ( z β ) < π / 2 . W e are then in a p osition to apply Lemma 5.1 to the ﬁltering family { B ( z β , r ) ∩ X α } α ∈ A . W e deduce tha t Y = T α X α is non-empt y . Moreo ve r, since d Y ( z β ) ≤ r < π 2 , w e deduce 13 b y considerin g the nearest p oin t pro jection o f z β to Y (see [ BH99 , I I.2.6(1)]) that rad( Y ) < π 2 .  5.2. CA T(0) case. W e start with the sp ecial case o f nested sequence s of con v ex sets. As p oin ted out to us by the referee, the use of the gradient ﬂow in the arg umen t b elo w is reminisce nt of the pro of of Lemma 5 on p. 217 in [ BGS85 ]. Lemma 5.4. L et X b e a c omplete CA T(0) sp ac e of telesc opic dime nsion n < ∞ and ( X i ) i ≥ 0 b e a neste d se quenc e of clo se d c onvex subsets such that T i ≥ 0 X i is empty. L et o ∈ X b e a b ase p oint and set f i : x 7→ d X i ( x ) − d X i ( o ) . Then the se quenc e ( f i ) i ≥ 0 sub-c on v e r ges to a 1 -Lipschitz c onvex function f which satisﬁes inf p ∈ X |∇ p ( − f ) | ≥ 1 2  1 − p n n +1  . Pr o of. The functions f i are 1 - Lipsc hitz and con v ex ( [ BH99 , I I.2.5(1)]), hence they are elemen ts of the space C 0 deﬁned in Section 4.3 . Since C 0 is compact, t he sequenc e ( f i ) indeed sub-conv erges to a f unction f ∈ C 0 . It remains to estimate the absolute gra dien t of f . Pic k a p oint p ∈ X . By assumption the inte rsection T i X i is empt y . Since b ounded closed con v ex sets enjo y the ﬁnite in tersection prop ert y (see Section 3.1 ), it follow s that d X i ( p ) tends to inﬁnit y with i . Th us for eac h t > 0 there is some N t suc h that d X i ( p ) > t for all i ≥ N t . W e ma y and shall assume without loss of generalit y that N 1 = 0 . Let x i denote the nearest p oint pro jection of p t o X i (see [ BH99 , Prop o si- tion I I.2.4]) and ρ i : [0 , d ( p, x i )] → X b e the geo desic path joining p to x i . Set D t = diam { ρ i ( t ) | i ≥ N t } . W e distinguish t w o cases. Assume ﬁrst that sup t D t < ∞ . It then follo ws that for all t > 0 , the sequence ( ρ i ( t )) i ≥ N t is Cauc h y . Denoting b y ρ ( t ) its limit, the map ρ : t 7→ ρ ( t ) is a geo desic ra y emanating from p . Therefore f = lim i f i is a Busemann function a nd w e ha v e |∇ p ( − f ) | = 1 . Th us w e are done in this case. Assume now that sup t D t = ∞ . Then D t tends to inﬁnity with t . Cho ose δ > 0 small enough so that √ 2 δ < 1 − p n n +1 and let D > 0 b e the constan t provid ed b y Theorem 1.3 . W e now pic k t large enough so that D t > D a nd set y i = ρ i ( t ) for a ll i ≥ N t . F or j > i we hav e ∠ x i ( p, x j ) ≥ π 2 and considering a comparison triangle for ∆( p, x i , x j ) w e deduce d ( y i , y j ) ≤ t √ 2 . Set Y = { y i | i ≥ 0 } . By Theorem 1.3 w e ha v e rad( Y ) ≤ t ( √ 2 δ + p n n +1 ) . Let z b e the circumcen tre of Y . W e hav e d X i ( z ) ≤ d X i ( y i ) + d ( y i , z ) and d ( y i , z ) ≤ rad( Y ) . Since moreo v er d X i ( p ) = d X i ( y i ) + t , w e deduce f i ( z ) − f i ( p ) = d X i ( z ) − d X i ( p ) ≤ d ( y i , z ) − t ≤ − t (1 − p n n +1 − √ 2 δ ) for eac h i ≥ 0 . Therefore f ( p ) − f ( z ) ≥ δ ′ t , where δ ′ = 1 − p n n +1 − √ 2 δ . On the other hand, w e ha ve d ( z , p ) ≤ d ( p, y i ) + d ( y i , z ) ≤ 2 t , thus f ( p ) − f ( z ) d ( p,z ) ≥ δ ′ / 2 . Since the restriction of − f to the geo desic segmen t [ p, z ] is conca v e b y assump- tion, w e deduce |∇ p ( − f ) | ≥ δ ′ / 2 . 14 Finally , recalling that δ ′ = 1 − p n n +1 − √ 2 δ and that δ > 0 may b e c hosen arbitrary small, the desired estimate follows .  Lemma 5.5. L et X b e a c omp l e te CA T(0) sp ac e of ﬁnite telesc opic dimension and ( X i ) i ≥ 0 b e a neste d se quenc e of clo se d c onvex subsets. If T i ≥ 0 X i is empty, then T i ≥ 0 ∂ X i is a non-e mpty subset of the visual b oundary ∂ X of in trinsic r adius ≤ π 2 . Pr o of. Let φ t : X → X denote the gradien t ﬂo w asso ciated to the con v ex function f deﬁned as in Lemma 5.4 . Prop osition 4.2 pro vides some p oin t ξ in the ideal b oundary ∂ X suc h that t he gradien t line t 7→ φ t ( p ) con v erges to ξ for an y starting p oin t p ∈ X . W e claim that ξ is con tained in ∂ X i for eac h i . T o this end, w e ﬁx an index i and consider the restriction h of f to X i . This is a con ve x function on X i and it is suﬃcien t to prov e that the gradien t ﬂow o f h coincides with the gr a dien t ﬂow of f starting at any p oin t of X i . Hence it is enough to pro ve t hat for all p ∈ X i the equalit y |∇ p ( − f ) | = | ∇ p ( − h ) | holds. Pic k a p oint x ∈ X and let x i denote the nearest p oint pro jection of x to X i . W e ha v e d X j ( x ) ≥ d X j ( x i ) and d ( p, x ) ≥ d ( p, x i ) for all p ∈ X i . Hence f or p ∈ X i and all j ≥ i w e get the inequalit y f j ( p ) − f j ( x ) d ( p, x ) ≤ f j ( p ) − f j ( x i ) d ( p, x i ) . Hence the same is true for the limiting function f , whic h implies the desired equalit y |∇ p ( − f ) | = | ∇ p ( − h ) | . This sho ws that ξ is con tained in the inters ection T i ∂ X i , whic h is th us non-empt y . F or any g eo desic ray η in X with endp oin t in T i ∂ X i , the restriction of f i to η is b ounded from ab ov e, hence non-increasing. Therefore the same holds true for the restriction of the limiting function f to the ray η . In other words the endp o in t of η is f -monoto ne. F rom Lemma 4.5 w e deduce that d ( ξ , ψ ) ≤ π / 2 for all ψ ∈ ∩ ∂ X i .  