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Dal’Bo, Françoise; Otal, Jean-Pierre; Peigné, Marc

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

Isr. J. Math. 118, 109-124 (2000).
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.
Reviewer: Paul Popescu (Craiova)

Cited in 2 Reviews
Cited in 38 Documents

MSC:

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

Keywords:

Poincaré series; geometrically finite group; parabolic group; divergent group of isometries
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\textit{F. Dal'Bo} et al., Isr. J. Math. 118, 109--124 (2000; Zbl 0968.53023)
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