Poincaré series of finite geometric groups. (Séries de Poincaré des groupes géométriquement finis.) (French) Zbl 0968.53023

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.


53C20 Global Riemannian geometry, including pinching
37A99 Ergodic theory
58D19 Group actions and symmetry properties
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