# zbMATH — the first resource for mathematics

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.

##### MSC:
 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)
Full Text: