aa r X i v : . [ m a t h . M G ] M a r Injective metrics on buildings and symmetric spaces
Thomas HaettelMarch 4, 2021
Abstract.
In this article, we show that the Goldman-Iwahori metric onthe space of all norms on a fixed vector space satisfies the Helly propertyfor balls.On the non-Archimedean side, we deduce that most classical Bruhat-Titsbuildings may be endowed with a natural piecewise ℓ ∞ metric which isinjective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs.This gives another proof of biautomaticity for their uniform lattices.On the Archimedean side, we deduce that most classical symmetricspaces of non-compact type may be endowed with a natural piecewise ℓ ∞ metric which is coarsely Helly. We also prove that most classical semisim-ple groups over Archimedean local fields act properly and cocompactlyon injective metric spaces.The only exception is the special linear group: if n > and K is a localfield, we show that SL( n, K ) does not act properly and coboundedly onan injective metric space. Introduction
In this article, we are interested in the relationship between symmetric spaces of non-compact type and Euclidean buildings, on one side, and injective metric spaces and Hellygraphs, on the other side.A geodesic metric space is called injective if the family of closed balls satisfies theHelly property, i.e. any family of pairwise intersecting balls has a non-empty global in-tersection. An injective metric space satisfies some properties of nonpositive curvature:it is contractible, any finite group action has a fixed point, and it has a conical geodesicbicombing. One key feature of injective metric spaces is that any metric space embedsisometrically in an essentially unique smallest injective metric space, called the injectivehull. Injective metric spaces in geometric group theory have been notably popularized byLang, who proved that any Gromov-hyperbolic group acts properly and cocompactly onan injective metric space, the injective hull of a Cayley graph (see [Lan13, Theorem 1.4]).A geodesic metric space is called coarsely Helly if any family of pairwise intersectingballs has a non-empty global intersection, up to increasing the radii by a uniform amount.If a finitely generated group acts properly and cocompactly on a coarsely Helly metricspace, we can deduce that it is semi-hyperbolic in the sense of Alonso-Bridson. Thisstrategy has been used by Hoda, Petyt and the author to prove that any hierarchicallyhyperbolic group, including any mapping class group of a surface, is coarsely Helly andsemi-hyperbolic.
Keywords : Injective metric, Helly graph, Bruhat-Tits building, symmetric space, biautomatic.
AMScodes : 20E42, 53C35, 52A35, 22E46 +
20] for the study of group actions on Helly graphs. Onenotable result is that a discrete group acting properly and cocompactly on a locally finiteHelly graph is biautomatic (see [CCG +
20, Theorem 1.5]).Symmetric spaces of non-compact type and Euclidean buildings already have a CAT(0)metric. Nevertheless, looking for injective metrics on those spaces may provide extrastructure. For instance, deciding which CAT(0) groups are biautomatic is very subtle, asLeary and Minasyan recently provided the first counter-examples (see [LM21]). On theother hand, any Helly group is biautomatic.Our work is based on a very simple remark that, given any set of norms on a vectorspace satisfying simple conditions, the Goldman-Iwahori metric satisfies the Helly propertyfor closed balls (see [GI63]). The fact that the metric is geodesic will be verified in concreteexamples.
Proposition A (Proposition 2.1) . Let K denote a valued field, let V denote a K -vectorspace, and let X denote a set of norms on V satisfying simple conditions (see Proposi-tion 2.1). For any two elements η, η ′ in X , let us define the Goldman-Iwahori metric d ( η, η ′ ) = sup v ∈ V \{ } (cid:12)(cid:12)(cid:12)(cid:12) log η ( v ) η ′ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) . The family of closed balls in the metric space ( X, d ) satisfies the Helly property. Bruhat-Tits buildings
The first example to which Proposition A applies is the Goldman-Iwahori space of allultrametrics norms (see [GI63]). It identifies with the Bruhat-Tits extended building of
GL( n, K ) , where K is a non-Archimedean valued field which is locally compact, or moregenerally spherically complete. Recall that the Bruhat-Tits building X of SL( n, K ) can bedescribed as the set of all homothety classes of ultrametric norms on K n (see [Par99] forinstance), and the Bruhat-Tits extended building X of GL( n, K ) can be described as theset of all ultrametric norms on K n , also called the Goldman-Iwahori space. Each apartmentin X naturally identifies with R n , and the Goldman-Iwahori metric from Proposition A isthe length metric associated to the standard piecewise ℓ ∞ metric on each apartment. Wetherefore have the following. Theorem B (Theorem 3.2) . Let K denote any non-Archimedean valued field K which isspherically complete, and consider the extended Bruhat-Tits building X of GL( n, K ) . Endow X with the Goldman-Iwahori metric, i.e. the length metric associated to the standardpiecewise ℓ ∞ metric on each apartment. Then ( X, d ) is injective. Note that a particular case of this result, when the valuation is discrete and the buildingis simplicial, was already known, combining works of Hirai and Chalopin et al.
