Skip to main content Accessibility help
×
×
Home

On the growth of quotients of Kleinian groups

  • FRANÇOISE DAL’BO (a1), MARC PEIGNÉ (a2), JEAN-CLAUDE PICAUD (a2) and ANDREA SAMBUSETTI (a3)
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.

Copyright
References
Hide All
[1]Arzhantseva, G. N. and Lysenok, I. G.. Growth tightness for word hyperbolic groups. Math. Z. 241(3) (2002), 597611.
[2]Babenko, I. K.. Asymptotic invariants of smooth manifolds. Russian Acad. Sci. Izv. Math. 41(1) (1993), 138.
[3]Babillot, M. and Peigné, M.. Asymptotic laws for geodesic homology on hyperbolic manifolds with cusps. Bull. Soc. Math. France 134(1) (2006), 119163.
[4]Bowditch, B.. Geometrical finiteness with variable negative curvature. Duke Math. J. 77 (1995), 229274.
[5]Corlette, K. and Iozzi, A.. Limit sets of isometry groups of exotic hyperbolic spaces. Trans. Amer. Math. Soc. 351(4) (1999), 15071530.
[6]Dal’bo, F., Otal, J. P. and Peigné, M.. Séries de Poincaré des groupes géométriquement finis. Israel J. Math. 118 (2000), 109124.
[7]Dal’bo, F., Peigné, M., Picaud, J. C. and Sambusetti, A.. On the growth of non-uniform lattices in pinched negatively curved manifolds. J. Reine Angew. Math. 627 (2009), 3152.
[8]de la Harpe, P.. Topics in Geometric Group Theory (Chicago Lectures in Mathematics). Chicago University Press, Chicago, 2000.
[9]Ghys, E. and de la Harpe, P.. Sur les groupes hyperboliques d’après Mikhael Gromov. Birkhäuser, Basel, 1990.
[10]Grigorchuk, R. and de la Harpe, P.. On problems related to growth, entropy and spectrum in group theory. J. Dyn. Control Syst. 3(1) (1997), 5189.
[11]Nicholls, P. J.. The Ergodic Theory of Discrete Groups (London Mathematical Society Lecture Note Series, 143). Cambridge University Press, Cambridge, 1989.
[12]Otal, J. P. and Peigné, M.. Principe variationnel et groupes Kleiniens. Duke Math. J. 125(1) (2004), 1544.
[13]Pólya, G. and Szegö, G.. Problems and Theorems in Analysis, Vol. I & II (Grundlehren der mathematischen Wissenschaften, 193 and 216). Springer, Berlin, 1972 & 1976.
[14]Robert, G.. Comptage pour des groupes co-compacts d’isométries d’un espace hyperbolique au sens de Gromov. Preprint.
[15]Roblin, T.. Ergodicité et equidistribution en courbure négative (Mémoires de la Société Mathématique de France (N.S.), 95). Société Mathématique de France, Paris, 2003.
[16]Sambusetti, A.. Growth tightness of surfaces groups. Expositiones Mathematicae 20 (2002), 335363.
[17]Sambusetti, A.. Growth tightness of free and amalgamated products. Ann. Sci. École Norm. Sup. (4) série 35 (2002), 477488.
[18]Sambusetti, A.. Asymptotic properties of coverings in negative curvature. Geom. Topol. 12(1) (2008), 617637.
[19]Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171202.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed