Trees and mapping class groups

If S is a hyperbolic surface andS the surface obtained from S by removing a point, the mapping class groups Mod(S) and Mod(S) fit into a short exact sequenceWe give a new criterion for mapping classes in the kernel to be pseudo-Anosov using the geometry of hyperbolic 3-manifolds. Namely, we show that if M is an ε-thick hyperbolic manifold homeomorphic to S × R, then an element of π 1 (M) ∼ = π 1 (S) represents a pseudo-Anosov element of Mod(S) if its geodesic representative is "wide." We establish similar criteria where M is replaced with a coarsely hyperbolic surface bundle coming from a δ -hyperbolic surface-group extension.

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“…In particular, this answers a special case of Question 1.5 of [9], and generalizes Theorem 6.1 of [13].…”

Section: Criterion 2 (Width Criterion)supporting

confidence: 65%

“…When k is nonzero, this element represents a pseudo-Anosov mapping class in Mod(S). When k is zero, this element lies in π 1 (S), and, by a theorem of Kra [15] (see also [13]), it is pseudo-Anosov in Mod(S) if and only if it fills S. These observations were first made by Ian Agol [2].…”

Section: Introduction: Mapping Classes From Fibrationsmentioning

confidence: 97%

A geometric criterion to be pseudo-Anosov

Kent

Leininger²

2014

Michigan Math. J.

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“…From [KLS09], the fiber over v 2 C 0 .S/ is 1 .S/-equivariantly isomorphic to the Bass-Serre tree T v determined by v. The action of 1 .S/ on C .S; z/ comes from the inclusion into the mapping class group Mod.S; z/ via the Birman exact sequence; see Section 1.2.3. We define a map W C .S/ H !…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…Associated to v there is an action of 1 .S/ on a tree T v , namely, the Bass-Serre tree for the splitting of 1 .S/ determined by v. We will make use of the following theorem of [KLS09]. Theorem 1.7 (Kent-Leininger-Schleimer).…”

Section: Curve Complexesmentioning

confidence: 99%

The universal Cannon–Thurston map and the boundary of the curve complex

Leininger¹,

Mj²,

Schleimer³

2011

Comment. Math. Helv.

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Abstract. In genus two and higher, the fundamental group of a closed surface acts naturally on the curve complex of the surface with one puncture. Combining ideas from previous work of Kent-Leininger-Schleimer and Mitra, we construct a universal Cannon-Thurston map from a subset of the circle at infinity for the closed surface group onto the boundary of the curve complex of the once-punctured surface. Using the techniques we have developed, we also show that the boundary of this curve complex is locally path-connected.Mathematics Subject Classification (2010). 20F67 Primary; Secondary 22E40, 57M50.

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“…It is easily seen to be true for subgroups of Veech groups, which preserve a hyperbolic disk isometrically embedded in Teichmüller space [KL07]. A more significant case is resolved by [DKL14] (generalizing [KLS09]) who answer Question 1.4 affirmatively for subgroups of certain hyperbolic 3-manifold groups embedded in Mod(S). Theorem 1.1 completes the affirmative answer for a third case, the family of mapping class subgroups studied by the second and third authors in [MT13].…”

Section: Theorem 13 ([Dt14 Ham05 Kl08])mentioning

confidence: 99%

The geometry of purely loxodromic subgroups of right-angled Artin groups

Koberda

Mangahas

Taylor

2017

Trans. Amer. Math. Soc.

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Abstract. We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group A(Γ) fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups Mod(S). In particular, such subgroups are quasiconvex in A(Γ). In addition, we identify a milder condition for a finitely generated subgroup of A(Γ) that guarantees it is free, undistorted, and retains finite generation when intersected with A(Λ) for subgraphs Λ of Γ. These results have applications to both the study of convex cocompactness in Mod(S) and the way in which certain groups can embed in right-angled Artin groups.

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Trees and mapping class groups

Cited by 31 publications

References 15 publications

A geometric criterion to be pseudo-Anosov

A geometric criterion to be pseudo-Anosov

The universal Cannon–Thurston map and the boundary of the curve complex

The geometry of purely loxodromic subgroups of right-angled Artin groups

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