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On the growth of quotients of Kleinian groups

Published online by Cambridge University Press:  26 May 2010

FRANÇOISE DAL’BO ,
MARC PEIGNÉ ,
JEAN-CLAUDE PICAUD  and
ANDREA SAMBUSETTI
FRANÇOISE DAL’BO
Affiliation:
IRMAR, Université de Rennes-I, Campus de Beaulieu, 35042 Rennes Cedex, France (email: francoise.dalbo@univ-rennes1.fr)
MARC PEIGNÉ
Affiliation:
LMPT, UMR 6083, Faculté des Sciences et Techniques, Parc de Grandmont, 37200 Tours, France (email: peigne@univ-tours.fr, jean-claude.picaud@univ-tours.fr)
JEAN-CLAUDE PICAUD
Affiliation:
LMPT, UMR 6083, Faculté des Sciences et Techniques, Parc de Grandmont, 37200 Tours, France (email: peigne@univ-tours.fr, jean-claude.picaud@univ-tours.fr)
ANDREA SAMBUSETTI
Affiliation:
Istituto di Matematica G. Castelnuovo Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Roma, Italy (email: sambuset@mat.uniroma1.it)
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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 of G and the ‘critical gap property’ . As a corollary, we prove that every geometrically finite Kleinian group satisfying the parabolic gap condition (i.e. δP<δG 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 31 , Issue 3 , June 2011 , pp. 835 - 851
Copyright
Copyright © Cambridge University Press 2010

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