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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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