Local connectivity, Kleinian groups and geodesics on the blowup of the torus

By Curtis T. McMullen. 2001

73 citations recorded by the source corpus.

Abstract.Let N=?3/Γ be a hyperbolic 3-manifold with free fundamental group π1(N)≅Γ≅, such that [A,B] is parabolic. We show that the limit set λ of N is always locally connected. More precisely, let Σ be a compact surface of genus 1 with a single boundary component, equipped with the Fuchsian action of π1(Σ) on the circle Sinfty1. We show that for any homotopy equivalence f:Σ?N, there is a natural continuous map¶¶F:Sinfty1?λ⊂Sinfty2,¶¶respecting the action of π1(Σ). In the course of the proof we determine the location of all closed geodesics in N, using a factorization of elements of π1(Σ) into simple loops.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/824538b78f59cb023967eaec8980fab85c32b928

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In Discrete Mathematics and Combinatorics

1. Anders Karlsson
2. V. Gerasimov
3. Michael Kapovich
4. Christoph Schwer
5. Arun Ram

In Geometry and Topology

1. V. Gerasimov
2. Anders Karlsson
3. Michael Kapovich
4. J. Porti
5. B. Leeb
6. Anna Wienhard
7. Jean-Marc Schlenker
8. Cornelia Drutu

In Mathematical Physics

1. V. Gerasimov
2. Anders Karlsson
3. Michael Kapovich
4. Anna Wienhard
5. B. Leeb
6. J. Porti
7. Rostislav I. Grigorchuk

Citations in the corpus

• Free Subgroups of Groups with Nontrivial Floyd Boundary (2003, background): “In the recent article (McMullen, 2001), a conjecture is stated concerning the existence of a continuous surjective map from the Floyd boundary of a finitely generated fundamental group of a hyperbolic 3-manifold onto its limit set on the boundary in hyperbolic 3-space.”
• Bounded geometry for Kleinian groups (2001, unknown): “Appendix: Hyperbolic geometry 36 References 47”
• Expansive Convergence Groups are Relatively Hyperbolic (2009, methodology): “It acts properly on M. If M is δ-hyperbolic [Gr1] then the action of G on the hyperbolic compactification M = M ∪ ∂∞M is 3-proper (=has convergence property).”
• KLEINIAN GROUPS IN HIGHER DIMENSIONS (2007, background): “nd the following problem is open even for n= 2: Problem 8.6. Suppose that Γ ⊂Mob(Sn) is a Kleinian group whose limit set is connected and 1-dimensional. Is it true that Λ(Γ) is locally connected? See [161], [163] and [166, 167] for partial results in dimension 2. 9. Groups whose limit sets are topol”