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A lower bound of maximal dilatations of quasiconformal automorphisms acting on Fuchsian groups. (English) Zbl 1089.30046

Let \(\Gamma \) be a torsion free Fuchsian group acting on the upper half-plane H and \(f\) a normalized quasiconformal automorphism of H such that \(f\circ \Gamma \circ f^{-1}=\Gamma \). Let \(L\) be the axis of a simple hyperbolic element \(\gamma \in \Gamma \), \(f(L)_{\ast }\) the axis of \( f\circ \gamma \circ f^{-1}\), such that \(f(L)_{\ast }\neq L\) and let \(a\) and \( b\) be the end points of \(L\). In the main result of the paper under review the author finds a lower bound for \(K(f)\), the maximal dilatation of \(f\): let us suppose that there exists an axis \(L\) of a simple hyperbolic element \(\gamma \) such that \( f(L)_{\ast }\neq L\), then there exists a constant \(A>1\) such that \(K(f)\geq A\) , where \(A\) depends only on the translation lengths of \(\gamma \) and \(f\circ \gamma \circ f^{-1}\) and the end points \(a\) and \(b\).

MSC:

30F60 Teichmüller theory for Riemann surfaces
30C62 Quasiconformal mappings in the complex plane
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
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