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Convergence groups, Hausdorff dimension, and a Theorem of Sullivan and Tukia. (English) Zbl 1067.30041
A \(K\)-quasiconformal group \(G\) acting on \(\overline{{\mathbb R}^n} = {\mathbb R}^n\cup\{0\}\) is a discrete group of homeomorphisms, each of which is a \(K-\)quasiconformal mapping. A quasiconformal Fuchsian group (QCF) is a \(K\)-quasiconformal group preserving the upper-half space \({\mathbb H}^n\). By previous work of the second and third authors of the paper under review, there is a well-defined exponent of convergence \(\delta(G)\) of the Poincaré series of a QCF \(G\), generalizing the corresponding notion in the Kleinian group setting. In this paper, the authors prove the following
Theorem: Let \(G\) be a discrete quasiconformal Fuchsian group acting on \(\overline{{\mathbb R}^n}\) and having nonempty regular set in \(\overline{{\mathbb R}^{n-1}}\) and having a purely conical limit set. Then \(\delta(G)<n-1\).
Theorem: Let \(G\) be a discrete quasiconformal group acting on \(\overline{{\mathbb R}^n}\) having non-empty regular set and a purely conical limit set. Then \(\text{dim} L(G)<n-1\). The results partially generalize results of Sullivan and of Tukia which they proved in the case of geometrically finite Kleinian groups acting isometrically on \({\mathbb H}^n\). As an application, the authors obtain geometric information about infinite-index subgroups within certain of these groups. The paper contains other useful results and some open questions.

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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