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Quasiconformal homogeneity of hyperbolic manifolds. (English) Zbl 1063.30020
The first result proved here shows that there are severe restrictions on the geometry of uniformly quasiconformally homogeneous hyperbolic manifolds [cf. F. W. Gehring and B. P. Palka, J. Anal. Math. 30, 172–199 (1976; Zbl 0349.30019)]. In particular, a geometrically finite hyperbolic \(n\)-manifold is uniformly quasiconformally homogeneous if and only if it is closed. For a hyperbolic \(n\)-manifold with \(n > 2\), it is shown that (i) it is uniformly quasi-conformally homogeneous if and only if it is a regular cover of a closed hyperbolic orbifold; (ii) there exists a uniform lower bound on the quasiconformal homogeneity constant. Deviations in the case of \(n=2\) are also highlighted.

MSC:
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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