<i>OE</i>and<i>W</i>* superrigidity results for actions by surface braid groups

Chifan, Ionuţ

doi:10.1112/plms/pdv058

Cited by 22 publications

(13 citation statements)

References 85 publications

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“…2.9]). Recently, some other examples of taut groups have arisen in the literature [CK15], [Bow17], and some other rigidity results have been obtained [Aus16], [Can17], [GL18], [Savb].…”

Section: Historical Backgroundmentioning

confidence: 99%

A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

Moraschini

Savini

2020

Transformation Groups

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Following the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

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“…2.9]). Recently, some other examples of taut groups have arisen in the literature [CK15], [Bow17], and some other rigidity results have been obtained [Aus16], [Can17], [GL18], [Savb].…”

Section: Historical Backgroundmentioning

confidence: 99%

A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

Moraschini

Savini

2020

Transformation Groups

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“…For example, one can let G be a direct product of mapping class groups: indeed, direct products of properly proximal groups are again properly proximal [BIP18, Proposition 4.10], and the orbit equivalence rigidity result from [Kid08] still holds for products of mapping class groups. Also, Chifan and Kida showed in [CK15] that many interesting subgroups of the mapping class groups which act nonelementarily on the curve graph -such as the Torelli subgroup -are rigid for measure equivalence, and therefore their ergodic actions are rigid for orbit equivalence. Such groups are properly proximal by Theorem 3.9, so an analogue of Theorem 4.3 also holds for these groups.…”

Section: C-rigidity Of Cat(0) Cubical Groupsmentioning

confidence: 99%

Proper proximality in non-positive curvature

Horbez¹,

Huang²,

Lécureux³

2020

Preprint

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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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“…[M : A 1 ∨ A 2 ] < ∞ and hence by [CK15,Proposition 8.5] we have L(Λ) ≺ M L(π −1 (Γ i )). This gives [Λ : π −1 (Γ i )] < ∞ which in turn implies [Γ : Γ i ] < ∞ which can hold only if Γ i = Σ, a contradiction.…”

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confidence: 92%

Tensor Product Decompositions of II1 Factors Arising from Extensions of Amalgamated Free Product Groups

Chifan

Santiago

Sucpikarnon

2018

Commun. Math. Phys.

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In this paper we introduce a new family of icc groups Γ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II 1 factor L(Γ) arise only from the canonical direct product decompositions of the underlying group Γ. Our groups are assembled from certain HNNextensions and amalgamated free products and include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As a consequence we obtain several new examples of groups that give rise to prime factors.of the corresponding II 1 factor L(Γ) can arise only from the canonical direct product decompositions of the underlying group Γ. Pant and the second author showed the same result holds when Γ is a poly-hyperbolic group with non-amenable factors in its composition series [dSP17]. In this paper we introduce several new classes of groups for which this tensor product rigidity phenomenon still holds. Our groups arise from natural algebraic constructions involving HNN-extensions or amalgamated free products thus including many remarkable groups intensively studied throughout mathematics such as graph products or poly-amalgams groups.Basic properties in Bass-Serre theory of groups show that the only way an amalgam Γ 1 * Σ Γ 2 could decompose as a direct product is through its core Σ.An interesting question is to investigate situations when this basic group theoretic aspect could be upgraded to the von Neumann algebraic setting. It is known this fails in general since there are examples of product indecomposable icc amalgams whose corresponding factors are McDuff and hence decomposable as tensor products (see Remarks 6.3). However, under certain indecomposability assumptions on the core algebra (see also Remarks 6.3) we are able to provide a positive answer to our question.Theorem A. Let Γ = Γ 1 * Σ Γ 2 be an icc group such that [Γ 1 : Σ] ≥ 2 and [Γ 2 : Σ] ≥ 3. Assume that Σ is finite-by-icc and any corner of L(Σ) is virtually prime. Suppose that L2 , for some groups Σ 0 Γ 0 1 , Γ 0 2 , and hence Γ = Ω × (Γ 0 1 * Σ 0 Γ 0 2 ). Moreover, there is a unitary u ∈ L(Γ), t > 0, and a permutation s of {1, 2} such that M s(1) = uL(Ω) t u * and M s(2) = uL(Γ 0 1 * Σ 0 Γ 0 2 ) 1/t u * .(1.1)The same question can be considered in the realm of non-degenerate HNN-extension groups and a similar approach leads to the following counterpart of Theorem A. Theorem B. Let Γ = HNN(Λ, Σ, θ) be an icc group such that Σ = Λ = θ(Σ). Assume that Σ is finite-by-icc and any corner of L(Σ) is virtually prime. Suppose that L(Γ) = M 1⊗ M 2 , for diffuse M i 's. Then there exist decompositions Σ = Ω × Σ 0 with Σ 0 finite and Λ = Ω × Λ 0 . In addition, there is ω ∈ Ω such that θ = ad(ω) |Ω × θ |Σ 0 : Ω × Σ 0 → Ω × Λ 0 and hence Γ = Ω × HNN(Λ 0 , Σ 0 , θ |Σ 0 ). Also, there is a unitary u ∈ L(Γ), t > 0, and a permutation s of {1, 2} such that M s(1) = uL(Ω) t u * and M s(2) = uL(HNN(Λ 0 , ...

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OEandW* superrigidity results for actions by surface braid groups

Cited by 22 publications

References 85 publications

A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

A Matsumoto–mostow Result for Zimmer’s Cocycles of Hyperbolic Lattices

Proper proximality in non-positive curvature

Tensor Product Decompositions of II1 Factors Arising from Extensions of Amalgamated Free Product Groups

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