Hyperbolic lengths of some filling geodesics on Riemann surfaces with punctures (Q953940)

Let \(\tilde A\) be a non-trivial closed curve on a Riemann surface \(\tilde S\) of type \((p, n)\) with \(3p- 3 + n > 0\). The length function, \(l_{\tilde A}: \mathcal T(\tilde S) \to \mathbb R^+\), on the Teichmüller space \(\mathcal T(\tilde S)\) is defined by sending each hyperbolic structure \(\sigma=\sigma(\tilde S)\) of \(\mathcal T(\tilde S)\) to the hyperbolic length \(l_{\tilde A}(\sigma)\) of the closed geodesic homotopic to \(\tilde A\) on \(\sigma(\tilde S)\). It is well known that the function \(l_{\tilde A}\) achieves its positive minimum value when \(\tilde A\) is a filling curve on \(\tilde S\) in the sense that every component of \(\tilde S \setminus \{\tilde A\}\) is either a disk or a once punctured disk. The extremal value, of course, depends only on the homotopy class of \(\tilde c\). In this paper, the author gives a quantitative common lower bound through \(\mathcal T(\tilde S)\) for the hyperbolic lengths of a certain kind of filling curves on~ \(\tilde S\). Note that if \(\tilde S\) is punctured, then some elements \([\alpha]\) in the fundamental group \(\pi_1(\tilde S, a)\), \(a\in\tilde S\), are represented by loops \(\alpha\) that pass through \(a\) and are boundaries of a once punctured disk. Let \(\mathcal F\) denote the set of those elements, \(\mathcal F\subset \pi_1(\tilde S, a)\). The main result of this paper is the following: Theorem 1. There are infinitely many homotopically independent filling curves \(\tilde A\) on \(\tilde S\) that can be expressed as products of two elements in \(\mathcal F\). For each such \(\tilde A\) and each hyperbolic structure \(\sigma\) on \(\tilde S\), we have: \[ l_{\tilde A}(\sigma)\geq 2 \log(\kappa^2 - 5) - 4 \log 2, \] where \(\kappa = 16 + 8n- 21\) if \(n\geq 3\), \(16p + 3\) if \(n = 2\), and \(16p + 7\) if \(n = 1\). Let \(\mathbb H = \{z\in\mathbb C:\,\text{Im}\, z > 0\}\) denote the upper half plane and \(\varrho:\mathbb H\to\tilde S\) be the universal covering with a covering group \(G\) being a torsion free Fuchsian group of fist kind of type \((p, n)\) so that \(\mathbb H/G \cong\tilde S\). The set \(\mathcal F\) is one-to-one correspondent with the set of parabolic elements of \(G\). Note that any hyperbolic element \(g\) is conjugate in PSL\(_2(\mathbb R)\) to \(z\mapsto \lambda_gz\), where \(\lambda_g > 1\) is called the multiplier of \(g\). Let \(A_g\) be the axis of \(g\). The hyperbolic element \(g\) is called essential if every component of \(\tilde S\setminus \varrho(A_g)\) is either a disk or possibly a once punctured disk. Theorem 1 can be restated as follows: {Theorem 2.} Let \(G\) be a finitely generated Fuchsian group of first kind of type \((p,n)\) with \(3 - 3+n > 0\) and \(n\geq 1\). There are infinitely many essential hyperbolic elements \(g\) of \(G\) that are generated by two parabolic elements of \(G\). Furthermore, for each such element \(g\), the multiplier \(\lambda_g\) of \(g\) satisfies: \[ \lambda_g\geq\left\{\frac{1}{4}(\kappa^2-5)\right\}^2, \] where \(\kappa\) is given in Theorem 1.