Pr o of of The or em 1.1 . Pic k a base p oin t o ∈ X . If the set { d X α ( o ) } α ∈ A is b ounded, then T α X α has a non-empt y in tersection b y the ﬁnite in tersection prop erty (see Section 3.1 ). W e assume henceforth that this is not the case. In particular there exists a sequence of indices ( α n ) n ≥ 0 suc h that lim n d X α n ( o ) = ∞ . Now for eac h α ∈ A , w e ma y apply Lemma 5.5 to the nested seque nce ( X α ∩ X α n ) n ≥ 0 . This sho ws that Y α = T n ≥ 0 ∂ ( X α ∩ X α n ) is a non- empt y subset of in trinsic r a dius ≤ π 2 of ∂ X . Notice that { Y α } α ∈ A is a ﬁltering family . Prop osition 2.1 then allows one to app eal to Prop osition 5.3 , whic h sho ws that T α Y α is a non-empt y subset of in trin- sic radius ≤ π 2 . This provide s the desired statemen t since T α ∂ X α = T α Y α .  W e end this section by an example illustrating that Theorem 1.1 fa ils if one assumes only that the Tits b o undary ∂ X b e ﬁnite-dimensional. Example 5.6 . Let H b e a separable (real) Hilb ert space with orthonormal basis { e i } and X ⊂ H b e the subset consisting of all po in ts P i a i e i with | a i | ≤ i for all i . Thu s X is a closed conv ex subset of H with empty (hence ﬁnite-dimensional) ideal b oundary . Let no w X n = { P i a i e i ∈ X | a i ≥ 1 for all i ≤ n } . Then { X n } is a nested family of closed con v ex subsets with empt y in tersection. 15 6. Applica t ions 6.1. P arab olic isometries. Pr o of of Cor ol lary 1.5 . By Prop osition 2.1 , the b oundary ∂ X is ﬁnite-dimensional. The sublev el sets of the displacemen t function of g deﬁne a Z Is( X ) ( g ) -in v ariant nested sequence of closed con v ex subspace. The in tersection of their b oundaries is nonempt y by Theorem 1.1 and p ossesses a barycen tre by [ BL0 5 , Prop. 1.4], whic h is the desired ﬁxed p oint.  6.2. Minimal and reduced actions. W e b egin with a de Rham t yp e decom- p osition prop ert y . It w as sho wn by F o ertsc h–Lytc hak [ FL0 8 ] that any ﬁnite- dimensional CA T(0) space (and more generally any geo desic metric space of ﬁnite aﬃne ra nk) admits a canonical isometric splitting in to a ﬂat factor and ﬁnitely man y non-ﬂat irreducible factors. Buildin g up on [ FL08 ], it w as then sho wn by Caprace–Mono d [ CM09 , Corollary 4.3(ii)] that the same conclusion holds for prop er CA T(0) spaces whose isometry group acts minimally , assuming that the Tits b oundary is ﬁnite-dimensional. W e shall need the follo wing ‘improper’ v a ri- ation of this result. Prop osition 6.1. L et X b e a c omplete CA T(0) sp ac e of ﬁnite telesc opic dime n - sion, such that Is( X ) acts minimal ly. The n ther e is a c a nonic a l maximal isometric splitting R n × X 1 × · · · × X m wher e e a c h X i is irr e d ucib le, unb ounde d and 6 ∼ = R . Every isometry pr es erves this de c omp osition up on p e rmuting p o s s ibly isometric factors X i . Pr o of. Let H b e a separable Hilb ert space with orthonormal basis { e i } i> 0 and denote b y C k the con v ex hu ll of the set { 0 } ∪ { 2 i e i | 0 < i ≤ k } . Let now X b e a CA T(0) space suc h that for ev ery isometric splitting X = X 1 × · · · × X p with each X i un b ounded, some factor X i admits an isometric splitting X i = X ′ i × X ′′ i with un b ounded factors. Then there is a p oin t o ∈ X and fo r eac h k > 0 an isometric em b edding ϕ k : C k → X with ϕ (0 ) = o . Since for all k > 0 the set 2 .C k em b eds isometrically in C k +1 , it follows that C k em b eds isometrically in the asymptotic cone lim ω ( 1 n X , o ) . In particular X do es not hav e ﬁnite telescopic dimension. This sho ws that an y CA T(0) space of ﬁnite telescopic dimension admits a m aximal isometric splitting in to a pro duct of ﬁnitely many unbounded (necessarily irre- ducible) subspaces. In view of the latter observ ation a nd giv en Prop osition 2.1 , the pro of of [ CM09 , Corollary 4.3(ii)] applies verb atim and yields t he desired conclusion.  Pr o of of Pr o p osition 1.8 . (i) W e claim that the statemen t of (i) follow s from (ii) and (iii). Indeed, if G has no ﬁxed p oin t at inﬁnit y , then there is a minimal non-empt y G -inv ariant subspace Y ⊆ X by (ii). Up on replacing G b y a ﬁnite index subgroup, this subspace Y admits a G - equiv a rian t decomposition as in Prop osition 6.1 . The induced action of G on eac h of these spaces is minimal without ﬁxed p oint at inﬁnit y . Therefore, it is non-ev anesce nt b y (iii), unless Y is b ounded, in whic h case it is reduced t o a single p o in t b y G -minimalit y . This means that G ﬁxes a p oint in X . (ii) Assume that G has no minimal in v a rian t subspace. By Zorn’s lemma this implies that there is a c hain of G -inv ariant subspaces with empty in tersection. 