Theorem ([Hir20],[CCHO21]) . Let X denote any extended Euclidean building of type e A n − . Endow X with the length metric associated to the standard piecewise ℓ ∞ metricon each apartment. Then ( X, d ) is injective. Our work has the advantage of being valid for a possibly non-discrete valuation if thefield K is spherically complete, and furthermore our proof is extremely simple.We can also wonder whether we can apply it to find a Helly graph related to Euclideanbuildings. This is indeed the case. 2 heorem C (Theorem 3.3) . Let K denote any non-Archimedean discretely valued field K ,and consider the extended Bruhat-Tits building X of GL( n, K ) . Then the thickening of thevertex set X (0) of X is a Helly graph. In particular, GL( n, K ) acts properly and cocompactlyby automorphisms on a Helly graph. The thickening of X (0) is the graph with vertex set X (0) , and with an edge betweentwo vertices if they are at ℓ ∞ distance in some apartment.For other classical groups, we can in fact deduce similar results using an embedding in GL( n, K ) . Corollary D (Theorems 3.4 and 3.5) . Let K denote a local field of characteristic differentfrom , and let G denote a classical connected semisimple group over K , realized as the iden-tity component of the fixed point set of an involution in the general linear group GL( n, K ) .Then the Bruhat-Tits building of G , endowed with the length metric induced from the ℓ ∞ metric on the extended Bruhat-Tits building of GL( n, K ) , is injective. Furthermore, thegroup G acts properly and cocompactly by automorphisms on a locally finite Helly graph. Note that Chalopin et al. proved that any cocompact lattice in a Euclidean building oftype e C n acts properly and cocompactly on a Helly graph (see [CCG +
20, Corollary 6.2]).We also easily deduce a result for all classical semisimple Lie groups and their cocompactlattices.
Corollary E (Corollary 3.6) . Let G denote a classical reductive Lie group over a non-Archimedean local field of characteristic different from , and let a > denote the numberof almost simple factors of type A . Then G × Z a acts properly and cocompactly by auto-morphisms on a locally finite Helly graph.For any cocompact lattice Γ in G , the group Γ × Z a acts properly and cocompactly byautomorphisms on a locally finite Helly graph, and the group Γ is biautomatic. Note that Swiatkowski proved that any group acting properly and cocompactly on anyEuclidean building is biautomatic (see [Ś06, Theorem 6.1]). Nevertheless, this providesanother perspective on this result.
Symmetric spaces
The second example to which Proposition A applies is the symmetric space X of GL( n, R ) / O ( n ) of GL( n, R ) , which may be described as the space of all Euclidean norms on R n . How-ever, it does not apply directly, since the supremum of two Euclidean norms is no longerEuclidean. So we apply Proposition A to the space ˆ X of all norms on R n , and use theJohn-Löwner ellipsoid to show that X is cobounded in ˆ X . Theorem F (Theorem 4.4) . Let X = GL( n, R ) / O ( n ) denote the symmetric space of GL( n, R ) , and endow X with the Finsler length metric associated to the standard piecewise ℓ ∞ metric on each apartment. Then ( X, d ) is coarsely Helly, and its injective hull is lo-cally compact. In particular, GL( n, R ) acts properly and cocompactly by isometries on aninjective metric space. For other classical groups, we can in fact deduce similar results using an embedding in
GL( n, R ) . Theorem G (Theorem 4.6) . Let G denote a classical almost simple non-compact real Liegroup which is not of type A , and let X denote its symmetric space. Then X has a naturalFinsler length metric d such that ( X, d ) is coarsely Helly, and its injective hull is locallycompact. In particular, G acts properly and cocompactly by isometries on an injectivemetric space.
3e also easily deduce a result for all classical semisimple Lie groups and their cocompactlattices.
Corollary H (Corollary 4.7) . Let G denote any reductive real Lie group, with classicalnon-compact almost simple factors. Let a > denote the number of almost simple factorsof type A . Then G × R a acts properly and cocompactly on an injective metric space. Inparticular, for any cocompact lattice Γ in G , the group Γ × Z a acts properly and cocompactlyon an injective metric space. Recall that Chalopin et al. proved that any Helly group is biautomatic. This motivatesthe question whether the non-discrete analogue of this result holds:
Question.
Assume that a finitely generated group Γ acts properly and cocompactly on aninjective metric space. Is Γ biautomatic ? The special linear group
We now turn to the special linear group. According to Theorems B and F, if K is a localfield, we have seen that GL( n, K ) acts properly and cocompactly on an injective metricspace. It is natural to ask what happens for SL( n, K ) . Inspired by the work of Hoda oncrystallographic Helly groups (see [Hod20]), we prove the following. Theorem I (Theorem 5.1) . Let K be a local field (with characteric different from if K isnon-Archimedean), and let n > . Then SL( n, K ) is not coarsely Helly: SL( n, K ) does notact properly and coboundedly on an injective metric space. This is also evidence that cocompact lattices in
SL( n, K ) are not expected to be coarselyHelly. Structure of the article
In Section 1, we review the notions of injective metric spaces, Helly graphs and groupactions. In Section 2, we present Proposition 2.1 stating that the Goldman-Iwahori metricon the space of all norms satisfies a Helly property for balls. In Section 3, we apply thisconstruction to Bruhat-Tits buildings, and in Section 4, we apply it to symmetric spacesof non-compact type. In the final Section 5, we prove that the special linear group is notcoarsely Helly.
Acknowledgments:
We would like to thank Victor Chepoi, François Fillastre, EliaFioravanti, Anthony Genevois, Hiroshi Hirai, Nima Hoda, Vladimir Kovalchuk, Urs Lang,Damian Osajda, Harry Petyt, Betrand Rémy and Constantin Vernicos for interesting dis-cussions and remarks on the first version of the article.
