Local connectivity, Kleinian groups and geodesics on the blowup of the torus
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- McMullen, C. Invent. math. (2001) 146: 35. doi:10.1007/PL00005809
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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 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.