Length inequalities for Riemann surfaces (Q2839354)

Let \(S=\mathbb{H}/G\) be a hyperbolic Riemann surface, where \(\mathbb{H}\) is the hyperbolic plane \(\{x+iy : y>0\}\) with the hyperbolic metric \(|dz|/y\) and hyperbolic distance \(\rho\). Let \(p\) be a point on \(S\), and let \(\alpha\) be a closed loop on \(S\) that starts and ends at \(p\). Then \(\alpha\) lifts to a curve in \(\mathbb{H}\) with endpoints, say, \(w\) and \(g(w)\), where \(g\in G\), and the hyperbolic length \(\ell(\alpha)\) of \(\alpha\) satisfies \(\ell(\alpha)\geq \rho(w,g(w))\). The author has proved in [The geometry of discrete groups. New York etc.: Springer (1983; Zbl 0528.30001), p. 198] that if \(g\) and \(h\) generate a non-cyclic group \(<g, h>\), then, for all \(z\) in \(\mathbb{H}\), NEWLINE\[NEWLINE\sinh\frac{1}{2}\rho(z,g(z))\sinh\frac{1}{2}\rho(z,h(z))\geq 1.NEWLINE\]NEWLINE This is valid for every hyperbolic Riemann surface \(S\) and every \(z\in \mathbb{H}\), and it is stronger than what is customarily known as the Collar Lemma. It shows, for example, that providing \(<g, h>\) is not cyclic, NEWLINE\[NEWLINE\sinh\frac{1}{2}(\ell(\alpha))\sinh\frac{1}{2}(\ell(\beta))\geq 1.NEWLINE\]NEWLINENEWLINENEWLINEIn this paper, stronger inequalities of a similar nature are established when \(S\) is a triply punctured sphere and a twice-punctured disc.