Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type

Horbez, Camille; Huang, Jingyin

doi:10.48550/arxiv.2004.09325

Cited by 8 publications

(22 citation statements)

References 73 publications

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“…Corollary 5 also shows that right-angled Artin groups are never superrigid for measure equivalence in the strongest possible sense. This is in strong contrast to the situation for some non-right-angled Artin groups with finite outer automorphism groups [HH20], or mapping class groups [Kid10]; this also strongly contrasts with the quasi-isometry superrigidity results obtained by the second named author in [Hua18].…”

Section: Introductioncontrasting

confidence: 73%

“…Also, Gaboriau proved that 2 -Betti numbers can be used to obtain measure equivalence invariants [Gab02a]; they have been computed by Davis and Leary for many Artin groups including all the right-angled ones [DL03]. Finally, we comment that Theorem 1 improves a result obtained as a corollary of our previous work on Artin groups [HH20, Corollary 7], where it was established with different techniques under the extra assumption that the defining graphs of the right-angled Artin groups G 1 and G 2 have girth at least 5 -this involved viewing those right-angled Artin groups as 2-dimensional Artin groups of hyperbolic type, and therefore the method in [HH20] cannot be used in the general case.…”

Section: Introductionsupporting

confidence: 55%

“…In the work of Kida [Kid10], this came in the form of a natural partition of the (compact) Thurston boundary of the Teichmüller space into arational and nonarational laminations. In our previous work on Artin groups of hyperbolic type [HH20], this came in the form of a partition of the horofunction compactification of a certain CAT(−1) simplicial complex on which the group is acting. In the present paper, we exploit the cubical geometry of right-angled Artin groups by considering the Roller boundary ∂ R S Γ of the universal cover of the Salvetti complex, and its partition, introduced by Fernós in [Fer18], into regular points and non-regular points.…”

Section: Introductionmentioning

confidence: 99%

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Measure equivalence classification of transvection-free right-angled Artin groups

Horbez,

Huang

2020

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We prove that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. This matches the quasi-isometry classification.However, in contrast with the quasi-isometry question, we observe that no rightangled Artin group is superrigid in the strongest possible sense, for two reasons. First, a right-angled Artin group G is always measure equivalent to any graph product of infinite countable amenable groups over the same defining graph. Second, when G is nonabelian, the automorphism group of the universal cover of the Salvetti complex of G always contains infinitely generated (non-uniform) lattices.

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Section: Introductioncontrasting

confidence: 73%

Section: Introductionsupporting

confidence: 55%

Section: Introductionmentioning

confidence: 99%

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Measure equivalence classification of transvection-free right-angled Artin groups

Horbez,

Huang

2020

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“…In contrast to the Ornstein-Weiss theorem exhibiting a wide class of groups that are indistinguishable from the viewpoint of orbit equivalence, several strong rigidity results have then been obtained for various classes of groups, like higher rank lattices (Furman [Fur99a,Fur99b]), mapping class groups (Kida [Kid10,Kid08]) and related groups (e.g. [CK15]), certain large type Artin groups [HH20a] or Out(F N ) with N ≥ 3 (as proved by Guirardel and the first named author in [GH21]). Interestingly, negative curvature features of the groups under consideration are often key ingredients in the proofs of orbit equivalence rigidity of their ergodic actions.…”

Section: Introductionmentioning

confidence: 99%

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Horbez,

Huang

2021

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Let G ΓX and G Λ Y be two free measure-preserving actions of one-ended centerless right-angled Artin groups on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably W *equivalent), then they are automatically conjugate through a group isomorphism between G Γ and G Λ . Through work of Monod and Shalom, we derive a superrigidity statement: if the action G Γ X is stably orbit equivalent (or merely stably W *equivalent) to a free, measure-preserving, mildly mixing action of a countable group, then the two actions are virtually conjugate. Independently, we also prove, using work of Popa, that all Bernoulli actions of one-ended nonabelian right-angled Artin groups are W * -superrigid.

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“…Finally, in the mapping class group setting, combining proper proximality with a theorem of Kida [Kid08,Kid10] on orbit equivalence rigidity for ergodic probability measurepreserving actions of Mod(Σ), we derive that the von Neumann algebra L ∞ (X)⋊Mod(Σ) associated to a weakly compact ergodic probability measure-preserving essentially free action of Mod(Σ) recovers the action (up to stable conjugation) among the class of weakly compact actions of countable groups (see Theorem 4.3 for a precise statement). See also [HH20,Section 7.4] for further applications to the context of Artin groups.…”

Section: Introductionmentioning

confidence: 99%

Proper proximality in non-positive curvature

Horbez¹,

Huang²,

Lécureux³

2020

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Proper proximality of a countable group is a notion that was introduced by Boutonnet, Ioana and Peterson as a tool to study rigidity properties of certain von Neumann algebras associated to groups or ergodic group actions. In the present paper, we establish the proper proximality of many groups acting on nonpositively curved spaces.First, these include many countable groups G acting properly nonelementarily by isometries on a proper CAT(0) space X. More precisely, proper proximality holds in the presence of rank one isometries or when X is a locally thick affine building with a minimal G-action. As a consequence of Rank Rigidity, we derive the proper proximality of all countable nonelementary CAT(0) cubical groups, and of all countable groups acting properly cocompactly nonelementarily by isometries on either a Hadamard manifold with no Euclidean factor, or on a 2-dimensional piecewise Euclidean CAT(0) simplicial complex.Second, we establish the proper proximality of many hierarchically hyperbolic groups. These include the mapping class groups of connected orientable finite-type boundaryless surfaces (apart from a few low-complexity cases), thus answering a question raised by Boutonnet, Ioana and Peterson. We also prove the proper proximality of all subgroups acting nonelementarily on the curve graph.In view of work of Boutonnet, Ioana and Peterson, our results have applications to structural and rigidity results for von Neumann algebras associated to all the above groups and their ergodic actions.

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Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type

Cited by 8 publications

References 73 publications

Measure equivalence classification of transvection-free right-angled Artin groups

Measure equivalence classification of transvection-free right-angled Artin groups

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Proper proximality in non-positive curvature

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