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

Let N = H-3/Gamma be a hyperbolic 3-manifold with free fundamental group pi (1)(N) congruent to Gamma congruent to [A, B], such that [A, B] is parabol ic. We show that the limit set Lambda of N is always locally connected. Mor e precisely, let Sigma be a compact surface of genus 1 with a single bounda ry component, equipped with the Fuchsian action of pi (1)(Sigma) on the cir cle S-infinity(1). We show that for any homotopy equivalence f : Sigma --> N, there is a natural continuous map F : S-infinity(1) --> Lambda subset of S-infinity(2), respecting the action of pi (1)(Sigma). In the course of the proof we deter mine the location of all closed geodesics in N, using a factorization of el ements of pi (1)(Sigma) into simple loops.