In this section, we recall some basic definitions about injective metric spaces and Hellygraphs. We refer the reader to [Lan13] and [CCG +
20] for more details.A metric space ( X, d ) is called injective if, for any family ( x i ) i ∈ I of points in X and ( r i ) i ∈ N of nonnegative real numbers satisfying ∀ i, j ∈ I, r i + r j > d ( x i , x j ) , the family of balls ( B ( x i , r i )) i ∈ N has a non-empty global intersection.4n case the metric space ( X, d ) is geodesic, it is injective of and only if the familyof balls satisfy the Helly property : any family of pairwise intersecting closed balls has anon-empty global intersection.Examples of geodesic injective metric spaces are normed vector spaces with the ℓ ∞ norm, and also finite-dimensional CAT(0) cube complexes with the piecewise ℓ ∞ metric(see [Bow20]).One key feature of the theory is that any metric space X embeds isometrically in aunique minimal injective metric space, called the injective hull of X and denoted EX (see [Isb64]).A metric space ( X, d ) is called coarsely Helly if there exists a constant C > such that,for any family ( x i ) i ∈ I of points in X and ( r i ) i ∈ N of nonnegative real numbers satisfying ∀ i, j ∈ I, r i + r j > d ( x i , x j ) , the family of balls ( B ( x i , r i + C )) i ∈ N has a non-empty global intersection.There is also a discrete version of injective metric spaces concerning graphs: a connectedgraph is called a Helly graph if the family of combinatorial balls satisfy the Helly property:any family of pairwise intersecting balls has a non-empty global intersection.Concerning actions of groups on injective metric spaces, we will distinguish three fam-ilies: • A group G is called coarsely Helly if it acts properly and coboundedly by isometrieson an injective metric space, or equivalently it acts properly and cocompactly byisometries on a coarsely Helly metric space (see [CCG +
20, Proposition 3.12]). • A group G is called metrically injective if it acts properly and cocompactly by isome-tries on an injective metric space. • A group G is called Helly if it acts properly and cocompactly by automorphisms ona Helly graph.Any Helly group is metrically injective, by considering the injective hull of a Hellygraph. And obvisouly, any metrically injective group is coarsely Helly.We now list examples of such groups.According to [BvdV91] (see also [HW09, Corollary 3.6]), the thickening of any CAT(0)cube complex is a Helly graph: in particular, any group acting properly and cocompactlyon a CAT(0) cube complex is Helly. More generally, any group acting properly and cocom-pactly on a finite rank metric median space is metrically injective (see [Bow20]). Urs Langmotivated the interest in group actions on injective metric spaces in [Lan13], notably prov-ing that any Gromov-hyperbolic group is Helly (see also [CE07]), and acts properly and co-compactly on the injective hull of any Cayley graph. Chalopin et al. proved (see [CCG + e C n is Helly. Huang and Osajda proved that any Artin group of type FC is Helly (see [HO19]).The author, Hoda and Petyt proved in [HHP20] that any hierarchically hyperbolicgroup, including any mapping class group of a surface, is coarsely Helly.The existence of such actions on injective metric spaces enables us to deduce manyproperties reminiscent of non-positive curvature, let us list some of them: Theorem 1.1.
Assume that a finitely generated group G is coarsely Helly. Then: • G is semi-hyperbolic in the sense of Alonso-Bridson, which has many consequences([BH99]). G has finitely many conjugacy classes of finite subgroups ([Lan13, Proposition 1.2]). • G satisfies the coarse Baum-Connes conjecture ([CCG +
20, Theorem 1.5]). • Asymptotic cones of G are contractible ([CCG +
20, Theorem 1.5]).Assume furthermore that G is metrically injective. Then: • G admits an EZ-boundary ([CCG +
20, Theorem 1.5]). • G satisfies the Farrell-Jones conjecture (see [KR17]).Assume in addition that G is a Helly group. Then: • G is biautomatic ([CCG +
20, Theorem 1.5]).
Note that all consequences are already known for CAT(0) groups, except the biauto-maticity (which does not hold for all CAT(0) groups, see [LM21]).However, not all non-positively curved groups are coarsely Helly: for instance, Hodaproved that the (3 , , triangle Coxeter group, which is virtually Z , is not Helly (see [Hod20]). Let K denote a field (or a division algebra) with an absolute value | · | : K → e H , where H is a non-zero additive subgroup of R . Let V denote a K -vector space. Recall that a normon V is a map η : V → e H that satisfies the following. • ∀ v ∈ V, η ( v ) = 0 ⇐⇒ v = 0 . • ∀ v ∈ V, ∀ α ∈ K , η ( αv ) = | α | η ( v ) . • ∀ u, v ∈ V, η ( u + v ) η ( u ) + η ( v ) .Note that there is a natural partial order on the set of all norms on V : we say that η η ′ if ∀ v ∈ V, η ( v ) η ′ ( v ) . If η η ′ , let us denote the interval I ( η, η ′ ) as the set of all norms θ such that η θ η ′ . Proposition 2.1.
