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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.

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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