The continuity of convex cores with respect to the geometric topology

Abstract

The volume of the convex core, the bottom of the spectrum of the Laplacian, and the Hausdorff dimension of the limit set are important invariants of hyperbolic 3-manifolds/Kleinian groups, which are closely related to one another. The behaviour of such invariants with respect to the convergence of Kleinian groups is an interesting problem to study, and also, it is expected that these invariants would be useful to investigate the topological structure of deformation spaces. The behaviour of these invariants with respect to algebraic and geometric convergence of Kleinian groups has been studied by Canary, Taylor, McMullen, among others. (See [10], [27], [31].) For studying this problem, it is essential to see whether convex cores of the corresponding hyperbolic 3-manifolds converge to the convex core of the limit geometrically in the sense of Gromov. Consider a sequence of Kleinian groups {Gi} converging geometrically to G∞. The limit set ΛG of a Kleinian group G coincides with the intersection of S2 ∞ and the closure of the Nielsen convex hull of G in H3 ∪ S2 ∞. If the convex cores C(H/Gi) converge geometrically to C(H/G∞), then the Nielsen convex hulls of Gi converge to that of G∞ with respect to the Hausdorff topology of H3. This implies that the limit sets ΛGi converge to ΛG∞ with respect to the Hausdorff topology on S2 ∞. Conversely, as was shown by Bowditch [4], if the limit sets ΛGi converge to ΛG∞ , then the convex cores C(H/Gi) converge to C(H/G∞) geometrically in the sense of Gromov. We are interested in the following question. Do the convex cores of H/Gi converge geometrically to that of H/G∞? Or equivalently, do the limit sets ΛGi converge to ΛG∞ with respect to the Hausdorff topology?