On the growth of quotients of Kleinian groups

Abstract

We study the growth and divergence of quotients of Kleinian groups G (i.e. discrete, torsionless groups of isometries of a Cartan-Hadamard manifold with pinched negative curvature). Namely, we give general criteria ensuring the divergence of a quotient group (G) over bar of G and the 'critical gap property' delta((G) over bar) < delta(G). As a corollary, we prove that every geometrically finite Kleinian group satisfying the parabolic gap condition (i.e. delta(P) < delta(G) for every parabolic subgroup P of G) is growth tight. These quotient groups naturally act on non-simply connected quotients of a Cartan-Hadamard manifold, so the classical arguments of Patterson-Sullivan theory are not available here; this forces us to adopt a more elementary approach, yielding as by-product a new elementary proof of the classical results of divergence for geometrically finite groups in the simply connected case. We construct some examples of quotients of Kleinian groups and discuss the optimality of our results.