## Maps on Boundaries of Hyperbolic Metric Spaces |

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3-manifold 6-hyperbolic Andrew Casson assume automorphism bi-infinite geodesic C",0)-quasi-isometric section Cannon-Thurston map Cayley graph Chabauty topology Choosing conjugacy class construct Corollary coset cyclic conjugate da(z define denote a geodesic depending dh(p dh(r dh(z distortion function e)-quasi-isometric section e)-quasigeodesic edge edge-path representative ending laminations exists finitely generated groups following Lemma free homotopy representative geodesic joining geodesic path geodesic ray geodesic representative geodesic subsegment Given K P H-invariant Hence homeomorphic hyperbolic 3-manifold hyperbolic group hyperbolic metric spaces hyperbolic subgroup i(TH IIA(y injectivity radius injw iterated exponential Kleinian groups leaf Lemma Let H lies main theorem metric spaces satisfying neighborhood normal subgroup º º Öſo path metric pleated surfaces Proof Proposition 3.1.1 quasi-isometrically embedded quasiconvex quasigeodesic segment in TH short exact sequence shortest single-valued quasi-isometric section subset Teichmuller TH joining Thurston map toog triangle inequality vertex set z e 6To