Let X denote a non-empty set of norms on V satisfying the followingproperties. • for every η ∈ X and every a ∈ H , we have e a η ∈ X . • for every η, η ′ ∈ X , there exist a ∈ H such that e − a η ′ η e a η ′ . • the set X is a join-semilattice: for every non-empty subset F ⊂ X such that thereexists η ∈ X with F η , the set { η ′ ∈ X | F η ′ } has a unique minimum ∧ F ∈ X .For any two elements η, η ′ in X , let us define the Goldman-Iwahori distance d ( η, η ′ ) = sup v ∈ V \{ } (cid:12)(cid:12)(cid:12)(cid:12) log η ( v ) η ′ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) . Then the family of closed balls in the metric space ( X, d ) satisfies the Helly property. roof. We will first describe balls in ( X, d ) . Fix η ∈ X and a ∈ R + . Then η ′ ∈ B ( η, a ) ifand only if, for every v ∈ V , we have − a log η ′ ( v ) η ( v ) a , hence e − a η ( v ) η ′ ( v ) e a η ( v ) .As a consequence, the ball B ( η, a ) coincides with the interval I ( e − a η, e a η ) .We will now prove that the intervals in X satisfy the Helly property. Consider a family ( I s = I ( η s , e a s η s )) s ∈ S of pairwise intersecting intervals in X , where a s ∈ H for each s ∈ S . Let F = { η s } s ∈ S ⊂ X : for any s, t ∈ S , since I s and I t are intersecting, we have η t e a s η s . According to the assumption on X , we can consider the join η = ∧ F ∈ X .For each s, t ∈ S , since η t e a s η s , we deduce that η e a s η s . In particular, for each s ∈ S , we have η s η e a s η s , so η ∈ I s . We have proved that the global intersection \ s ∈ S I s is non-empty. We will now apply Proposition 2.1 to define an injective metric on classical Bruhat-Titsbuildings.
GL( n, K ) Let K be a field, with a non-Archimedean absolute value | · | : K → R + . Assume that K is alocal field, or more generally that K is spherically complete: any decreasing intersection ofballs in F has non-empty intersection. Let V denote a n -dimensional vector space over K .Let us say that a map η : V → R + is an ultrametric norm on V if it satisfies thefollowing. • ∀ v ∈ V, η ( v ) = 0 ⇐⇒ v = 0 . • ∀ v ∈ V, ∀ α ∈ K , η ( αv ) = | α | η ( v ) . • ∀ u, v ∈ V, η ( u + v ) max( η ( u ) , η ( v )) .An ultrametric norm η on V is called diagonalizable if there exists a basis ( v , . . . , v n ) of V such that ∀ v = n X i =1 x i v i ∈ V, η ( v ) = max i n | x i | . According to [RTW12, Proposition 1.20], if K is a local field, any ultrametric norm on V is diagonalizable. This holds more generally if K is spherically complete, see [RTW12,Remark 1.24].Say that two ultrametric norms η, η ′ : V → R + are homothetic if there exists a ∈ R such that η ′ = e a η . The set X of homothety classes of ultrametric norms on V is calledthe Bruhat-Tits building of SL( n, K ) (see [Par99] for instance).Let X denote the space of all (diagonalizable) ultrametric norms on V , it has beenstudied by Goldman and Iwahori (see [GI63]) and can be identified with the extendedBruhat-Tits building of GL( n, K ) . It is homeomorphic to the product X × R .For any two elements η, η ′ in X , let us define the Goldman-Iwahori distance d ( η, η ′ ) = sup v ∈ V \{ } (cid:12)(cid:12)(cid:12)(cid:12) log η ( v ) η ′ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) . We have an explicit description of the distance d in terms of apartments of X . Thisdescription can also be found in [GI63] without the building point of view, but we will givehere a simple description using the building.7et us recall the description of apartments in the Bruhat-Tits building X of GL( n, K ) .For each basis v , . . . , v n of V (up to homotheties and permutations), there is an associatedapartment in X . For each m ∈ R n , let us consider the following ultrametric norm on V : ∀ v = n X i =1 x i v i ∈ V, η m ( v ) = max i n e m i | x i | . Then the set of such homothety classes identifies with { x ∈ R n | x + x + · · · + x n = 0 } ≃ R n − . It is a model of the standard Euclidean apartment of type ] A n − .Let us now describe the apartments of the extended Bruhat-Tits building X of GL( n, K ) .For each basis v , . . . , v n of V (up to homotheties and permutations), there is an associatedapartment in X : the set of all norms { η m | m ∈ R n } identifies with R n , which is a modelof the extended Euclidean apartment of type ] A n − . Proposition 3.1.
The metric d on X coincides with the length metric associated to the ℓ ∞ metric on each extended apartment. Proof.
Let d ∞ denote the length metric on X associated to the standard ℓ ∞ metric oneach extended apartment. Since the building X admits retractions onto apartments, andas the two metrics d and d ∞ are invariant under the action of GL( n, K ) , it is sufficient toprove that the two metrics coincide on a given apartment.Fix a basis v , . . . , v n of V , and the associated apartment A = { η m , m ∈ R n } in X . Fixany m ∈ R n . Let i n such that | m i | = k m k ∞ , then we have (cid:12)(cid:12)(cid:12)(cid:12) log η m ( v i ) η ( v i ) (cid:12)(cid:12)(cid:12)(cid:12) = | log e m i | = | m i | = k m k ∞ , hence d ∞ ( η , η m ) = k m k ∞ d ( η , η m ) .On the other hand, for any v = P ni =1 x i v i ∈ V , we have (cid:12)(cid:12)(cid:12)(cid:12) log η m ( v ) η ( v ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) log max i n e m i | x i | max i n | x i | (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log max i n e k m k ∞ | x i | max i n | x i | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k m k ∞ , so we deduce that d ( η , η m ) d ∞ ( η , η m ) .So we have proved that d ( η , η m ) = d ∞ ( η , η m ) , for any m ∈ R n . Hence we deducethat d = d ∞ .We can now apply Proposition 2.1 to prove that the metric d is injective. Theorem 3.2.
