Distortion of the exponent of convergence in space. (English) Zbl 1069.30031

Bonfert-Taylor, Petra; Bridgeman, Martin; Taylor, Edward C.

Ann. Acad. Sci. Fenn., Math. 29, No. 2, 383-406 (2004).

For a discrete quasiconformal (qc) group \(G\) acting on \(\overline {\mathbb R}^n\) having regular set \( \Omega(G)\) and set of discontinuity \(\Lambda(G)\) the authors define the chordal exponent of convergence as \[ \delta_{\text{chord}}(G) = \inf \left\{ s > 0: \sum_{g\in G} {\text{ dist}}_{\text{chord}}\left( g(z_0),\Lambda(G)\right)^s <\infty \right\} \] for a fixed \(z_0\in \Omega(G)\). If \( \Omega(G)\neq \Phi\) and \(|\Lambda(G)| \geq 2,\) the authors prove that \(\delta_{\text{chord}}(G) = \delta_{\text{hyp}}(G)\) where \(\delta_{\text{hyp}}(G)\) is the exponent defined in terms of the Poincaré series.
In their main theorem the authors study the conjugation by a \(K\)-qc mapping \(\varphi: \overline {\mathbb R}^n \to \overline {\mathbb R}^n\) and prove, with \(H= \varphi G \varphi^{-1},\) \[ \delta_{\text{chord}}(H) \leq (n+c) \delta_{\text{chord}}(G)/(c+ \delta_{\text{chord}}(G)). \] The constant \(c\) comes from the integrability of the \(K\)-qc mapping and only depends on \(n\) and \(K\).

Reviewer: Matti Vuorinen (Turku)

MSC:

30C62	Quasiconformal mappings in the complex plane
30F40	Kleinian groups (aspects of compact Riemann surfaces and uniformization)