16 By Theorem 1.1 the in tersection o f the b oundaries at inﬁnit y of the subspaces in this c hain provi de a closed con v ex G -in v ariant set Y ⊆ ∂ X o f radius ≤ π 2 . By Prop osition 2.1 , the set Y is ﬁnite-dimensional. Hence it p ossesses a unique barycen tre by [ BL05 , Prop. 1 .4], whic h is th us ﬁxed b y G . By Prop osition 2.1 the b oundary ∂ X is ﬁnite-dimensional. Therefore, for (iii) and (iv), Theorem 1.1 (in fact, Lemma 5.5 is suﬃcien t) allows one to rep eat verb atim the pro ofs of the corresponding statemen ts that are giv en in [ CM09 ], namely Theorem 1.6 in l o c. cit. for the fa ct that normal subgroups act minimally without ﬁxed p oint a t inﬁnit y , Corollary 2.8 in lo c. cit. for the f act that the G -action is reduced and Prop osition 1.3(i) in lo c. c it. for the fact tha t X is b oundary-minimal prov ided Is( X ) a cts minimally .  Pr o of of Cor ol lary 1.9 . By Prop osition 6.1 the space X admits a canonical decom- p osition as a pro duct of ﬁnitely man y irreducible fa ctor s. The lat t ice Γ admits a ﬁnite index normal subgroup Γ ∗ whic h acts comp onen t wise on t his decomp osition (the ﬁnite quotien t Γ / Γ ∗ acts by p ermuting p o ssibly isometric irreducible factors). Let G ∗ i b e the closure o f the pro jection of Γ ∗ to G i and set G ∗ = G ∗ 1 ×· · ·× G ∗ n . Th us G ∗ is a closed normal subgroup o f ﬁnite index of G and w e ha v e G = Γ · G ∗ . In par- ticular it is suﬃcien t to show that the Γ ∗ -action extends to a contin uous G ∗ -action. T o this end, w e w ork one irreducible factor at a time. Given Prop osition 1.8 (iii), the desired contin uous extension is pro vided by [ Mon06 , Theorem 6].  6.3. Isometric actions of amenable groups. Pr o of of The or em 1.6 . Assume that G has no ﬁxed p oin t at inﬁnity . Th us there is a minimal closed conv ex in v a rian t subset b y Prop osition 1.8 (ii) and we ma y assume tha t this subset coincides with X . In other w ords G acts minimally o n X . Let X = E × X ′ b e the maximal Euclidean decomp osition (see [ BH99 , I I.6.15(6)]). Th us G preserv es the splitting X = E × X ′ and the induced G -action on b oth E and X ′ is minimal and do es not ﬁx any p oin t at inﬁnit y . W e need t o sho w that X ′ is reduced t o a single p oint. T o this end, it is thu s suﬃcien t to establish the follo wing claim. If an amenab l e lo c a l ly c omp act gr o up G acts c ontinuously, minimal ly and wi th- out ﬁxe d p o i n ts at inﬁ nity on a CA T(0) sp ac e X of ﬁnite telesc opic dime n sion without Euclide a n factor, then X i s r e duc e d to a si n gle p oint. Assume that this is not the case. Pic k a base p o int o ∈ X and consider the spaces C ⊂ C 0 deﬁned in Subsection 4.3 . Let A denote the closed con v ex h ull of C in the lo cally conv ex top olo g ical vec tor space B of all f unctions v anishing at o . By Prop osition 6.1 the subset C do es not con tain any a ﬃne function. It follo ws from Lemma 4.10 t ha t A do es not con tain an y aﬃne function either. The induced action of G on B is contin uous and preserv es the compact con v ex set A . By t he deﬁnition of amenabilit y G has a ﬁxed p oin t in A . Th us w e ha v e found some non -c onstant 1 - L ipsch itz con v ex function f whic h is quasi-in v arian t with respect to G in the sense that, for eac h g ∈ G , one has f ( g x ) = f ( x ) + f ( g o ) . (In other w ords, this means that for eac h g , the map x 7→ f ( g x ) − f ( x ) is constan t.) The follow ing lemma, analogous to [ AB98 , Lemma 2.4], implies that G has a ﬁxed p oin t at inﬁnit y , whic h is absurd.  17 Lemma 6.2. L et a gr oup G act minimal ly by isom etries on a c omplete CA T(0) sp ac e X of ﬁ n ite telesc op ic dim e nsion. Ther e is a G -quasi-invariant c on tinuous non-c onstant c onvex function f on X i f and only if G ﬁx e s a p oin t in ∂ X . Pr o of. If G ﬁxes a p oin t in ∂ X , then the Busemann function of this p oint (that is uniquely deﬁned up to a p ositiv e constan t ) is quasi-in v arian t. Assume that f is quasi-in v arian t a nd deﬁne a : G → R b y a ( g ) = f ( g x ) − f ( x ) . By assumption a do es not dep end on x ; f urthermore a is a homomorphism. If a w ere constan t, then f would b e G -in v arian t and, hence, so would b e any sub-lev el set of f . This contradicts the minimalit y assumption on the G -action. Therefore a is non-constan t; mor e precisely the image of a is unbounded and inf f = −∞ . F or each r ∈ R set X r := φ − 1 ( −∞ , − r ] . Then ( X r ) r ∈ R is a ch ain of closed conv ex subspaces with empt y in tersection; furthermore eve ry elemen t of G p erm utes the sets X r . It follo ws that C = T r ∈ R ∂ X r is G -inv ariant. Theorem 1.1 now sho ws that C is no nempty of radius ≤ π 2 , and [ BL05 , Prop. 1.4] implies that G ﬁxes a p oin t in C ⊂ ∂ X .  