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.

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