Dimensions of limit sets of Kleinian groups

Abstract

In this paper we give a brief survey of results on dimension gaps for limit sets of geometrically infinite Kleinian groups. We concentrate on an important notion from geometric group theory, amenability, as a criterion for the existence of such gaps. 1. Dimensions and invariants in conformal dynamics One goal often encountered in mathematics is to prove equality of independently defined invariants for large classes of a certain mathematical object. An instance of this in conformal dynamics has been the attempt to show that the critical exponent of the Poincaré series associated to a conformal dynamical system , e.g. a Kleinian group or a rational map on the Riemann sphere, coincides with the Hausdorff dimension of the corresponding limit set or Julia set, respectively. Since here we will be dealing mainly with Kleinian groups, i.e. discrete, torsion-free subgroups of the group of orientation preserving isometries of (n+ 1)-dimensional hyperbolic space H, n ∈ N, let us first explain briefly what these invariants are. The limit set L(G) of a Kleinian group G is the set of accumulation points of some and thus any orbit Gx of G, x ∈ H, and is a subset of the boundary ∂H of H due to the discreteness of G. The Poincaré series of G is defined as a Dirichlet series