Pr o of of The or em 1.7 . The pro of mimic ks the arguments giv en in [ Cap09 ]; we do not repro duce all the details. As in lo c. cit. the ke y p oint is to establish that ev ery p oint of the reﬁned b oundary ∂ ﬁne X (deﬁned in lo c. cit. , §4.2) has an amenable stabiliser in G and that, con ve rsely , a ny amenable subgroup of G p ossesses a ﬁnite index subgroup whic h ﬁxes a p oin t in X ∪ ∂ ﬁne X . The pro of that amenable groups stabilise p oin t in X ∪ ∂ ﬁne X uses Theorem 1.6 together with an induction on the geometric dimension (see the remark following Corollary 4.4 in lo c. cit. sho wing that there is a uniform upp er-b o und on the lev el of a p oint in the reﬁned b o undary). F or the conv erse, one sho ws directly that the G -stabiliser of a p o in t in ∂ ﬁne X is (top ologically lo cally ﬁnite)-b y-(virtually Ab elian); the co compactness argumen t used in Prop osition 4.5 of lo c . cit. is replaced b y a compactness argumen t relying on the hypothesis that X has ﬁnitely man y t yp es of cells, all of whic h are compact.  Pr o of of Cor ol lary 1.10 . By Prop osition 6.1 , the space X admits a canonical de- comp osition as a pro duct of a maximal Euclidean f actor and a ﬁnite n um b er of irreducible non-Euclidean factors. The Euclidean factor is G -in v a ria n t and G p o s- sesse s a closed normal subgroup of ﬁnite index G ∗ that acts comp onen t wise on the ab ov e pro duct. By hypothesis, the G ∗ -action on eac h non-Euclidean factor is minimal and do es not ﬁx any p oin t at inﬁnity . Theorem 1.6 and Prop osi- tion 1.8 (iii) therefore imply that the amenable ra dical o f G ∗ acts trivially . This implies that the amenable radical of G acts as a ﬁnite group o n the pro duct of all non-Euclidean factors of X . Th us this action is trivial since G acts minimally .  References [AB98] Scot Adams and W erner B a llmann, Amenable isometry gr oups of Hadamar d sp ac es , Math. 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Lytchak, Op en map t he or em for metric sp ac es , Algebra i Analiz 17 (2005), no. 3, 139–1 59. ↑ 7 , 8 [Lyt05b] , Rigi dity of spheric al buildings and joins , Geo m. F unct. Anal. 15 (2005), no. 3, 720–7 52. ↑ 5 , 13 [May98] Uwe F. Mayer, Gr adient ﬂows on nonp ositively curve d metric sp ac es and harmonic maps , Comm. Anal. Geom. 6 (199 8 ), no . 2, 199– 2 53. ↑ 7 , 8 [Mon06] Nicolas Mono d, Sup errigidity for irr e ducible lattic es and ge ometric splitting , J. Amer. Math. So c. 1 9 (2006 ), no . 4, 781–8 14. ↑ 2 , 4 , 6 , 13 , 1 7 [Šar77] V. A. Šarafutdinov, The Po gor elov-Klingenb er g the or em for manifolds that ar e home- omorphic to R n , Sibirsk. Mat. Ž. 18 (197 7), no. 4, 915– 925, 95 8 (Russian). ↑ 8 19 Erratum to ‘A t inﬁnit y of ﬁnite-di m ensional CA T(0) spaces’ Pierre-Emman uel Caprace and Alexander Lytc hak First draft Dece mber 2012 ; revised Ma y 2014 A ga p in the pro ofs of Prop ositions 1.8 and 6 .1 from our pap er [ CL10 ] w as p oin ted out to us b y Pierre Py: w e ov erlo oke d the p ossibilit y that a complete CA T(0) space X of ﬁnite telescopic dimension could b e un b ounded, ha v e a n isom- etry group acting minimally , and nev ertheless hav e an empty visual b oundary . Although it is still not clear to us whether this situation can actually o ccur 1 , w e sho w in this note how to ov ercome the question and complete the pro ofs whic h w ere ﬂa w ed in the orig inal pap er (only the pro ofs of Prop. 1.8 and 6.1, a nd of Corollaries 1.9 and 1.10, whic h we re all g iv en in the ﬁnal section of the original pap er [ CL10 ], are concerned b y t hese adjustmen ts). Besides Prop osition 1.8 whic h has to b e sligh tly corrected (see Prop osition 1 b elow ), all the other statemen ts of the pap er are correct and remain unc hanged. W e thank Pierre Py fo r p oin ting o ut the gap. W e a r e also gra t eful to him f or letting us know of the unpublished manus cript [ KS99 ] b y K orev a ar–Sc ho en, which w e w ere not aw are of while writing our pap er. In [ KS99 ], a class of CA T(0) spaces called FR-spaces , is in tro duced. It turns out that a CA T(0) space is an FR-space in the sense of K orev aar– Sho en if and only if it has ﬁnite telescopi c dimension in the sense of our pap er [ CL10 ]: this equiv alence can b e deduced from Theorem 1.3 from [ CL10 ]. Giv en this equiv alence, one sees that Prop osition 1 fro m [ KS99 ] o v erlaps somewhat Theorem 1 .1 from [ CL10 ]. The condition of ﬁnite telescopic dimension seems ho w ev er somewhat easier to c hec k in concrete settings; for exam- ple it w as used by B. Duc hesne [ Duc12 ], in com bination with the aforemen tioned equiv a lence, to show t ha t inﬁnite-dimensional symmetric spaces of ﬁnite rank are FR-spaces, thereb y answ ering a ques tion ask ed b y K orev aar– Sho en in [ KS99 ]. By conv enti on, all the num b ered references of the form ‘Theorem 1.1 ’ in this note refer to the corresp onding results from [ CL10 ]. W e moreo v er retain the terminology used in lo c. cit. Prop osition 1.8 m ust b e replaced by t he follo wing. Prop osition A.1. L et X b e a c omplete CA T(0) sp ac e of ﬁnite telesc opi c dimen- sion and G < Is( X ) b e any gr oup of isometries. (i) If the G -action is minimal, then it is evan esc ent if and