The extended Bruhat-Tits building X of GL( n, K ) , endowed with the metric d , is injective. Proof.
We first have to check that X satisfies the three assumptions of Proposition 2.1. • For every η ∈ X and every a ∈ R , we know that e a η is an ultrametric norm on V ,hence e a η ∈ X . • For every η, η ′ ∈ X , let a = d ( η, η ′ ) = sup v ∈ V \{ } (cid:12)(cid:12)(cid:12) log η ( v ) η ′ ( v ) (cid:12)(cid:12)(cid:12) ∈ R + . For each v ∈ V ,we have η ( v ) e a η ′ ( v ) and η ′ ( v ) e a η ( v ) , hence e − a η ′ η e a η ′ .8 For every non-empty subset F ⊂ X such that there exists η ∈ X with F η ,let θ = sup F . It it cleat that θ is a well-defined norm on V , we will check thatit is ultrametric: fix u, v ∈ V . For every ε > , there exists η ′ ∈ F such that θ ( u + v ) η ′ ( u + v ) + ε . Then θ ( u + v ) η ′ ( u + v ) + ε max( η ′ ( u ) , η ′ ( v )) + ε max( θ ( u ) , θ ( v )) + ε. This holds for any ε > , hence θ ( u + v ) max( θ ( u ) , θ ( v )) . So θ is an ultrametricnorm on V : θ ∈ X , and it is the unique minimum of the set { η ′ ∈ X | F η ′ } . Alsorecall that, since K is spherically complete, any ultrametric norm on V is diagonaliz-able.According to Proposition 2.1, the balls in ( X, d ) satisfy the Helly property.We also know by Proposition 3.1 that the metric space ( X, d ) is geodesic. So we deducethat the metric space ( X, d ) is injective. We will show that, if we further assume that the valuation is discrete, we can improveTheorem 3.2 by finding a Helly graph.Assume now that the absolute value is discrete: | · | ( K ) = q Z ⊂ R + , where q is thecardinality of the residue field. Then the Bruhat-Tits building X of GL( n, K ) has a nat-ural simplicial structure, where the vertex set X (0) is given by the homothety classes ofultrametric norms with values in q Z .Similarly, the extended Bruhat-Tits building X of GL( n, K ) has a natural simplicialstructure, where the vertex set X (0) is given by the ultrametric norms with values in q Z .To be consistent, we will in this case define the metric d on X (0) as d ( η, η ′ ) = sup v ∈ V \{ } (cid:12)(cid:12)(cid:12)(cid:12) log q η ( v ) η ′ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) ∈ N . Let us define the thickening X ′ of X as the graph with vertex set X (0) , and with anedge between two vertices η, η ′ if they satisfy d ( η, η ′ ) = 1 . Theorem 3.3.
The thickening X ′ of the extended Bruhat-Tits building of GL( n, K ) is aHelly graph. Proof.
Following the same proof as Theorem 3.2, with H = log( q ) Z , we prove that theinteger-valued metric space ( X (0) , d ) has the Helly property for balls.It now suffices to prove that the distance d is a graph distance. According to Proposi-tion 3.1, on each extended apartment, the metric d coincides with the standard ℓ ∞ metricon R n . Since the restriction of the ℓ ∞ metric on R n to the vertex set Z n is a graph distance,we deduce that d is a graph distance on X (0) . This proves that the thickening X ′ is a Hellygraph. We now show how to apply the previous results concerning the general linear group to theother classical groups.Fix a local non-Archimedean field K with residual characteristic different from , andconsider a classical connected semisimple group G over K , realized as the identity compo-nent of the fixed point set of an involution Φ in a general linear group GL( n, K ) . According9o Bruhat and Tits (see [BT84] and [PY02]), the Bruhat-Tits building X of G identifieswith the set of Φ -fixed points in the Bruhat-Tits extended building Y of GL( n, K ) .More generally, we may consider a finite group F of automorphisms of GL( n, K ) suchthat the residual characteristic of K does not divide the order of F . Then, accordingto [PY02], the Bruhat-Tits building X of G = (GL( n, K ) F ) o identifies with the F -fixedpoints in the Bruhat-Tits extended building Y of GL( n, K ) .Endow X with the induced piecewise ℓ ∞ metric d from Y . Theorem 3.4.
The Bruhat-Tits building X of G , with the metric d , is injective. Proof.
According to [Lan13, Proposition 1.2], the fixed point set X = Y F of any finitegroup action on an injective metric space is non-empty and injective. So the metric space ( X, d ) is injective.We can also strengthen this result by looking for an action of G on a Helly graph. Theorem 3.5.
The group G acts properly and cocompactly by automorphisms on a Hellygraph. Proof.
Let Y ′ denote the thickening of the -skeleton of Y , which is a Helly graph accordingto Theorem 3.3. Let F ( Y ′ ) denote the face complex of Y ′ : it is the simplicial complex withvertex set the set of cliques of Y ′ , and with simplices the set of cliques contained in a givenclique of Y ′ . According to [CCG +
20, Lemma 5.30], the face complex F ( Y ′ ) is clique-Helly(i.e. the family of maximal cliques satisfies the Helly property).The group GL( n, K ) acts properly and cocompactly on Y ′ . Let X ′ denote the fixedpoint set of F inside F ( Y ′ ) : according to [CCG +
20, Theorem 7.1, Corollary 7.4], it isa non-empty clique-Helly graph. According to [CCHO21], the underlying graph of X ′ isHelly, and G acts properly and cocompactly on X ′ .The following is immediate. Corollary 3.6.
