A note on geometric upper bounds for the exponent of convergence of convex cocompact Kleinian groups
Let \(\Gamma\) be a convex cocompact Kleinian group acting on the \((N+1)\)-dimensional hyperbolic space. Let \(\delta(\Gamma)\) be its ``exponent of convergence''. Then it is known if \(\Gamma\) is of the second kind \(\delta(\Gamma)< N\). In this paper the author provides an explicit lower bound for \(N-\Delta(\Gamma)\) in terms of two geometric quantities. The first of these is the diameter of the quotient of the convex core of \(\Gamma\) by \(\Gamma\). The second is less conventional and depends on the geometry of the ends and depends on the geometry of geodesics from the ordinary set and perpendicular to the boundary of the convex core. The proof depends on a refinement of Sullivan's ``Shadow Lemma''. A similar result was given by \textit{M. Burger} and \textit{R. D. Canary} [J. Reine Angew. Math. 454, 37-57 (1994; Zbl 0806.53046)] but this does not use the second invariant introduced in this paper. This estimate has consequences for the eigenvalues of the Laplace operator, for the orbital counting problem and for the fractal geometry of the limit set.
scientific article; zbMATH DE number 1893645
scientific article