only if it ﬁxes a p oint in ∂ X . (ii) If G do es not ﬁx a p oint in ∂ X , then ther e is a non-empty G -invariant close d c onv e x subset Y ⊆ X on which G acts minimal ly. (iii) Assume that X is irr e ducible, not isometric to the r e al line R . If G a c ts minimal ly without a ﬁ xe d p oint at inﬁn i ty, then so do es every non-trivial normal sub gr oup. Mor e over, for any unb ounde d close d c on v e x subset T which is mapp e d at b ounde d Hausdo rﬀ distanc e fr om itself b y e ach element of G , we must have X = T unless ∂ T = ∅ and T is not Is( T ) -m i n imal. 1 This question has recent ly been answered neg atively by Bader –Duchesne–Lécureux in [ BDL14 , Th. 1 .2 ]. This ﬁlls in the gap from [ CL10 ] in an optimal wa y , thereby making the arguments develop e d in the present er ratum obsolete. This note is ther e fo re not intended fo r publication. (iv) If Is( X ) acts m i nimal ly, then X admits a unique p r o duct de c omp osition X = X ′ × T wher e X ′ is b oundary-minimal and T has empty b oundary. A closed con v ex subset Y of a CA T(0) space X is called b oundary-mi nimal if ∂ Z ( ∂ Y for ev ery closed conv ex subse t Z ( Y . The pro of o f Prop osition A.1 requires some preparation. Ultracompletions. W e shall use the ultracompletion of a CA T(0) space X , with resp ect to an ultraﬁlter ω . It is denoted b y X ω and deﬁned a s the ultralimit lim ω ( X , o ) o f the constan t sequence of p ointed metric spaces ( X , o ) . The ultra com- pletion is also called ultrapro duct or ultrap ow er by some authors; w e prefer the term ultr ac ompletion , notably b ecause of its basic prop erties, collected in the follo wing, whic h strongly suggest t hat the space X ω is some kind of completion of X . Lemma A.2. L et X b e a CA T(0) sp ac e. (i) X ω is a c omplete CA T(0) sp a c e in whic h X emb e ds c anonic al ly as a c onv e x subsp ac e. Mor e over any isometric action of a gr oup G o n X extends to an isometric action of G on X ω . (ii) X is unb ounde d if and only if ∂ ( X ω ) 6 = ∅ . (iii) If ω 1 and ω 2 ar e ultr aﬁlters then the iter ate d ultr ac omple tion ( X ω 1 ) ω 2 is (c ano nic al ly) isome tric to an ultr ac om pletion of X w ith r esp e c t to the ul- tr aﬁlter ω 1 × ω 2 . (iv) The telesc opic dimensions of X a nd X ω c oinc ide. (v) If X h a s a ﬁnite telesc opic dime n sion d , then for any ultr aﬁlter ω , the ge o- metric dimension of ∂ ( X ω ) is at most d − 1 . Pr o of. F or (i) and (ii), see Cor. I I.3.10 in [ BH ]. F or (iii) see [ DS05 , §3.2]. (iv) is a consequence of Theorem 1.3. T aking ultralimits of CA T(0) spaces do es not in- crease the geometric dimension b y Lemma 11 .1 from [ Lyt05 ]; therefore (v) follows from Prop. 2.1 .  Lemma A .3. L et X b e a CA T(0) sp ac e of ﬁnite telesc opi c dimensi o n. Then X ω do es not c ontain an y c opy of X × I with I an interval of p ositive length. Pr o of. Otherwise, an iterated ultracompletion con tains X × I n . Notice that the n -dimensional Euclidean cub e of edge length √ 2 c em b eds isometrically in the 2 n - dimensional Euclidean cub e of edge length c . It follo ws t ha t for n large, the space X × I n con tains an arbitrar y larg e Euclidean ball of arbitrary large dimension. But an iterated ultracompletion is an ultracompletion for a nother ultraﬁlter ω ′ b y Lemma A.2 (iii). Therefore X cannot ha v e ﬁnite telescopic dimension (b y Theorem 1.3), a contradic tion.  Lemma A .4. L et G < Is( X ) act minimal l y on a CA T(0) sp ac e X of ﬁnite tele- sc opi c dimensio n, with no ﬁxe d p oin ts in ∂ X . Then the extende d action of G on X ω do es not have ﬁxe d p oints in the (lar ge r) b oundary ∂ ( X ω ) . Pr o of. Otherwise, the correspo nding Busemann function on X ω is quasi-in v ariant under G . By Lemma 6.2, it m ust b e constan t on X . It follo ws that the pro duct X × [0 , ∞ ) em b eds isometrically in X ω , con tradicting Lemma A.3 .  Lemma A.5. L et G < Is( X ) ac t min i m al ly on a c omple te CA T(0) sp ac e X of ﬁnite telesc opic dim ension. The n any G -invariant close d c on v ex subset of X ω c ontains (the c a n onic al) subset X o f X ω . Pr o of. Let C b e the collection of all G - in v ariant closed con ve x subspace s of X ω on whic h G -a cts minimally . Then C is non-empt y since it contains X . An easy argumen t using the Sandwic h Lemma ([ BH , I I.2.12 (2)]) then sho ws that ev ery G -in v ar ia nt closed con v ex subspace con tains a minimal one, whic h is thus an elemen t of C . Moreov er, the union S C is a closed con v ex subspace whic h splits as a pro duct o f the fo r m X × T (see [ Mon06 , R em. 39]). By Lemma A.3 , the cross-section T m ust b e reduced to a single p oin t. Hence C = { X } .  