Median structures on asymptotic cones and homomorphisms into mapping class groups

Behrstock, Jason; Druţu, Cornelia; Sapir, Mark

doi:10.1112/plms/pdq025

Cited by 30 publications

(27 citation statements)

References 52 publications

(164 reference statements)

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“…Now putting these four sums (3-2), (3-3), (3)(4), (3)(4)(5) together, and recalling that Y ı open. i / if and only if Y 6 t i , it follows that each Y S appears in exactly one of these four sums.…”

Section: Product Regionsmentioning

confidence: 99%

Geometry and rigidity of mapping class groups

et al. 2012

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We study the large scale geometry of mapping class groups MCG.S/, using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG.S / (outside a few sporadic cases) is a bounded distance away from a leftmultiplication, and as a consequence obtain quasi-isometric rigidity for MCG.S /, namely that groups quasi-isometric to MCG.S / are equivalent to it up to extraction of finite-index subgroups and quotients with finite kernel. (The latter theorem was proved by Hamenstädt using different methods).As part of our approach we obtain several other structural results: a description of the tree-graded structure on the asymptotic cone of MCG.S/; a characterization of the image of the curve complex projections map from MCG.S/ to Q Y ÂS C.Y /; and a construction of †-hulls in MCG.S /, an analogue of convex hulls.

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Section: Product Regionsmentioning

confidence: 99%

Geometry and rigidity of mapping class groups

et al. 2012

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“…One might conjecture that it applies to a much broader class of spaces that are in some sense non-positively curved, such as CAT(0) spaces. Much of this work is inspired by the results in [BehM1,BesBF,BehM2,BehBKM,BehDS,ChaDH]. It seems a natural general setting in which to view some of this work.…”

Section: Introductionmentioning

confidence: 99%

Coarse median spaces and groups

Bowditch¹

2013

Pacific J. Math.

205

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Abstract. We introduce the notion of a coarse median on a metric space. This satisfies the axioms of a median algebra up to bounded distance. The existence of such a median on a geodesic space is quasi-isometry invariant, and so applies to finitely generated groups via their Cayley graphs. We show that asymptotic cones of such spaces are topological median algebras. We define a notion of rank for a coarse median and show that this bounds the dimension of a quasi-isometrically embedded euclidean plane in the space. Using the centroid construction of Behrstock and Minsky, we show that the mapping class group has this property, and recover the rank theorem of Behrstock and Minsky and of Hamenstädt. We explore various other properties of such spaces, and develop some of the background material regarding median algebras.

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“…In contrast, note that a recent result of [8] shows that the rank 2 lattice SL 3 (Z) contains infinitely many pairwise non-conjugate copies of the triangle group Δ(3, 3, 4) = a, b | a 3 = b 3 = (ab) 4 = 1 . Also, as was pointed out to us by Kassabov, although the group SL 3 (Z[x]) has property (T) (see [12]), it has infinitely many pairwise non-conjugate homomorphisms into SL 3 (Z) induced by ring homomorphisms Z[x] → Z.The following proposition contains one of the main auxiliary results in [3] and the key ingredient missed in our treatment of groups with many homomorphisms into mapping class groups in [1].Proposition 2 (Bestvina, Bromberg and Fujiwara [3, Proposition 5.9]). There exists an explicitly defined finite index torsion-free subgroup BBF(S) of MCG(S) such that the set of all sub-surfaces of S can be partitioned into a finite number of subsets C 1 , C 2 , .…”

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The goal of this addendum to [1] is to show that our methods together with a result of Bestvina, Bromberg and Fujiwara [3, Proposition 5.9] yield a proof of the following theorem. Theorem 1.

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Addendum: Median structures on asymptotic cones and homomorphisms into mapping class groups

Behrstock

Druţu²,

Sapir

2011

Proceedings of the London Mathematical Society

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The goal of this addendum to [1] is to show that our methods together with a result of Bestvina, Bromberg and Fujiwara [3, Proposition 5.9] yield a proof of the following theorem. Theorem 1. If a finitely presented group Γ has infinitely many pairwise non-conjugate homomorphisms into MCG(S), then Γ virtually splits (virtually acts non-trivially on a simplicial tree).This theorem is a particular case of a result announced by Groves. † From private emails received by the authors, it is clear that the methods used by Groves are significantly different. Note that the same new methods allow us to give another proof of the finiteness of the set of homomorphisms from a group with property (T) to a mapping class group [1, Theorem 1.2], which is considerably shorter than our original proof; see Corollary 6 and the discussion following it. Theorem 1.2 in [1] may equally be obtained from Theorem 1 and the fact that every group with property (T) is a quotient of a finitely presented group with property (T) (see [11, Theorem p. 5]).The property of the mapping class groups given in Theorem 1 can be viewed as another 'rank 1' feature of these groups. In contrast, note that a recent result of [8] shows that the rank 2 lattice SL 3 (Z) contains infinitely many pairwise non-conjugate copies of the triangle group Δ(3, 3, 4) = a, b | a 3 = b 3 = (ab) 4 = 1 . Also, as was pointed out to us by Kassabov, although the group SL 3 (Z[x]) has property (T) (see [12]), it has infinitely many pairwise non-conjugate homomorphisms into SL 3 (Z) induced by ring homomorphisms Z[x] → Z.The following proposition contains one of the main auxiliary results in [3] and the key ingredient missed in our treatment of groups with many homomorphisms into mapping class groups in [1].Proposition 2 (Bestvina, Bromberg and Fujiwara [3, Proposition 5.9]). There exists an explicitly defined finite index torsion-free subgroup BBF(S) of MCG(S) such that the set of all sub-surfaces of S can be partitioned into a finite number of subsets C 1 , C 2 , . . . , C s , each of which is an orbit of BBF(S), and any two sub-surfaces in the same subset overlap and have the same complexity.

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Median structures on asymptotic cones and homomorphisms into mapping class groups

Cited by 30 publications

References 52 publications

Geometry and rigidity of mapping class groups

Geometry and rigidity of mapping class groups

Coarse median spaces and groups

Addendum: Median structures on asymptotic cones and homomorphisms into mapping class groups

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