Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

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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“….S / as a tree-graded space. We note that by results of [4] such pieces can not be realized as asymptotic cones of subgroups of M.S/.…”

Section: Classification Of Piecesmentioning

confidence: 88%

“…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

Self Cite

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“…On the other hand, as observed in [12], the presence of numerous mutually intersecting free abelian subgroups tends to rule out altogether the possibility of relative hyperbolicity in the stronger sense for most Artin groups. More precisely, it is shown in [2] and also follows easily from Lemma 4 of [1] that if an Artin group G is strongly relatively hyperbolic with respect to a family of groups H then each freely indecomposable free factor of G must be included in H. In particular, a freely indecomposable Artin group is strongly relatively hyperbolic in only the most trivial of senses, namely with respect to itself.…”

Section: Introductionmentioning

confidence: 99%

Relative hyperbolicity and Artin groups

Charney

Crisp

2007

Geom Dedicata

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Abstract. This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a 2-dimensional Artin group the Deligne complex is Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for hyperbolicity of the Deligne complex which applies to a large class of these groups for which the underlying Coxeter group is hyperbolic.

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“…(2) Collection of maximal abelian subgroups of the mapping class group (Behrstock-Drutu-Mosher [1]). …”

Section: Relative Rigidity and Statement Of Resultsmentioning

confidence: 99%

Relative rigidity, quasiconvexity andC–complexes

2008

Algebr. Geom. Topol.

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1691Relative rigidity, quasiconvexity and C -complexes MAHAN MJWe introduce and study the notion of relative rigidity for pairs .X; J / where (1) X is a hyperbolic metric space and J a collection of quasiconvex sets, (2) X is a relatively hyperbolic group and J the collection of parabolics, (3) X is a higher rank symmetric space and J an equivariant collection of maximal flats.Relative rigidity can roughly be described as upgrading a uniformly proper map between two such J to a quasi-isometry between the corresponding X . A related notion is that of a C -complex which is the adaptation of a Tits complex to this context. We prove the relative rigidity of the collection of pairs .X; J / as above. This generalises a result of Schwarz for symmetric patterns of geodesics in hyperbolic space. We show that a uniformly proper map induces an isomorphism of the corresponding C -complexes. We also give a couple of characterizations of quasiconvexity of subgroups of hyperbolic groups on the way.

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Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

Cited by 123 publications

References 70 publications

Geometry and rigidity of mapping class groups

Geometry and rigidity of mapping class groups

Relative hyperbolicity and Artin groups

Relative rigidity, quasiconvexity andC–complexes

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