Local Connectivity, Kleinian Groups and Geodesics on the Blowup of the Torus

Abstract

Let N=?3/Γ be a hyperbolic 3-manifold with free fundamental group π1(N)≅Γ≅<A,B>, 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 S infty 1. We show that for any homotopy equivalence f:Σ?N, there is a natural continuous map¶¶F:S infty 1?λ⊂S infty 2,¶¶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.