J. Aust. Math. Soc. 83 (2007), no. 1, pp. 1–9.

Maximal convergence groups and rank one symmetric spaces

Ara Basmajian^†	Mahmoud Zeinalian
Department of Mathematics Hunter College and Graduate Center City University of New York 365 Fifth Avenue New York NY 10016-4309 USA abasmaji@hunter.cuny.edu	Department of Mathematics C.W. Post Campus Long Island University 720 Northern Boulevard Brookville, NY 11548 USA mzeinalian@liu.edu

Received 12 February 2005; revised 16 March 2006
Communicated by C. D. Hodson

Abstract

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.

Download the article in PDF format (size 112 Kb)

2000 Mathematics Subject Classification: primary 30F40; secondary 30G65, 37F30
(Metadata: XML, RSS, BibTeX)	MathSciNet: MR2354???	Z'blatt-MATH: pre05231328
^†indicates author for correspondence

References

A. Basmajian and R. Miner, ‘Discrete subgroups of complex hyperbolic motions’, Invent. Math. 131 (1998), 85–136. MR1489895
F. W. Gehring and G. J. Martin, ‘Discrete quasiconformal groups I’, Proc. London Math. Soc. (3) 55 (1987), 331–358. MR896224
M. Gromov and P. Pansu, Rigidity of lattices: an introduction, Lecture Notes in Math. 1504 (Springer, Berlin, 1991) pp. 39–137. MR1168043
J. Heinonen, Lectures on analysis on metric spaces, Universitext (Springer-Verlag, New York, 2001). MR1800917
J. Heinonen and P. Koskela, ‘Quasiconformal maps in metric spaces with controlled geometry’, Acta Math. 181 (1998), 1–61. MR1654771
A. Koranyi and H. M. Reimann, ‘Foundations for the theory of quasiconformal mappings on the Heisenberg group’, Adv. Math. 111 (1995), 1–87. MR1317384
G. Mostow, Strong rigidity of locally symmetric spaces, Ann. Math. Stud. 78 (Princeton University Press, Princeton, 1978). MR385004
P. Pansu, Quasiconformal mappings and manifolds of negative curvature, Lecture Notes in Math. 1201 (Springer, Berlin, 1986). MR859587
P. Pansu, Quasiisometries des varietes a courbure negative (Ph.D. Thesis, Paris, 1987).
P. Pansu, ‘Metrique de Carnot-Carathéodory et quasi-isométries des espaces symmetrique de rang un’, Ann. of Math. (2) 129 (1989), 1–60. MR979599
P. Tukia, ‘Convergence groups and Gromov's metric hyperbolic spaces’, New Zealand J. Math. 23 (1994), 157–187. MR1313451

Australian Mathematical Publishing Association Inc.