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Riemann mappings of invariant components of Kleinian groups. (English) Zbl 1184.30036

The author investigates complex analytic properties of Riemann mappings of simply connected invariant components of Kleinian groups. In particular, he considers the growth of the derivatives of Riemann mappings to understand Kleinian groups that are quasi-Fuchsian groups, regular \(b\)-groups, and Kleinian groups with bounded geometry.
Theorem 1.1. Let \(G\) be a finite generated non-elementary Kleinian group with an invariant component \(\Omega_{0}.\) Then, the following conditions are equivalent:
(1) \(\Omega_{0}\) is a Hölder domain;
(2) \(\Omega_{0}\) is a John domain;
(3) \(G\) is geometrically finite and a parabolic element stabilizes a round disk in \(\Omega_{0}\). Furthermore, if \(\Omega_{0}\) is simply connected, then \(\Omega_{0}\) is a Hölder domain if and only if it is a quasi-disk. Hence, \(G\) is a quasi-Fuchsian group.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30C35 General theory of conformal mappings
30C62 Quasiconformal mappings in the complex plane
30C20 Conformal mappings of special domains
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