Shadows Of Mapping Class Groups: Capturing Convex Cocompactness

Kent, Richard P.; Leininger, Christopher J.

doi:10.1007/s00039-008-0680-9

Cited by 70 publications

(86 citation statements)

References 46 publications

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“…Peter Storm conjectured that @C.S/ is path connected if S is any other finite type hyperbolic surface. See Question 10 of Kent and Leininger [9]. Saul Schleimer [17] showed if…”

Section: Introductionmentioning

confidence: 99%

Almost filling laminations and the connectivity of ending lamination space

Gabai

2009

Geom. Topol.

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We show that if S is a finite type orientable surface of negative Euler characteristic which is not the 3-holed sphere, 4-holed sphere or 1-holed torus, then EL.S/ is path connected, locally path connected and cyclic. 57M50; 57R30, 51M10, 20F65 IntroductionThe main result of this paper is the following:Theorem 0.1 If S is a finite type orientable surface of negative Euler characteristic which is not the 3-holed sphere, 4-holed sphere or 1-holed torus, then EL.S/, the space of ending laminations, is path connected, locally path connected and cyclic.EL.S/ is the space of filling, minimal geodesic laminations with the topology induced from ML.S/ by forgetting the measure. Equivalently, it has the coarse Hausdorff topology. See Definition 1.12. Interestingly, with respect to the Hausdorff topology, EL.S/ is totally disconnected as shown by Thurston [19, Section 10] and Zhu and Bonahon [22].Erica Klarreich [10] showed that if S is hyperbolic, then the boundary of the curve complex C.S/ is homeomorphic to the space of ending laminations on S . Therefore we have:Corollary 0.2 If S is a finite type orientable surface of negative Euler characteristic which is not the 3-holed sphere, 4-holed sphere or 1-holed torus, then @C.S/ is path connected, locally path connected and cyclic, where C.S/ is the curve complex of S .History For the thrice-punctured sphere S , C.S/ is trivial. It is well known that @C.S/ is totally disconnected when S is the 4-holed sphere or 1-holed torus. S is a once-punctured surface of genus at least two, then C.S / has exactly one end. With Leininger, he then showed [12] that EL.S/ is connected if S is any punctured surface of genus at least two or S is closed of genus at least 4. Leininger, Mj and Schleimer [11] showed that if S is a once-punctured surface of genus at least two then EL.S/ is both path connected and locally path connected.The idea of the proof is as follows. Given 0 ; 1 2 EL.S/ let 0 and 1 be measured laminations whose underlying laminations are 0 and 1 . Next, construct a path in ML.S / from 0 to 1 . A generic PL approximation of this path will yield a new path f 1 W OE0; 1 ! ML.S/ such that for each t , f 1 .t/ is an almost almost minimal almost filling measured lamination. That is, it has a sublamination f 1 .t/ without proper leaves whose complement supports at most a single simple closed geodesic. A measure of the complexity of this lamination is the length of the complementary geodesic, if one exists. We now find a sequence f 1 ; f 2 ; f 3 ; : : : such that the minimal length of all complementary geodesics to the f n .t/'s approaches infinity as n ! 1, provided such geodesics exist at all. After forgetting the measures, in the limit, we obtain the desired path in EL.S/ from 0 to 1 . (We don't worry about whether or not the f i 's converge to a path in ML.S/.) Since the final path can, in the appropriate sense, be taken arbitrarily close to the original (see Lemma 5.1), we obtain local path connectivity. Proving that the limit path is actually continuous requires control of the...

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“…Peter Storm conjectured that @C.S/ is path connected if S is any other finite type hyperbolic surface. See Question 10 of Kent and Leininger [9]. Saul Schleimer [17] showed if…”

Section: Introductionmentioning

confidence: 99%

Almost filling laminations and the connectivity of ending lamination space

Gabai

2009

Geom. Topol.

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“…This distinguishes our characterization of convex cocompact subgroups of mapping class groups from those appearing in [FM02,KL08,Ham05]. We prove Theorem 1.1.…”

Section: Introductionmentioning

confidence: 88%

“…Kent-Leininger [KL08] and, independently, Hamenstädt [Ham05] gave a characterization of convex cocompactness in terms of the curve graph CpSq:…”

Section: Hyperbolic Geometrymentioning

confidence: 99%

“…Our starting point is the following characterization of convex cocompact subgroups of the mapping class group, which follows easily from [KL08] or by combining results of [FM02] and [Raf10]. We provide a few details using these references.…”

Section: Convex Cocompactness Implies Stabilitymentioning

confidence: 99%

“…The first statement is contained in the proof of Theorem 7.4 of [KL08], where the assumption on G is that the orbit map from G into CpSq is a quasi-isometric embedding.…”

Section: Convex Cocompactness Implies Stabilitymentioning

confidence: 99%

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Convex cocompactness and stability in mapping class groups

Durham¹,

Taylor²

2015

Algebr. Geom. Topol.

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Abstract. We introduce a strong notion of quasiconvexity in finitely generated groups, which we call stability. Stability agrees with quasiconvexity in hyperbolic groups and is preserved under quasi-isometry for finitely generated groups. We show that the stable subgroups of mapping class groups are precisely the convex cocompact subgroups. This generalizes a well-known result of Behrstock [Beh06] and is related to questions asked by .

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A Combination Theorem for Metric Bundles

Sardar

2012

Geom. Funct. Anal.

149

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We define metric bundles/metric graph bundles which provide a purely topological/coarse-geometric generalization of the notion of trees of metric spaces a la Bestvina-Feighn in the special case that the inclusions of the edge spaces into the vertex spaces are uniform coarsely surjective quasi-isometries. We prove the existence of quasi-isometric sections in this generality. Then we prove a combination theorem for metric (graph) bundles (including exact sequences of groups) that establishes sufficient conditions, particularly flaring, under which the metric bundles are hyperbolic. We use this to give examples of surface bundles over hyperbolic disks, whose universal cover is Gromov-hyperbolic. We also show that in typical situations, flaring is also a necessary condition.Comment: v3: Major revision: 56 pages 5 figures. Many details added. Characterization of convex cocompact subgroups of mapping class groups of surfaces with punctures in terms of relative hyperbolicity given v4: Final version incorporating referee comments: 63 pages 5 figures. To appear in Geom. Funct. Ana

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Shadows Of Mapping Class Groups: Capturing Convex Cocompactness

Abstract: We strengthen the analogy between convex cocompact Kleinian groups and convex cocompact subgroups of the mapping class group of a surface in the sense of B. Farb and L. Mosher.

Cited by 70 publications

References 46 publications

Almost filling laminations and the connectivity of ending lamination space

Almost filling laminations and the connectivity of ending lamination space

Convex cocompactness and stability in mapping class groups

A Combination Theorem for Metric Bundles

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