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

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