Poincaré series of finite geometric groups (Q1580500)

A Poincaré series can be associated with a discrete group of isometries (in particular with a finite geometric group \(\Gamma \) considered in this paper) which acts on a Hadamard manifold (i.e., a negative pinched Riemannian manifold). The case when the group contains parabolic elements is studied in the paper under review. A sufficient condition on the parabolic subgroup of \(\Gamma \), in order that \(\Gamma \) be of divergent type, is given. If \(\Gamma \) is of divergent type, a necessary and sufficient condition for the Sullivan measure on the unit tangent bundle of the orbit space to be finite is also given; this condition expresses the convergence of certain series, which involves only parabolic elements of \(\Gamma \). Some suggestive examples are also given.