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Gaps in the exponent spectrum of subgroups of discrete quasiconformal groups. (English) Zbl 1148.30025
The authors study a class of convergence groups, discrete quasiconformal groups, acting on the $$n$$-dimensional unit ball $${\mathbb B}^n$$. A group $$G$$ of quasiconformal homeomorphisms is a discrete quasiconformal group if there exists a uniform bound for the dilatation coefficients of all elements of $$G$$. For example, a Kleinian group acting on $${\mathbb B}^n$$ can be regarded as a discrete $$1$$-quasiconformal group. The main result is the following. Denote by $$\Lambda^s_c(G)$$ the set of strong conical limit points of $$G$$, i.e. points $$\zeta\in S^{n-1}$$ for which there exists a sequence $$(g_j)$$ in $$G$$ such that $$(g_j(0))$$ converges to $$\zeta$$ within a Euclidean cone based at $$\zeta$$, and $$(g_j^{-1}(0))$$ converges to a point $$b\neq \zeta$$. The set of all $$K$$-fat horospherical limit points of $$G$$, $$\Lambda_K(G)$$, is the set of all points $$\zeta\in S^{n-1}$$ for which there exists a sequence $$(g_j)$$ in $$G$$ and a constant $$C>0$$ such that
$\frac{1-| g_j(0)| }{| g_j(0)-\zeta| ^{K+1}} \geq C$ for all $$j$$. Suppose that $$G$$ is a discrete $$K$$-quasiconformal group acting on $${\mathbb B}^n$$ with empty regular set, and let $$\widehat G$$ be a non-elementary normal subgroup of $$G$$. Then
$\Lambda^s_c(G) \subset \Lambda_K(\widehat G).$ For $$n=3$$, the authors show that for a discrete quasiconformal group $$G$$ acting on $${\mathbb B}^3$$, with empty regular set and purely conical limit set, the strong conical limit set of $$G$$ has full $$2$$-dimensional Lebesgue measure in $$S^2$$. As an application, a lower bound for the exponent of convergence of a non-elementary normal subgroup $$\widehat G$$ of $$G$$ is given.
##### MSC:
 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations
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