Actions of higher-rank lattices on free groups

Bridson, Martin R.; Wade, Richard D.

doi:10.1112/s0010437x11005598

Cited by 17 publications

(23 citation statements)

References 29 publications

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“…To finish the discussion of Theorem 2.1, we sketch an alternative beautiful argument due to Bridson-Wade [20]. Suppose that φ : Γ → Map(X) is a homomorphism with infinite image.…”

Section: Homomorphisms Between Lattices and Map(x)mentioning

confidence: 99%

“…In section 2 we discuss a result due to Farb-Masur [29], which states that every homomorphism from a higher rank lattice to Map(X) has finite image, sketching a proof due to Bridson-Wade [20]. We also discuss briefly homomorphisms from Map(X) to lattices, proving for example that, under any such homomorphism, Dehn twists are mapped to roots of unipotent elements.…”

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confidence: 99%

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Rigidity phenomena in the mapping class group

Aramayona

Souto

2016

IRMA Lectures in Mathematics and Theoretical Physics

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Throughout this article we will consider connected orientable surfaces of negative Euler characteristic and of finite topological type, meaning of finite genus and with finitely many boundary components and/or cusps. We will feel free to think about cusps as marked points, punctures or topological ends. Sometimes we will need to make explicit mention of the genus and number of punctures of a surface: in this case, we will write S g,n for the surface of genus g with n punctures and empty boundary. Finally, we define the complexity of a surface X as the number κ(X) = 3g − 3 + p, where g is the genus and p is the number of cusps and boundary components of X.In order to avoid too cumbersome notation, we denote byf is an orientation-preserving homeomorphism fixing pointwise the boundary and each puncture of X    the group of orientation-preserving self-homeomorphisms of X relative to the boundary and the set of punctures. We endow Homeo(X) with the compactopen topology, and denote by Homeo 0 (X) the connected component of the identity Id : X → X. It is well-known that Homeo 0 (X) consists of those elements in Homeo(X) that are isotopic to Id : X → X relative to ∂X and the set of punctures of X. The mapping class group Map(X) of X is the group Map(X) = Homeo(X)/ Homeo 0 (X). In the literature, Map(X) is sometimes referred to as the pure mapping class group. We will also need to consider the extended mapping class group Map * (X), i.e. the group of all isotopy classes of self-homeomorphisms of X. Note that if X has r boundary components and n punctures, we have an exact sequencewhere Sym s is the group of permutations of the set with s elements. Let T (X) and M(X) = T (X)/ Map(X) be, respectively, the Teichmüller and moduli spaces of X. The triad formed by Map(X), T (X) and M(X) is often compared with the one formed, for n ≥ 3, by SL n Z, the symmetric space SO n \ SL n R, and the locally symmetric space SO n \ SL n R/ SL n Z.The second author has been partially supported by NSERC Discovery and Accelerator Supplement grants.

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“…To finish the discussion of Theorem 2.1, we sketch an alternative beautiful argument due to Bridson-Wade [20]. Suppose that φ : Γ → Map(X) is a homomorphism with infinite image.…”

Section: Homomorphisms Between Lattices and Map(x)mentioning

confidence: 99%

mentioning

confidence: 99%

Rigidity phenomena in the mapping class group

Aramayona

Souto

2016

IRMA Lectures in Mathematics and Theoretical Physics

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“…Later, Bridson and Wade [BW11] showed that the same superrigidity remains true if the target group is replaced with Out(F N ), the outer automorphism group of a (non-abelian) free group F N of finite rank N. In an unpublished manuscript of [Mim], the present author obtained a similar homomorphism superrigidity from (commutative) universal lattices and symplectic universal lattices, that means, groups of the form SL(n, Z[x 1 , . .…”

Section: Resultsmentioning

confidence: 99%

“…Remark 1.3. In [BW11], Bridson and Wade defined that a group is Z-averse if no finite index subgroup admits a normal subgroup that surjects onto Z. Then, they showed that for every Z-averse group, the homomorphism superrigidity into MCG(Σ g ) and into Out(F N ) holds true.…”

Section: (I) For Every Acylindrically Hyperbolic Group G Every Groupmentioning

confidence: 99%

Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles

Mimura

2017

J. Eur. Math. Soc.

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Abstract. We prove that every homomorphism from the elementary Chevalley group over a finitely generated unital commutative ring associated with a reduced irreducible classical root system of rank at least 2, and ME analogues of such groups, into acylindrically hyperbolic groups has an absolutely elliptic image. This result provides a non-arithmetic generalization of homomorphism superrigidity of Farb-Kaimanovich-Masur and Bridson-Wade.

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“…Right-angled Artin groups (RAAGs) have delivered considerable applications to geometric group theory (two examples are the association with special cube complexes [18] and Bestvina-Brady groups [2]). Another recent area of progress has been the extension of rigidity properties held by irreducible lattices in semisimple Lie groups to involve mapping class groups and automorphism groups of free groups [5,6,17]. In this paper, we look to what extent this phenomenon extends to the automorphism group of an RAAG A Γ , where Γ is the defining graph.…”

Section: Introductionmentioning

confidence: 99%

Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups

Wade

2013

Journal of the London Mathematical Society

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Abstract. Let G be a real semisimple Lie group with no compact factors and finite centre, and let Λ be an irreducible lattice in G. Suppose that there exists a homomorphism from Λ to the outer automorphism group of a right-angled Artin group A Γ with infinite image. We give a strict upper bound to the real rank of G that is determined by the structure of cliques in Γ. An essential tool is the Andreadakis-Johnson filtration of the Torelli subgroup T (A Γ ) of Aut(A Γ ). We answer a question of Day relating to the abelianisation of T (A Γ ), and show that T (A Γ ) and its image in Out(A Γ ) are residually torsion-free nilpotent.

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Actions of higher-rank lattices on free groups

Abstract: If G is a semisimple Lie group of real rank at least two and Γ is an irreducible lattice in G, then every homomorphism from Γ to the outer automorphism group of a finitely generated free group has finite image.

Cited by 17 publications

References 29 publications

Rigidity phenomena in the mapping class group

Rigidity phenomena in the mapping class group

Superrigidity from Chevalley groups into acylindrically hyperbolic groups via quasi-cocycles

Johnson homomorphisms and actions of higher-rank lattices on right-angled Artin groups

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