Ev anescen t actions. Lemma A .6. L et G < Is( X ) act minimal l y on a CA T(0) sp ac e X of ﬁnite tele- sc opi c dimension. The G -action is evanesc ent if and only if G ﬁxes a p oint in ∂ X . Pr o of. The ‘if ’ direction is clear. F or the rev erse implication, consider an un- b ounded subset T ⊂ X , on whic h all elemen ts o f G hav e ﬁnite displacemen t functions. Since the displacemen t functions of isometries are conv ex, they r emain ﬁnite on the con v ex hull of T , and we ma y thus assume that T is con v ex. Then G ﬁxes all p oin ts in ∂ ( T ω ) ⊂ ∂ X ω , whic h con tradicts Lemma A.4 since ∂ ( T ω ) is non-empt y b y Lemma A.2 (ii).  Minimal actions and b oundary-mini malit y. Lemma A.7. L et X b e a c ompl e te CA T(0) sp ac e of ﬁnite telesc opic d imension and Y ⊆ X b e a cl o se d c o nvex subsp ac e. Assume that the b oundary ∂ Y has r adius > π / 2 . Then X c ontains some b oundary-minimal c l o se d c onvex subsp ac e Y ′ with ∂ Y ′ = ∂ Y . Mor e over, the union of al l such subsp ac es spli ts as a pr o duct Y ′ × T , and if ∂ Y = ∂ X then T has empty visual b ounda ry. Pr o of. An y chain of closed con v ex subspaces with visual b oundary equal to ∂ Y has a non-empt y inte rsection, since otherwise ∂ Y w ould ha v e radius ≤ π / 2 b y Theorem 1.1. Th us the existence of Y ′ follo ws from Zorn’s lemma. That the union of a ll b oundary-minimal subspaces with a giv en b oundary is a pro duct space is true in arbitrary CA T(0) spaces, a nd can b e shown as in [ CM09 , Prop. 3.6].  Lemma A.8. L et X b e a c omplete CA T(0) s p ac e of ﬁnite telesc opic dimension, such that Is( X ) acts min i m al ly. Then either ∂ X ha s r adius > π / 2 , or ∂ X is empty. Pr o of. Assume for a contradiction that ∂ X is non-empt y of radius ≤ π / 2 . By Prop osition 2.1 and [ BL05 , Prop. 1.4], the gro up G = Is( X ) ﬁxes some p oin t ξ ∈ ∂ X , and the ball of radius π / 2 around ξ is the entire b oundary . As in [ CM09 , Prop. 3.11], one then sho ws that G preserv es all hor o balls cen tered at ξ , con tradicting minimalit y .  Lemma A.9. L et X b e a c ompl e te CA T(0) sp ac e of ﬁnite telesc opic d imension not r e duc e d to a si ngle p oint, such that Is( X ) ac ts m i nimal ly. Then ∂ ( X ω ) has r adius > π / 2 . If in a d dition the ultr aﬁlter ω is chosen so that the di m ension o f ∂ ( X ω ) is maximal (s uch a choi c e of ω exists b y L emma A.2 (v)), then ther e is a unique b oundary-mi nimal subsp ac e X ′ of X ω with ∂ X ′ = ∂ ( X ω ) . Mor e over X ′ c ontains X . Pr o of. If X is b ounded, then it is a p oint b y minimalit y , whic h is excluded b y h yp othesis. W e assume henceforth that X is unbounded, hence ∂ ( X ω ) is non- empt y a nd of ﬁnite geometric dimensi on by Lemma A.2 (ii) and (v). Assume for a con tradiction that the ra dius of ∂ ( X ω ) is smaller than or equal to π / 2 . Then G = Is( X ) ﬁxes some circumcen t re ξ in ∂ ( X ω ) by [ BL05 , Prop. 1.4]; the whole b oundary ∂ ( X ω ) is thus con tained in the ball of radius π / 2 around ξ . By [ CM09 , Lem. 3.12], this implies that G stabilizes each horoball around ξ . W e deduce f ro m L emma A.5 that X ⊂ X ω is contained in the inters ection of a ll these horoballs, whic h is absurd. Hence ∂ ( X ω ) has radius > π / 2 . By Lemma A.7 there is a b o undary minimal subspace X ′ of X ω with full b ound- ary , and the union of a ll suc h subspaces has the f o rm Y = X ′ × T , b y Lemma A.7 . W e no w assume that ω is chos en so that dim( ∂ ( X ω )) is maximal. If T is un b ounded, then t he ideal b oundary of an y ultra completion of Y contains the spherical join o f ∂ X ′ = ∂ ( X ω ) with a singleton. Suc h a subspace has larger dimension than ∂ X ω , whic h con tradicts the c hoice of ω . Therefore, the cross-section T is b ounded. Notice that the G -action on X ω preserv es Y and its pro duct decomp osition. Since T is b ounded, the induced action of G o n T has a ﬁxed p oin t. Th us G preserv es some ﬁb er along X ′ in the pro duct X ′ × T . By Lemma A.5 this ﬁb er m ust con tain X , hence T m ust b e reduced to a single p oint by Lemma A.3 . Thus Y = X ′ and X ⊂ X ′ .  Lemma A.10. L et X b e a c omplete CA T(0) sp ac e of ﬁnite telesc opic d imension. If Is( X ) acts minim al ly, then X ad m its a unique pr o duct de c om p osition X = X ′ × T wher e X ′ is b oundary-minimal and T has empty b oundary. Pr o of. If ∂ X is non-empt y , then it has radius > π / 2 b y Lemma A.8 . W e ma y then in v ok e L emma A.7 , whic h provide s a canonical pro duct decomp osition X = X ′ × T , where X ′ is b oundary-minimal and T is unbounded and has empt y b oundary . If ∂ X is empt y , the desired decomp osition holds trivially by setting T = X and X ′ a singleton.  