Algebraic convergence of Schottky groups

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 $K$.