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Hausdorff dimension and limit sets of quasiconformal groups. (English) Zbl 0999.30029
There is an extensive theory of Kleinian groups concerning the connections of the exponent of convergence of the Poincaré series and the Hausdorff dimension of the limit set or the conical limit set of the group. This paper extends this study to quasiconformal Fuchsian groups (QCF) of \(\overline{\mathbb{R}^n}\). These are groups of quasiconformal maps of \(\overline{\mathbb{R}^n}\) with uniformly bounded dilatation which preserve the unit ball \(\mathbb{B}^n\) and act discontinuously on \(\mathbb{B}^n\). It is possible to define the exponent of convergence \(\delta(G)\) for a QCF group like for Kleinian groups. It is shown that \(\delta(G)\) is at least the Hausdorff dimension of the conical limit set; if \(G\) is Kleinian, there is strict equality here but a counterexample shows that this is not the case for QCF groups. Another main result states that if \(G_i\) is a series of QCF groups tending toward \(G\) (in a suitable sense), it need not be true that \(\delta(G)= \lim\delta(G_i)\) or that \(\dim L(G)\leq\lim\inf\dim L(G_i)\) (here dim is the Hausdorff dimension), contrary to the behavior of Kleinian groups. There are some other results in this vein and the final section contains some counterexamples for convergence groups. This class of groups is still more general than QCF groups and here, it seems, nothing can be said.

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
37F35 Conformal densities and Hausdorff dimension for holomorphic dynamical systems
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