The median class and superrigidity of actions on CAT(0) cube complexes

Chatterji, Indira; Fernós, Talia; Iozzi, Alessandra

doi:10.1112/jtopol/jtu025

Cited by 61 publications

(77 citation statements)

References 45 publications

(72 reference statements)

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“…An analogue of Corollary D for CAT(0) cube complexes was proved in [CFI16]. We remark however that the full strength of Corollary D reveals a surprising new phenomenon.…”

Section: Remarksmentioning

confidence: 75%

“…Let a median algebra (M, m) and two gate-convex subsets C 1 , C 2 be fixed throughout this section. All the following results have analogues in the context of CAT(0) cube complexes; see Section 2.G in [CFI16].…”

Section: 2mentioning

confidence: 91%

“…Since X lacks a cellular structure in general, we are forced to concoct a more elaborate proof for Lemma 4.2, if compared to analogues in the context of trees (pp. 44-45 in [Sha00]) and cube complexes (Claim 6.1 in [CFI16]).…”

Section: Superrigiditymentioning

confidence: 99%

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Superrigidity of actions on finite rank median spaces

Fioravanti¹

2019

Advances in Mathematics

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Finite rank median spaces are a simultaneous generalisation of finite dimensional CAT(0) cube complexes and real trees. If Γ is an irreducible lattice in a product of rank-one simple Lie groups, we show that every action of Γ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces, where even proper cocompact actions can arise.The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated topological groups.We exploit Roller compactifications of median spaces; these were introduced in [Fio17] and generalise a well-known construction in the case of cube complexes. We show that the Haagerup cocycle provides a reduced 1-cohomology class that detects group actions with a finite orbit in the Roller compactification. This is new even for CAT(0) cube complexes and has interesting consequences involving Shalom's property HF D . For instance, in Gromov's density model, random groups at low density do not have HF D .

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“…An analogue of Corollary D for CAT(0) cube complexes was proved in [CFI16]. We remark however that the full strength of Corollary D reveals a surprising new phenomenon.…”

Section: Remarksmentioning

confidence: 75%

Section: 2mentioning

confidence: 91%

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Superrigidity of actions on finite rank median spaces

Fioravanti¹

2019

Advances in Mathematics

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“…Recall that an action G X on a CAT(0) cube complex X is essential if, for every point x ∈ X and every halfspace D, the orbit G · x does not lie in a neighborhood of D. It is worth noticing that an action can often be made essential thanks to [CS11, Proposition 3.5]. The boundary which is considered in this statement is the CAT(0) boundary; see [CFI16,Proposition 2.26] to compare with the Roller boundary. For the combinatorial boundary, see [Gen16c], where it is proved that, under some assumptions on the action, the existence of contracting isometries is equivalent to the existence of an isolated point in the combinatorial boundary.…”

Section: Contracting Isometriesmentioning

confidence: 99%

Hyperbolicities in CAT(0) cube complexes

Genevois

2020

Enseign. Math.

