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.
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Petra Bonfert-Taylor, Gaven J. Martin, Quasiconformal groups of compact type. Rev. Mat. Iberoam. 21 (2005), no. 3, pp. 997–1012DOI 10.4171/RMI/442