The lengths of the closed geodesics on a Riemann surface with self- intersections.
Let H be the upper half plane with the Poincaré metric \(ds^ 2=(dx^ 2+dy^ 2)/y^ 2\). PSL(2,\({\mathbb{R}})\) acts on H as the group of orientation preserving isometries. The geodesic \(a_{\gamma}\) connecting the two fixed points of a hyperbolic element \(\gamma\in PSL(3,{\mathbb{R}})\) is called the axis of \(\gamma\). Let \(\Gamma\) be a Fuchsian group in PSL(2,\({\mathbb{R}})\) and \(\phi\) :H\(\to H/\Gamma\) the natural projection on the quotient space. \(\phi\) induces a 1-1-correspondence between the equivalence classes \(\{\eta (a_{\gamma}):\) \(\eta\in \Gamma \}\) of axes and the closed geodesics on H/\(\Gamma\). A hyperbolic element \(\gamma\in PSL(2,{\mathbb{R}})\) is said to satisfy condition (\(\infty)\) if there exists a Fuchsian group \(\Gamma\) containing \(\gamma\) and another element \(\delta\) such that \(\delta\) does not preserve the axis of \(\gamma\) and that \(a_{\gamma}\) and \(\delta (a_{\gamma})=a_{\delta \gamma \delta -1}\) intersect each other. It is shown that \(| trace(\gamma)| \geq 2 \cos (2\pi /7)+1\) for every hyperbolic transformation \(\gamma\in PSL(2,{\mathbb{R}})\) satisfying condition (\(\infty)\). This constant cannot be replaced by any greater value unless some restrictions are put on the Fuchsian group \(\Gamma\). The proof is more geometric compared to the one given by Pommerenke and Purzitsky in 1983.
scientific article; zbMATH DE number 4136315
scientific article