Let G denote a classical reductive Lie group over a non-Archimedean localfield of characteristic different from , and let a > denote the number of almost simplefactors of type A . Then G × Z a acts properly and cocompactly by automorphisms on a Hellygraph.For any cocompact lattice Γ in G , the group Γ × Z a acts properly and cocompactly byautomorphisms on a Helly graph, and the group Γ is biautomatic. Proof.
This is a direct consequence of Theorem 3.5. According to [CCG +
20, Theorem 1.5],any Helly group is biautomatic. And according to [Mos97, Theorem B], every direct factorof a biautomatic group is biautomatic.Swiatkowski proved that any group acting properly and cocompactly on any Euclideanbuilding is biautomatic (see [Ś06, Theorem 6.1]). So we obtain another point of view onthis result, for uniform lattices in classical groups.
We will apply Proposition 2.1 to define a coarsely Helly metric on classical symmetricspaces of non-compact type. 10 .1 The symmetric space of
GL( n, R ) Fix K = R , C or H (the division algebra of quaternions), fix n > , and let V denote a n -dimensional vector space over K .Say that two Euclidean norms η, η ′ : V → R + are homothetic if there exists a ∈ R such that η ′ = e a η . The set X of homothety classes of hermitian norms on V is calledthe symmetric space of SL( n, K ) , and it identifies naturally with the homogeneous space SL( n, K ) / SU( n, K ) .Let X denote the space of all hermitian norms on V , it is called the symmetric spaceof GL( n, K ) and it identifies naturally with the homogeneous space GL( n, K ) / U( n, K ) . Itis homeomorphic to the product X × R .Let ˆ X denote the space of all norms on V , it contains X as the subset of hermitiannorms. The space ˆ X can also be described as the space of all compact convex subsets of V with non-empty interior, which are invariant under the linear diagonal action of the unitgroup U of K . We will call it the augmented symmetric space of GL( n, K ) . The group GL( n, K ) acts naturally on ˆ X , by precomposing the norms, or by the linear action onconvex subsets of V .For any two elements η, η ′ in X , let us define the distance d ( η, η ′ ) = sup v ∈ V \{ } (cid:12)(cid:12)(cid:12)(cid:12) log η ( v ) η ′ ( v ) (cid:12)(cid:12)(cid:12)(cid:12) . It is a lift of the Banach-Mazur distance, which is defined on the set of isometry classes ofsuch norms.We have an explicit description of the distance d in terms of apartments of X .Let us recall the description of maximal flats in the symmetric space X of SL( n, K ) .For each basis v , . . . , v n of V (up to homotheties and permutations), there is an associatedmaximal flat in X . For each m ∈ R n , let us consider the following hermitian norm on V : ∀ x = n X i =1 x i v i ∈ V, η m ( x ) = vuut n X i =1 e m i | x i | . Then the set of such homothety classes identifies with { m ∈ R n | m + m + · · · + m n =0 } ≃ R n − . It is a model of the standard Euclidean flat of type ] A n − .Let us now describe the maximal flats of the symmetric space X of GL( n, K ) . For eachbasis v , . . . , v n of V (up to homotheties and permutations), there is an associated maximalflat in X , the set { η m | m ∈ R n } is a model of the extended Euclidean flat of type ] A n − . Proposition 4.1.
The metric d on X coincides with the Finsler length metric associatedto the ℓ ∞ metric on each extended maximal flat. Proof.
Let d ∞ denote the length metric on X associated to the standard ℓ ∞ metric oneach extended apartment. Since the symmetric space X admits retractions onto maximalflats, and as the two metrics d and d ∞ are invariant under the action of GL( n, K ) , it issufficient to prove that the two metrics coincide on a given maximal flat.Fix a basis v , . . . , v n of V , and the associated maximal flat A = { η m , m ∈ R n } in X .Fix any m ∈ R n . Let i n such that | m i | = k m k ∞ , then we have (cid:12)(cid:12)(cid:12)(cid:12) log η m ( v i ) η ( v i ) (cid:12)(cid:12)(cid:12)(cid:12) = | log e m i | = | m i | = k m k ∞ , d ∞ ( η , η m ) = k m k ∞ d ( η , η m ) .On the other hand, for any v = P ni =1 x i v i ∈ V , we have (cid:12)(cid:12)(cid:12)(cid:12) log η m ( v ) η ( v ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log pP ni =1 e m i | x i | pP ni =1 | x i | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) log qP ni =1 e k m k ∞ | x i | pP ni =1 | x i | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k m k ∞ , so we deduce that d ( η , η m ) d ∞ ( η , η m ) .So we have proved that d ( η , η m ) = d ∞ ( η , η m ) , for any m ∈ R n . Hence we deducethat d = d ∞ . Proposition 4.2.
The symmetric space X of GL( n, K ) is cobounded in ˆ X . Proof.
Let K ∈ ˆ X . Let B ⊂ K denote the unique John-Löwner ellipsoid of maximalvolume. Since K is invariant under the linear diagonal action of the unit group U , byuniqueness of B , we deduce that B is also invariant under the linear diagonal action ofthe unit group U . So the convex B is the unit ball of a hermitian norm on K n : B ∈ X .According to [Joh48], we know that d ( B, K ) log( √ an ) , where a = dim R ( K ) . Thereforeany point of ˆ X is at distance at most log( √ an ) from X .We do not know whether the metric space ( ˆ X, d ) of all norms on K n is geodesic: ac-cording to Proposition 4.1, we only know that the subspace ( X, d ) of all hermitian normsis geodesic. Nevertheless, the Helly property for balls is enough to study the injective hullof ( X, d ) . Proposition 4.3.
