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Algebraic convergence of Schottky groups


Author: Richard D. Canary
Journal: Trans. Amer. Math. Soc. 337 (1993), 235-258
MSC: Primary 30F40; Secondary 30F60, 32G15, 57M07, 57S30
DOI: https://doi.org/10.1090/S0002-9947-1993-1137257-9
MathSciNet review: 1137257
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Abstract: A discrete faithful representation of the free group on $ g$ generators $ {F_g}$ into $ \operatorname{Isom}_ + ({{\mathbf{H}}^3})$ is said to be a Schottky group if $ ({{\mathbf{H}}^3} \cup {D_\Gamma })/\Gamma $ is homeomorphic to a handlebody $ {H_g}$ (where $ {D_\Gamma }$ is the domain of discontinuity for $ \Gamma $'s action on the sphere at infinity for $ {{\mathbf{H}}^3}$). Schottky space $ {\mathcal{S}_g}$, the space of all Schottky groups, is parameterized by the quotient of the Teichmüller space $ \mathcal{T}({S_g})$ of the closed surface of genus $ g$ by $ {\operatorname{Mod} _0}({H_g})$ where $ {\operatorname{Mod} _0}({H_g})$ is the group of (isotopy classes of) homeomorphisms of $ {S_g}$ which extend to homeomorphisms of $ {H_g}$ which are homotopic to the identity. Masur exhibited a domain $ \mathcal{O}({H_g})$ of discontinuity for $ {\operatorname{Mod} _0}({H_g})$'s action on $ PL({S_g})$ (the space of projective measured laminations on $ {S_g}$), so $ \mathcal{B}({H_g}) = \mathcal{O}({H_g})/{\operatorname{Mod} _0}({H_g})$ may be appended to $ {\mathcal{S}_g}$ as a boundary. Thurston conjectured that if a sequence $ \{ {\rho _i}:{F_g} \to \operatorname{Isom}_ + ({{\mathbf{H}}^3})\} $ of Schottky groups converged into $ \mathcal{B}({H_g})$, then it converged as a sequence of representations, up to subsequence and conjugation. In this paper, we prove Thurston's conjecture in the case where $ {H_g}$ is homeomorphic to $ S \times I$ and the length $ {l_{{N_i}}}({(\partial S)^\ast})$ in $ {N_i} = {{\mathbf{H}}^3}/{\rho _i}({F_g})$ of the closed geodesic(s) in the homotopy class of the boundary of $ S$ is bounded above by some constant $ K$.


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DOI: https://doi.org/10.1090/S0002-9947-1993-1137257-9
Article copyright: © Copyright 1993 American Mathematical Society

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