Quasiconformal homogeneity of hyperbolic manifolds (Q1772070)

The first result proved here shows that there are severe restrictions on the geometry of uniformly quasiconformally homogeneous hyperbolic manifolds [cf. \textit{F. W. Gehring} and \textit{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.