Reduced actions. F ollowing N. Mono d [ Mon06 ], the action of a gro up G on a CA T(0) space X is called reduced if there is no unbounded closed conv ex subset T ( X whic h is mapp ed a t b ounded Hausdorﬀ distance f rom itself by eve ry elemen t of G . Lemma A.11. L et G < Is ( X ) act minima l ly on a c omplete CA T(0) sp ac e X of ﬁnite telesc opic di m ension, w ithout a ﬁxe d p oint at in ﬁ nity. Assume tha t X is irr e ducible, and let T b e an unb ounde d close d c onvex subset, such that d ( g T , T ) < ∞ , fo r al l g ∈ G . I f ∂ T is non-em pty, or if Is( T ) acts min i mal ly on T , then T = X . Pr o of. If the b oundary o f T is non-empt y , our old pro of applies: ∂ T is a G - in v ariant closed con ve x subset of the b oundary , and m ust b e of radius > π / 2 b y [ BL05 , Prop. 1.4]. Lemma A.7 then show s that the (non-empt y) union of all b oundary-minimal closed con v ex subsets with b oundary ∂ T is a pro duct space, whic h is G - inv a r ia nt. Since G acts minimally and X is irreducible, it follo ws that X is b oundary-minimal and that ∂ X = ∂ T . In particular X = T . Assume now that T is Is( T ) - minimal. Ev ery elemen t of G maps the subset T ω of X ω at b ounded Hausdorﬀ distance from itself. Hence an y elemen t of G preserv es the b oundary S = ∂ T ω , whic h is non-empt y by Lemma A.2 (ii) and has radius > π / 2 b y [ BL05 , Prop. 1.4] and Lemma A.4 . W e are thus in a p osition to apply Lemma A.7 to the subspace T ω ⊆ X ω . This sho ws that the (non-empt y) union of all b oundary-minimal closed conv ex subse ts with b oundary S is a pro duct space, whic h is G -inv ariant. Th us this union has the form T ′ × T ′′ , where T ′ is some b o undary-minimal subset of T ω . By Lemma A.9 , and up to replacing ω b y an ultraﬁlter making dim( ∂ T ω ) maximal, the h yp o thesis that T is Is( T ) - minimal implies that T ′ is unique. More imp ort a n tly , it contains T . Remark that for an y CA T(0) space X and an y con v ex subspace T , the nearest p oin t pro jection of X to T ω inside X ω has T as its image, due t o the ve ry deﬁnition of distances in ultracompletions. Th us we ha v e T ⊆ T ′ ∩ X ⊆ π T ′ ( X ) ⊆ π T ω ( X ) = T , where π Z denotes the nearest p oin t pro j ection to Z . It fo llo ws t hat π T ′ ( X ) = T . On the other hand, the space X is con tained in the G -in v ariant space T ′ × T ′′ , whose pro duct decompo sition is G -inv ariant. Moreov er, on all p o in ts of T ′ × T ′′ , the pro jection to the ﬁrst fa ctor T ′ × T ′′ → T ′ coincides with the nearest p oin t pro jection π T ′ . W e deduce from the preceding paragraph that X is in fa ct con tained in T × T ′′ . Let ﬁnally Z ′′ ⊂ T ′′ b e the set of those z ∈ T ′′ suc h that the T -ﬁb er thro ug h z in the pro duct T × T ′′ is entirely con tained in X . Since T ⊂ X , the set Z ′′ is non-empt y . Moreov er Z ′′ is con v ex (b y the Sandwic h Lemma) and closed (b ecause X is complete). By construction the pro duct T × Z ′′ is then a G -inv ariant closed con v ex subset contained in X . By minimalit y of the G -action it coincides with X , hence T = X b y irreducibil ity .  Pro of of Pr op osition A.1 . (i) is prov ed in Lemma A.6 . (ii) f ollo ws from The- orem 1.1 and [ BL05 , Prop 1.4], as in the original argumen t. (iv) was pro ve d in Lemma A.10 . The part o f assertion (iii) concerning reduced actions w as prov ed in Lemma A.11 . It remains to prov e the statemen t on normal subgroups. Giv en a normal subgroup N < G , there are tw o p ossibilities: either there is some non- empt y closed con v ex subset Y ⊆ X whic h is N -in v ar ia nt and on whic h N acts minimally , or there is no suc h, whic h implies by (ii) that N ﬁxes a p oint in ∂ Y for any N - in v ariant closed con v ex subset Y ⊆ X b y (ii). In particular ∂ Y is non-empt y . Giv en this observ ation, one can pro ceed with the same argumen t a s in the original pro of.  de R ham decomp ositions. In order to ﬁll in the gap in the pro of of Prop osi- tion 6.1, w e ﬁrst record an elemen tary fact. Lemma A .12. The telesc opic dimension of a pr o duct o f two sp ac es is the sum of the c o rr esp onding telesc opic di m ensions. Pr o of. The statemen t is true for the geometric dimension. The pro duct decom- p osition is stable under rescalings and ultralimits.  The follo wing statemen t is Prop osition 6.1, whic h w e repro duce here for the reader’s con v enience. Prop osition A .13. L et X b e a c omplete CA T(0) s p ac e of ﬁnite telesc op i c dime n- sion, such that Is( X ) acts minima l ly on X . Then X ha s a c anonic al ﬁn i te p r o duct de c omp osition X = R n × X 1 × ... × X m , wher e e ach X i is irr e ducible, unb ounde d and not isometric to R . Pr o of. By Lemma A.10 , it is therefore suﬃcien t to prov e the prop osition for b oundary-minimal spaces, and fo r spaces with an empt y b oundary . In the former case, w e may pro ceed as in the original argument and g et the conclusion. In order to treat the latter case, w e ma y assume that X is un b ounded and has empt y visual b oundary . W e claim that X do es not ha v e an y b o unded non-trivial factor. Assume the con trary and write X = X 0 × C with a b ounded space C . Then ∂ ( X ω 0 ) = ∂ ( X ω ) . Th us X ω 0 con tains a b oundary-minimal subset X ′′ of X ω with full b oundary . F or diﬀeren t p oints c 1 , c 2 ∈ C we obtain disjoint b oundary-minimal subsets X ′′ × { c i } of X ω , a contradiction. This prov es the claim. In par ticular any non-trivial factor of X has strictly p ositiv e telescopic dimens ion. Due to Lemma A.12 , an y pro duct decomp osition o f X has at most k non- trivial factors, where k is the telescopic dimens ion of