Floyd maps for relatively hyperbolic groups

Gerasimov, Victor

doi:10.1007/s00039-012-0175-6

Cited by 50 publications

(71 citation statements)

References 22 publications

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“…H is the set of ideal points of lifts of almost minimizing geodesic rays. Theorem 1.3 answers an issue that has come up in works of several authors [Kap95,Ger12,JKLO16] who tried to relate the injective points of the Cannon-Thurston map to the conical limit set. They concluded that the conical limit set is strictly contained in the set of injective points of the Cannon-Thurston map.…”

Section: Introductionmentioning

confidence: 92%

Horospheres in Degenerate 3-Manifolds

Lecuire

2016

Int Math Res Notices

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Abstract. We study horospheres in hyperbolic 3-manifolds M all whose ends are degenerate. Deciding which horospheres in M are properly embedded and which are dense reduces to a) studying the horospherical limit set; b) deciding which almost minimizing geodesics in M go through arbitrarily thin parts. As an answer to (a), we show that the horospherical limit set consists precisely of the injective points of the Cannon-Thurston map. Addressing (b), we provide characterizations, sufficient conditions as well as a number of examples and counterexamples.

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Section: Introductionmentioning

confidence: 92%

Horospheres in Degenerate 3-Manifolds

Lecuire

2016

Int Math Res Notices

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“…Only recently did the work of Baker and Riley [BR1] produce the first example of a word-hyperbolic subgroup H of a word-hyperbolic group G for which the inclusion H ≤ G does not extend to a Cannon-Thurston map. Analogs and generalizations of the Cannon-Thurston map have been studied in many other contexts, see for example [Kla,McM,Miy,LLR,LMS,Ger,Bow1,Bow2,MP,Mj2,Mj1,JKLO]. The best understood case concerns discrete isometric actions of surface groups on H 3 , where the most general results about Cannon-Thurston maps are due to Mj [Mj1].…”

Section: Introductionmentioning

confidence: 99%

Cannon–Thurston maps for hyperbolic free group extensions

2016

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This paper gives a detailed analysis of the Cannon-Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free groups of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E Γ of F, and the main theorem of Dowdall-Taylor [DT1] states that in this situation E Γ is hyperbolic if and only if Γ is purely atoroidal.Here, we give an explicit geometric description of the Cannon-Thurston maps ∂F → ∂E Γ for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon-Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig [KL5] which treats the special case where Γ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of Γ to the space of laminations of the free group (with the Chabauty topology) is not continuous.

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“…Recently, Mj [77] has shown that for any properly discontinuous action on H 3 without accidental parabolics, there exists a Cannon-Thurston map, using the theory of model manifolds which were developed by Minsky. There are extensions of the Cannon-Thurston maps also for subgroups of mapping class groups [62], and in other related contexts [39,41].…”

Section: Introductionmentioning

confidence: 99%

Conical limit points and the Cannon-Thurston map

Jeon

Kapovich

Leininger

et al. 2016

Conform. Geom. Dyn.

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Abstract. Let G be a non-elementary word-hyperbolic group acting as a convergence group on a compact metrizable space Z so that there exists a continuous G-equivariant map i : ∂G → Z, which we call a Cannon-Thurston map. We obtain two characterzations (a dynamical one and a geometric one) of conical limit points in Z in terms of their pre-images under the Cannon-Thurston map i. As an application we prove, under the extra assumption that the action of G on Z has no accidental parabolics, that if the map i is not injective then there exists a non-conical limit point z ∈ Z with |i −1 (z)| = 1. This result applies to most natural contexts where the Cannon-Thurston map is known to exist, including subgroups of wordhyperbolic groups and Kleinian representations of surface groups. As another application, we prove that if G is a non-elementary torsion-free word-hyperbolic group then there exists x ∈ ∂G such that x is not a "controlled concentration point" for the action of G on ∂G.

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Floyd maps for relatively hyperbolic groups

Cited by 50 publications

References 22 publications

Horospheres in Degenerate 3-Manifolds

Horospheres in Degenerate 3-Manifolds

Cannon–Thurston maps for hyperbolic free group extensions

Conical limit points and the Cannon-Thurston map

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