Hyperbolicity in Teichmüller space

We study the smallest positive eigenvalue λ1false(Mfalse) of the Laplace–Beltrami operator on a closed hyperbolic 3‐manifold M which fibers over the circle, with fiber a closed surface of genus g⩾2. We show the existence of a constant C>0 only depending on g so that λ1false(Mfalse)∈false[C−1/ vol false(Mfalse)2,Clog vol (M)/ vol false(Mfalse)22g−2/(22g−2−1)false] and that this estimate is essentially sharp. We show that if M is typical or random, then we have λ1false(Mfalse)∈false[C−1/ vol false(Mfalse)2,C/ vol false(Mfalse)2false]. This rests on a result of independent interest about reccurence properties of axes of random pseudo‐Anosov elements.

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“…Thus, τ (t i ) are within κ of some point of γ n and so by [27], τ ([t 1 , t 2 ]) is within κ of γ n where κ depends only on κ .…”

Section: Random Mapping Torimentioning

confidence: 99%

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

Baik

Gekhtman

Hamenstädt

2019

Proc. London Math. Soc.

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“…Combined with Proposition A.4, we have the following proposition about the structure of active segments of hierarchy paths, which resembles [26,Theorem 5.3] for Teichmüller geodesics.…”

Section: A1 Active Segmentsmentioning

confidence: 60%

The augmented marking complex of a surface

Durham

2016

J. London Math. Soc.

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We build an augmentation of the Masur-Minsky marking complex by Groves-Manning combinatorial horoballs to obtain a graph we call the augmented marking complex, AM(S). Adapting work of Masur-Minsky, we show that this augmented marking complex is quasiisometric to Teichmüller space with the Teichmüller metric. A similar construction was independently discovered by Eskin-Masur-Rafi. We also completely integrate the Masur-Minsky hierarchy machinery to AM(S) to build flexible families of uniform quasigeodesics in Teichmüller space. As an application, we give a new proof of Rafi's distance formula for T (S) with the Teichmüller metric. We have included an appendix, in which we prove a number of facts about hierarchies that we hope will be of independent interest.Theorem 1.1. The augmented marking complex, AM(S), is MCG(S)-equivariantly quasiisometric to T (S) in the Teichmüller metric.A large part of this paper is spent adapting the Masur-Minsky hierarchy machinery for M(S) and P(S) to AM(S). We use these augmented hierarchies for AM(S) to build families of uniform quasigeodesics called augmented hierarchy paths, and derive a version of Rafi's distance formula for the Teichmüller metric (Theorem 2.10), thereby completing the unification of the coarse geometries of MCG(S) and T (S) in the Weil-Petersson, and Teichmüller metrics by a common framework developed in [7,19,20,24,25]. In a recent paper, Eskin-Masur-Rafi [12] used AM(S) and augmented hierarchy paths, which they independently discovered,

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“…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%

“…then the Teichmüller geodesics τ i joining x and g i¨x become i -thin for i Ñ 0 [Theorem 5.5, [Raf10]]. (See also Theorem 4.1 of [RS09].)…”

Section: Convex Cocompactness Implies Stabilitymentioning

confidence: 99%

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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Hyperbolicity in Teichmüller space

Cited by 61 publications

References 17 publications

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

The smallest positive eigenvalue of fibered hyperbolic 3‐manifolds

The augmented marking complex of a surface

Convex cocompactness and stability in mapping class groups

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