Maximal Convergence Groups and Rank One Symmetric Spaces

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 Mobius 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.