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Quasiconformal groups of compact type. (English) Zbl 1281.30033
Summary: We establish that a quasiconformal group is of compact type if and only if its limits set is purely conical and find that the limit set of a quasiconformal group of compact type is uniformly perfect. A key tool is the result of Bowditch-Tukia on compact-type convergence groups. These results provide crucial tools for studying the deformations of quasiconformal groups and in establishing isomorphisms between such groups and conformal groups.

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
57S30 Discontinuous groups of transformations
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