Assume that ( ˆ
X, d ) is a locally compact metric space satisfying the Hellyproperty for balls, and assume that X is a geodesic subspace of ˆ X that is cobounded in ˆ X .Then X is cobounded in the injective hull EX of ( X, d ) , and EX is locally compact. Proof.
We first prove that X is cobounded in its injective hull EX , so according to [CCG + ( X, d ) is coarsely Helly: consider a family ( x i ) i ∈ I ofpoints in X , and a family ( r i ) i ∈ I of nonnegative real numbers, such that ∀ i, j ∈ I, r i + r j > d ( x i , x j ) . Since X is geodesic, the balls ( B X ( x i , r i )) i ∈ I pairwise intersect in X . Sinceballs in ( ˆ X, d ) satisfy the Helly property, we deduce that the balls ( B ˆ X ( x i , r i )) i ∈ I have aglobal intersection in ˆ X . Since X is cobounded in ˆ X , there exists a real number C > such that every point of ˆ X is at distance at most C from a point in X . Therefore theballs ( B X ( x i , r i + C )) i ∈ I have a global intersection in X . Hence the metric space ( X, d ) iscoarsely Helly.We now prove that the injective hull EX is locally compact metric space, by provingthat any bounded sequence in EX has a convergent subsequence. According to [Lan13], ifwe denote ∆ X = { f : X → R + | ∀ x, y ∈ X, f ( x ) + f ( y ) > d ( x, y ) } , then EX can be realized as the minimal set of ∆ X .For each f ∈ EX ⊂ ∆ X , note that the intersection K f = T x ∈ X B ˆ X ( x, f ( x )) is non-empty in ˆ X by injectivity. Furthermore, by minimality of f , we have ∀ x ∈ X, f ( x ) = min { r > | K f ⊂ B ˆ X ( x, r ) } . Let ( f k ) k ∈ N be a bounded sequence in EX ⊂ ∆ X . We have a corresponding boundedsequence ( K f k ) k ∈ N of non-empty compact subspaces of ˆ X . Since ˆ X is locally compact,we deduce that, up to passing to a subsequence, the sequence ( K f k ) k ∈ N converges, with12espect to the Chabauty topology on the space of closed subsets of ˆ X , to a non-emptycompact subset K of ˆ X . For any x ∈ X , let us denote f ( x ) = min { r > | K f ⊂ B ˆ X ( x, r ) } .For each k ∈ N , let d k > denote the Hausdorff distance between K f k and K . Sincethe bounded sequence ( K f k ) k ∈ N converges to K , we know that ( d k ) k ∈ N converges to .Also, for any x ∈ X , we have d ( f k , f ) = sup x ∈ X | f k ( x ) − f ( x ) | d k . As a consequence, wededuce that ( f k ) k ∈ N converges to f in ∆ X . Since EX is closed in ∆ X , we conclude that f ∈ EX . This concludes that EX is locally compact. Theorem 4.4.
The symmetric space X of GL( n, K ) , endowed with the distance d , iscoarsely Helly. Moreover, the injective hull of ( X, d ) is locally compact. Proof.
We will first check that ˆ X satisfies the three assumptions of Proposition 2.1. • For every η ∈ ˆ X and every a ∈ R , we know that e a η is a norm on V , hence e a η ∈ ˆ X . • For every η, η ′ ∈ ˆ X , let a = d ( η, η ′ ) = sup v ∈ V \{ } (cid:12)(cid:12)(cid:12) log η ( v ) η ′ ( v ) (cid:12)(cid:12)(cid:12) ∈ R + . For each v ∈ V ,we have η ( v ) e a η ′ ( v ) and η ′ ( v ) e a η ( v ) , hence e − a η ′ η e a η ′ . • For every non-empty subset F ⊂ ˆ X such that there exists η ∈ ˆ X with F η , let θ = sup F . It it clear that θ is a well-defined norm on V , so θ ∈ ˆ X , and it is theunique minimum of the set { η ′ ∈ ˆ X | F η ′ } .According to Proposition 2.1, the balls in ( ˆ X, d ) satisfy the Helly property.We also know by Proposition 4.1 that the metric space ( X, d ) is geodesic, and byProposition 4.2 we know that X is cobounded in ˆ X , which is locally compact. So we canapply Proposition 4.3 to deduce that ( X, d ) is coarsely Helly, and that the injective hull of ( X, d ) is locally compact.This raises the following natural questions: is the space ˆ X of all norms geodesic ? Ifnot, can we describe the injective hull EX inside ˆ X ? We now show how to apply the previous results concerning the general linear group to theother classical groups.Fix a classical almost simple non-compact real Lie group G over R which is not of type A , i.e. G is commensurable to one of Sp( n, R ) , Sp( n, C ) , Sp( n, H ) , O ( n, C ) , O ( n, H ) , O ( p, q ) , U( p, q ) , Sp( p, q ) .There exists n > and K = R , C or H , and a finite group F of automorphisms of GL( n, K ) such that G embeds in GL( n, K ) and identifies with the fixed point subgroup GL( n, K ) F . Furthermore, if we denote by K a maximal compact subgroup of G , we canassume that K = U( n ) F , and that the corresponding embedding of the symmetric space X = G/K of G into the symmetric space Y = GL( n, K ) / U( n ) has image the fixed pointset X = Y F of F . We endow X with the induced piecewise ℓ ∞ metric d from Y . Let usdenote ˆ X = ˆ Y F . Proposition 4.5.
