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

Let N = H/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 ∞ . We show that for any homotopy equivalence f : Σ → N , there is a natural continuous map