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 dsphere 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

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MathSciNet:
MR2354??? 
Z'blattMATH:
pre05231328 
^{†}indicates author for correspondence 
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