Any classical irreducible symmetric space of non-compact type X , whichis not of type A , is cobounded in ˆ X . Proof.
Let K ∈ ˆ Y F . Let B ⊂ K denote the unique John-Löwner ellipsoid of maximalvolume. By uniqueness, we deduce that B is invariant under F , i.e. B ∈ ˆ X F . Accordingto [Joh48], we know that d ( B, K ) log( √ an ) , where a = dim R ( K ) . Therefore any pointof ˆ X F is at distance at most log( √ an ) from X .13 heorem 4.6. Let X denote a classical irreducible symmetric space of non-compact type,which is not of type A . Then the Finsler metric space ( X, d ) is coarsely Helly, and itsinjective hull is locally compact. Proof.
According to Theorem 4.4, the symmetric space Y = GL( n, K ) / U( n ) , endowedwith the piecewise ℓ ∞ distance, is coarsely Helly, and its injective hull EY is locallycompact. The isometric action of the finite group F on Y extends to an isometric actionon EY .According to [Lan13, Proposition 1.2], the fixed point set ( EY ) F of F on EY is aninjective metric space. Therefore, the injective hull EX of X may be realized as anisometric closed subspace of ( EY ) F , so EX is locally compact.On the other hand, since X = Y F is geodesic and cobounded in ˆ X = ˆ Y F , which satisfythe Helly property for balls, we deduce by Proposition 4.3 that the metric space X iscoarsely Helly.The following consequence is immediate. Corollary 4.7.
Let G denote any reductive Lie group over R , with classical non-compactalmost simple factors. Let a > denote the number of almost simple factors of type A .Then G × R a acts properly and cocompactly on an injective metric space. In particular,for any cocompact lattice Γ in G , the group Γ × Z a acts properly and cocompactly on aninjective metric space. As we will see below, the factors R a and Z a are necessary. We now turn to the case of the special linear group. We will prove that it is not coarselyHelly, inspired by the result of Hoda that the (3 , , triangle Coxeter group W , which isvirtually Z , is not Helly (see [Hod20]). However, the group W is a subgroup of Z ⋊ S ,which is Helly. This situation is analogous to the inclusion of SL( n, K ) in GL( n, K ) : Theorem 5.1.
Let K be a local field (with characteric different from if K in non-Archimedean), and let n > . Then SL( n, K ) is not coarsely Helly: SL( n, K ) does notact properly and coboundedly on an injective metric space. Proof.
By contradiction, assume that G = SL( n, K ) acts properly and coboundedly on aninjective metric space X .Let A ⊂ SL( n, K ) denote the diagonal subgroup, and let M ⊂ PSL( n, K ) denote themonomial subgroup of PSL( n, K ) : M ≃ A ⋊ A n is the subgroup of matrices with exactlyone non-zero entry on each row and each column (and A n denotes the alternating group).Let F ⊂ A denote the finite diagonal subgroup with entries in {− , } . Since K hascharacteristic different from , we know that the subgroup of G fixed by the conjugationby F is G F = A . According to [Lan13, Proposition 1.2], the fixed point set X F of F in X is non-empty and injective. We will prove that M acts properly and coboundedly onthe injective metric space X F . Firstly, since F is normalized by M , we deduce that M stabilizes X F , and acts properly on X F . We will prove that A acts coboundedly on X F ,which will imply that M also acts coboundedly on X F .Fix x ∈ X F , and let C X > such that any x ∈ X is at distance at most C X from apoint in G · x .Fix x ∈ X F , there exists g ∈ G such that d ( x, g · x ) C X . So we deduce that, forany f ∈ F , we have d ( g · x , f g · x ) C X . Let d G denote a proper left-invariant metric14n G . Since the action of G on X is proper, we deduce that there exists C G > suchthat, for any f ∈ F , we have d G ( g, f gf − ) C G . Let Y denote the symmetric space orBruhat-Tits building of G , endowed with the CAT(0) metric, choose a basepoint y ∈ Y fixed by F , and let y = g · y . Then there exists C Y > such that, for any f ∈ F , we have d ( y, f · y ) C Y . Let y ∈ Y denote the CAT(0) barycenter of the finite orbit F · y : it isfixed by F , and also d ( y, y ) C Y . Since G acts coboundedly on Y , there exists a constant C ′ G > and g ∈ G F = A such that d G ( g, g ) C ′ G . Let us denote x = g · x ∈ X F : thereexists a constant C ′ X such that d ( x, x ) C ′ X . This proves that the action of A on X F iscobounded.So we have proved that the group M ≃ A ⋊ A n acts properly and coboundedly onan injective metric space. By passing to an asymptotic cone, we deduce that the group R n − ⋊ A n has a left-invariant injective metric. In particular, this defines an injective normon the vector space R n − , with linear isometry group containing the alternating group A n .According to [Nac50], the only ( n − -dimensional injective normed vector spaces arelinearly isometry to ℓ n − ∞ : this is a contradiction.This concludes the proof that SL( n, K ) is not coarsely Helly.However, this leaves the following question open: are uniform lattices in SL( n, K ) coarsely Helly ? Thomas HaettelIMAG, Univ Montpellier, CNRS, [email protected] References [BH99]
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