X . Th us there is some ﬁnite decompo sition of X with irreducible non-trivial fa ctors, whic h are all un b ounded. It remains to pro v e that suc h a decomp osition is canonical. By Lemma A.2 (ii) the space X ω has non-empt y visual b oundary , and Lemma A.9 implies (up on changing the ultraﬁlter ω ) that there is a unique b oundary-minimal subspace X ′ of X ω . The b oundary-minimal subset X ′ of X ω admits a canonical pro duct decom- p osition b y the ﬁrst part o f the pro o f. W e next claim that X ′ do es not hav e a Euclidean factor. Otherwise, the pro j ection to a line factor of X ′ is an aﬃne function f . Since X do es not hav e Euclidean f actors, the restriction of f to X is constan t, due to Prop osition 4.2. Hence X is contained in a non-Euclidean factor of X ′ and it follo ws that X ′ con tains X × R , in con tradiction to Lemma A.3 . Therefore, X ′ has a unique decomp osition with irreducible non-Euclidean factors. Let no w X = X 1 × X 2 b e a pro duct decomp osition. W e hav e seen that X 1 and X 2 are b oth unbounded. Th us ∂ X ω 1 and ∂ X ω 2 are b oth non-empt y , and of radius > π / 2 since o therwise ∂ X ω = ∂ ( X ω 1 × X ω 2 ) w ould ha v e radius ≤ π / 2 , con tradicting Lemma A.9 . By Lemma A.7 , w e can ﬁnd a b oundary-minimal subspace X ′ i in X ω i with ∂ X ′ i = ∂ X ω i . Then X ′ 1 × X ′ 2 has ∂ X ω as its b oundary , hence X ′ 1 × X ′ 2 con tains X ′ . Ho w ev er, the in tersection of X ′ with an y X ′ i -ﬁb er of the decomp osition X ′ 1 × X ′ 2 is either empt y or the whole X ′ i -ﬁb er b y b oundary-minimalit y . W e deduce X ′ = X ′ 1 × X ′ 2 . Let no w X = Y × ¯ Y = Z × ¯ Z b e tw o decomp ositions. Consider the corresp onding decompo sitions X ′ = Y ′ × ¯ Y ′ = Z ′ × ¯ Z ′ constructed as ab ov e. By the canonicit y of the decomposition of X ′ , w e infer that for x ∈ X , the factor Y ′ is a pro duct of Y ′ x ∩ Z ′ x and Y ′ x ∩ ¯ Z ′ x , where the subscript x is used to denote the corresp o nding ﬁb er through x . Moreov er, the pro j ection Y ′ x → Y ′ x ∩ Z ′ x coincides with the nearest p oin t pro jection Y ′ x → Z ′ x (and the same f o r the other factor). Hence the image of Y x under this pro j ections is Z x ∩ Y x and ¯ Z x ∩ Y x respectiv ely . This implies that Y x splits as a pro duct of Y x ∩ Z x and Y x ∩ ¯ Z x . The canonicit y of the decomp osition of X follo ws.  Final adjustmen t s. The pro of of Corollary 1.9 on sup errigidit y remains v alid, since the weak er notion of reduced actions established in Prop osition A.1 (iii) is suﬃcien t to apply Mono d’s theorem. Coro lla r y 1.10 asserting that the amenable radical R acts trivially on eac h non Euclidean factor of X remains v alid; indeed, one only needs to discuss the action of R on the ‘bad’ f actor T from Lemma A.10 . Ho w ev er, since T has a n empty b oundary , it follo ws from Theorem 1 .6 that R m ust stabilize a Euclidean subspace of T , whic h mu st b e a p oin t since ∂ T is empt y . Since the G -action is minimal and R is normal, the R -a ction on T m ust b e trivial, as desired. References [BDL14] Uri Bader, Bruno Duchesne, and J ean Lécureux, F urst enb er g maps for CA T(0) tar gets of ﬁnite telesc opic dimension , 2014. preprint arXiv:140 4.3187 . ↑ 20 [BL05] Andreas Balser and Alexa nder Lytchak, Centers of c onvex subsets of bu ildings , Ann. Global Anal. Geo m. 28 (2 005), no. 2, 201 –209. ↑ 22 , 23 , 2 4 [BH] Martin R. Bridson and André Haeﬂiger, Metric sp ac es of non- p ositive curvature , Grundlehren der Ma thema tischen Wissenschaften [F undamental Principles o f Mathe- matical Sciences], vol. 319 . ↑ 21 , 22 [CL10] Pierre-Emmanuel Caprace and Alexa nder Ly tc hak, At inﬁn ity of ﬁnite-dimensional CA T(0) sp ac es , Math. Ann. 346 (2010), no. 1 , 1– 21. ↑ 20 [CM09] Pierre-Emmanuel Capr ace and Nicola s Mono d, Isometry gr oups of non-p ositively curve d sp ac es: structur e t he ory , J. T op ol. 2 ( 20 09), no. 4, 661 – 700. ↑ 22 , 23 [DS05] Cor nelia Druţu and Mark Sapir, T r e e-gr ade d sp ac es and asymptotic c ones of gr oups , T o po lo gy 44 (2005), no . 5, 959– 1058. With an a ppendix by Denis O sin and Sapir. ↑ 21 [Duc12] Bruno Duchesne, Inﬁnite dimensional Riemannian symmetric sp ac es with ﬁxe d-sign curvatur e , 2012 . Preprint. ↑ 20 [KS99] Nicholas J. Korev aar and Ric har d M. Schoen, Glob al existenc e the or ems for harmonic maps: ﬁnite r ank sp ac es and an appr o ach to rigidity for smo oth actions , 1999. Unpub- lished manuscript. ↑ 20 [Lyt05] A. Lytchak, Rigidity of spheric al buildings and joins , Geom. F unct. Anal. 15 (2005), no. 3, 7 20–75 2. ↑ 21 [Mon06] Nicolas Mono d, Sup errigidity for irr e ducible lattic es and ge ometric splitting , J. Amer. Math. So c. 1 9 (2006 ), no . 4, 781–8 14. ↑ 22 , 23 UCL – Ma th, Chemin du Cyclotron 2, 13 4 8 Louv ain-la-Neuve, Belgium E-mail addr ess : pe.ca prace@ uclouvain.be Ma thema tisches Institut, Universit ä t Bonn, Beringstrasse 1, D-53115 Bonn E-mail addr ess : lytch ak@mat h.uni-bonn.de

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