Geodesics in the mapping class group

Rafi, Kasra; Verberne, Yvon

doi:10.2140/agt.2021.21.2995

Cited by 7 publications

(5 citation statements)

References 20 publications

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“…Given a finite set of points F X in an HHS, the standard notions of a hull for F are very difficult to analyze. For example, while little is known about geodesics in the mapping class group, Rafi and Verberne [50] proved that geodesics do not always interact well with the curve graph machinery. In Teichmüller space with the Teichmüller metric, geodesics are unique, but it is an open question of Masur whether the classical convex hull of a set of three points can be the whole space.…”

Section: Stable Cubical Models For Hulls In Hhssmentioning

confidence: 99%

Stable cubulations, bicombings, and barycenters

Durham,

Minsky,

Sisto

2023

Geom. Topol.

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We prove that the hierarchical hulls of finite sets of points in mapping class groups and Teichmüller spaces are stably approximated by CAT(0) cube complexes, strengthening a result of Behrstock, Hagen and Sisto. As applications, we prove that mapping class groups are semihyperbolic and Teichmüller spaces are coarsely equivariantly bicombable, and both admit stable coarse barycenters. Our results apply to the broader class of "colorable" hierarchically hyperbolic spaces and groups.

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Section: Stable Cubical Models For Hulls In Hhssmentioning

confidence: 99%

Stable cubulations, bicombings, and barycenters

Durham,

Minsky,

Sisto

2023

Geom. Topol.

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“…I would like to thank Dan Margalit for suggesting I generalize the pseudo-Anosov map from [12], as well as many helpful conversations. I would like to thank Balázs Strenner for suggesting I analyze the number-theoretic properties associated to the stretch factors of the maps produced, and Joan Birman for suggesting I apply the construction to surfaces of higher genus.…”

Section: Acknowledgementsmentioning

confidence: 99%

A construction of pseudo-Anosov homeomorphisms using positive twists

Verberne

2023

Algebr. Geom. Topol.

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We introduce a construction of pseudo-Anosov homeomorphisms on n-times punctured spheres and surfaces of higher genus using only positive half-twists and Dehn twists. These constructions produce explicit examples of pseudo-Anosov maps with various number-theoretic properties associated to the stretch factors. For instance, we produce examples where the trace field is not totally real and the Galois conjugates of the stretch factor are on the unit circle. It follows that for these examples, no power of these maps can arise from either Thurston's or Penner's constructions.

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“…We give an example showing that the answer to this question is no. In order to do that, we will use the following theorem by Rafi and Verberne (Theorem 1.1 in [RV18]). For a surface S, we let Map(S) denote the mapping class group of S and C(S) denote the curve graph of S.…”

Section: A Remark On Hierarchically Hyperbolic Groupsmentioning

confidence: 99%

“…Therefore, it's natural to wonder if the same conclusion of Theorem 1.1 holds in the settings of hierarchically hyperbolic spaces. Using work of Rafi and Verberne [RV18], we show that the answer to this questions is negative.…”

mentioning

confidence: 92%

Sublinearly Morse boundaries from the viewpoint of combinatorics

Incerti-Medici¹,

Zalloum²

2021

Preprint

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We prove that the sublinearly Morse boundary of every known cubulated group continuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for all CAT(0) cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.

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Geodesics in the mapping class group

Cited by 7 publications

References 20 publications

Stable cubulations, bicombings, and barycenters

Stable cubulations, bicombings, and barycenters

A construction of pseudo-Anosov homeomorphisms using positive twists

Sublinearly Morse boundaries from the viewpoint of combinatorics

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