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This paper is a survey dedicated to the following question: given a group acting on a CAT(0) cube complex, how to exploit this action to determine whether or not the group is Gromov / relatively / acylindrically hyperbolic? As much as possible, the different criteria we mention are illustrated by applications. We also propose a model for universal acylindrical actions of cubulable groups, and give a few applications to Morse, stable and hyperbolically embedded subgroups. arXiv:1709.08843v2 [math.GR] 26 Apr 2019 PreliminariesA cube complex is a CW complex constructed by gluing together cubes of arbitrary (finite) dimension by isometries along their faces. It is nonpositively curved if the link of any of its vertices is a simplicial flag complex (ie., n + 1 vertices span a n-simplex if and only if they are pairwise adjacent), and CAT(0) if it is nonpositively curved and simply-connected. See [BH99, page 111] for more information. Proposition 2.2. [Gen16b, Proposition 2.9] Let X be a CAT(0) cube complex and A, B ⊂ X two convex subcomplexes. The projection proj B (A) is a geodesic subcomplex of B. Moreover, the hyperplanes intersecting proj B (A) are precisely those which intersect both A and B. Lemma 2.3. [HW08, Lemma 13.8] Let X be a CAT(0) cube complex, Y ⊂ X a convex subcomplex and x ∈ X a vertex. Any hyperplane separating x from its projection onto Y must separate x from Y . Lemma 2.4. [Gen16c, Proposition 2.6] Let X be a CAT (0) cube complex, C ⊂ X a convex subcomplex and x, y ∈ X two vertices. The hyperplanes separating the projections of x and y onto C are precisely the hyperplanes separating x and y which intersect C.Lemma 2.5. [HW08, Corollary 13.10] Let X be a CAT(0) cube complex and Y 1 , Y 2 ⊂ X two convex subcomplexes. If x ∈ Y 1 and y ∈ Y 2 are two vertices minimising the distance between Y 1 and Y 2 , then the hyperplanes separating x and y are precisely the hyperplanes separating Y 1 and Y 2 .Cycles of subcomplexes. Given a CAT(0) cube complex, a cycle of subcomplexes is a sequence of subcomplexes (C 1 , . . . , C r ) such that, for every i ∈ Z/rZ, the subcomplexes C i and C i+1 intersects.Proposition 2.7. Let (A, B, C, D) be a cycle of four convex subcomplexes. There exists a combinatorial isometric embedding [0, p] disjoint from B and D (resp. A and C).Proof. First of all, let us record a statement which is contained into the proof of [Gen17b, Proposition 2.111] (in the context of quasi-median graphs, a class of graphs including median graphs, i.e., one-skeleta of CAT(0) cube complexes).Fact 2.8. If a is a vertex of A ∩ D minimising the distance to B ∩ C and if b (resp. c, d) denotes the projection of a onto B (resp. B ∩ C, C), then there exists a combinatorial isometric embedding [0, p] × [0, q] → X such that (0, 0) = a, (p, 0) = b, (p, q) = c and (0, q) = d.By convexity of A, B, C and D, this implies that [0, p] Let J be a hyperplane intersecting [0, p] × {0}. We know from Lemma 2.3 that J must be disjoint from B. Moreover, if J intersects D, then it follows from Helly's property, satisfied by...

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“…Monod and Shalom showed the importance of bounded cohomology with coefficients in 2 (G) in the study of rigidity of G [28,31,32], and proposed the condition H 2 b (G, 2 (G)) = 0 as a cohomological definition of negative curvature for groups. More in general, bounded cohomology with coefficients in p (G), 1 p < ∞ has been widely studied as a powerful tool to prove (super)rigidity results (see, for example, [7,16]). However, little is known in this context about degrees higher than 2.…”

Section: Applications To Bounded Cohomologymentioning

confidence: 99%

Extending higher-dimensional quasi-cocycles

2015

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Abstract. Let G be a group admitting a non-elementary acylindrical action on a Gromov hyperbolic space (for example, a non-elementary relatively hyperbolic group, or the mapping class group of a closed hyperbolic surface, or Out(Fn) for n ≥ 2). We prove that, in degree 3, the bounded cohomology of G with real coefficients is infinite-dimensional. Our proof is based on an extension to higher degrees of a recent result by Hull and Osin. Namely, we prove that, if H is a hyperbolically embedded subgroup of G and V is any R[G]-module, then any n-quasi cocycle on H with values in V may be extended to G. Also, we show that our extensions detect the geometry of the embedding of hyperbolically embedded subgroups, in a suitable sense.Bounded cohomology of discrete groups is very hard to compute. For example, as observed in [Mon06], there is not a single countable group G whose bounded cohomology (with trivial coefficients) is known in every degree, unless it is known to vanish in all positive degrees (this is the case, for example, of amenable groups). Even worse, no group G is known for which the supremum of the degrees n such that H n b (G, R) = 0 is positive and finite.In degree 2, bounded cohomology has been extensively studied via the analysis of quasi-morphisms (see e.g. [Bro81, EF97, Fuj98, BF02, Fuj00] for the case of trivial coefficients, and [HO13, BBF13] for more general coefficient modules). In this paper we exploit quasi-cocycles, which are the higherdimensional analogue of quasi-morphisms, to prove non-vanishing results for bounded cohomology in degree 3. To this aim, we prove that quasi-cocycles may be extended from a hyperbolically embedded family of subgroups to the ambient group. We also discuss the geometric information carried by our extensions of quasi-cocycles, showing in particular that projections on hyperbolically embedded subgroups may be reconstructed from the extensions of suitably chosen quasi-cocycles.Quasi-cocycles and bounded cohomology. Let G be a group, and V be a normed R[G]-space, i.e. a normed real vector space endowed with an

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The median class and superrigidity of actions on CAT(0) cube complexes

Cited by 61 publications

References 45 publications

Superrigidity of actions on finite rank median spaces

Superrigidity of actions on finite rank median spaces

Hyperbolicities in CAT(0) cube complexes

Extending higher-dimensional quasi-cocycles

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