Growth and cogrowth tightness of Kleinian and hyperbolic groups (Geometry and Analysis of Discrete Groups and Hyperbolic Spaces)

Abstract

Let G be a non-elementary discrete isometry group of the hyperbolic space or more generally a proper geodesic Gromov hyperbolic space X. We say that G is growth tight if for any non-trivial normal subgroup H the critical exponent k(H＼G) of the quotient group is strictly smaller than k(G). Moreover, G is cogrowth tight if the critical exponent δ(H) of any such H is strictly greater than δ(G)/2. We review recent results on these properties of G with the addition of certain new observation. In particular, we see that a non-elementary quasi-convex cocompact discrete isometry group G of X is growth tight.