By Allen Hatcher

ISBN-10: 052179160X

ISBN-13: 9780521791601

ISBN-10: 0521795400

ISBN-13: 9780521795401

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D. thesis, University of Michigan, 1996. [62] M. B. Shalen, “Varieties of group representations and splittings of 3-manifolds,” Annals of Math. 117(1983), 109–146. [63] M. B. S. 5(1992), 231–288. [64] W. Dunbar and R. Meyerhoff, “Volumes of hyperbolic 3-orbifolds,” Ind. Math. J. 43(1994), 611–637. A. Epstein, “Transversely hyperbolic 1-dimensional foliations,” Astérisque 116(1984), 53–69. A. Epstein and A. Marden, “Convex hulls in hyperbolic space, a Theorem of Sullivan, and measured pleated surfaces,” in Analytical and Geometrical Aspects of Hyperbolic Space, Cambridge University Press, 1987, 113–253.

It can be derived from the work of Bonahon [24, 25] but is not explicitly stated in any of his papers. We say that an isomorphism τ: → is weakly type-preserving if whenever γ ∈ is parabolic, then τ(γ) is parabolic. 1: Let S be a finite area hyperbolic surface and let ρ: π1 (S) → PSL2 (C) be a discrete, faithful, geometrically tame, weakly type-preserving representation. There exist two geodesic laminations λ+ and λ− on S, such that α ∈ GL (S) is realizable if and only if α contains no component of λ+ or λ− .

2 Suppose {Mi } converges to M in (N). Then it is clear that we may choose associated developing maps {Di } and D such that Di converges to D in the ˜ X)(0 ≤ r ≤ ∞). Intuitively, any compact set in Ñ compact C r -topology of C(N, is covered by a finite number of lifts of coordinate charts, and one may control the behaviour on each coordinate chart. 3. Theorem: Limit is an embedding. Suppose M i converges to M in (N). Let Di and D be the associated developing maps such that Di converges ˜ to D, and suppose that D|K is an embedding, for some compact subset K of N.

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