Hostname: page-component-cb9f654ff-mwwwr Total loading time: 0 Render date: 2025-08-22T23:18:48.213Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximal convergence groups and rank one symmetric spaces

Part of: Complex dynamical systems Riemann surfaces

Published online by Cambridge University Press:  09 April 2009

Ara Basmajian  and
Mahmoud Zeinalian
Ara Basmajian
Affiliation:
Department of MathematicsHunter College and Graduate CenterCity University of New York 365 Fifth Avenue New York NY 10016-4309USAabasmaji@hunter.cuny.edu
Mahmoud Zeinalian
Affiliation:
Department of MathematicsC.W. Post CampusLong Island University720 Northern Boulevard Brookville, NY 11548USAmzeinalian@liu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the group of conformal homeomorphisms of the boundary of a rank one symmetric space (except the hyperbolic plane) of noncompact type acts as a maximal convergence group. Moreover, we show that any family of uniformly quasiconformal homeomorphisms has the convergence property. Our theorems generalize results of Gehring and Martin in the real hyperbolic case for Möbius groups. As a consequence, this shows that the maximal convergence subgroups of the group of self homeomorphisms of the d–sphere are not unique up to conjugacy. Finally, we discuss some implications of maximality.

MSC classification

Secondary: 30F40: Kleinian groups 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 1 , August 2007 , pp. 1 - 10
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Basmajian, A. and Miner, R., ‘Discrete subgroups of complex hyperbolic motions’, Invent. Math. 131 (1998), 85–136.CrossRefGoogle Scholar
[2]Gehring, F. W. and Martin, G. J., ‘Discrete quasiconformal groups I’, Proc. London Math. Soc. (3) 55 (1987), 331–358.CrossRefGoogle Scholar
[3]Gromov, M. and Pansu, P., Rigidity of lattices: an introduction, Lecture Notes in Math. 1504 (Springer, Berlin, 1991) pp. 39–137.Google Scholar
[4]Heinonen, J., Lectures on analysis on metric spaces, Universitext (Springer-Verlag, New York, 2001).Google Scholar
[5]Heinonen, J. and Koskela, P., ‘Quasiconformal maps in metric spaces with controlled geometry’, Ada Math. 181 (1998), 1–61.Google Scholar
[6]Koranyi, A. and Reimann, H. M., ‘Foundations for the theory of quasiconformal mappings on the Heisenberg group’, Adv. Math. 111 (1995), 1–87.CrossRefGoogle Scholar
[7]Mostow, G., Strong rigidity of locally symmetric spaces, Ann. Math. Stud. 78 (Princeton University Press, Princeton, 1978).Google Scholar
[8]Pansu, P., Quasiconformal mappings and manifolds of negative curvature, Lecture Notes in Math. 1201 (Springer, Berlin, 1986).Google Scholar
[9]Pansu, P., Quasiisometries des varietes a courbure negative (Ph.D. Thesis, Paris, 1987).Google Scholar
[10]Pansu, P., ‘Metrique de Carnot-Carathéodory et quasi-isométries des espaces symmetrique de rang un’, Ann. of Math. (2) 129 (1989), 1–60.CrossRefGoogle Scholar
[11]Tukia, P., ‘Convergence groups and Gromov's metric hyperbolic spaces’, New Zealand J. Math. 23 (1994), 157–187.Google Scholar