A lower bound of maximal dilatations of quasiconformal automorphisms acting on Fuchsian groups (Q2574386)

Let \(\Gamma \) be a torsion free Fuchsian group acting on the upper half-plane \textbf{H} and \(f\) a normalized quasiconformal automorphism of \textbf{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 \(b\).