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.