Realizability of torsion free subgroups with prescribed signatures in Fuchsian groups (Q1026960)

A Fuchsian group is a discrete subgroup of the group \(PSL(2,\mathbb{R})\) of orientation-preserving isometries of the hyperbolic plane \(\mathbb{H}^2\). Let \(G\) be a finitely generated Fuchsian group so that \(\mathbb{H}^2/G\) is a Riemann surface. The signature of \(G\) is denoted by \((g;t;m_1,...,m_r)\), and \(\mathbb{H}^2/G\) is a surface of genus \(g\) with \(t\) punctures. Let \(\mathbb{Z}_p\) denote a finite cyclic group of order \(p\), and let \(\prod^*\) stand for the free product. The author in the paper under review studies the realizability of signatures by torsion free subgroups of finite index in a Fuchsian group \(\Gamma=\prod_{j=1}^{*n}\mathbb{Z}_{p_j}\), where \(n\geq3\) and each \(p_j\geq2\). In general, the Riemann-Hurwitz and diophantine conditions are not sufficient for realizability if \(n\geq3\). An additional necessary end-condition for the existennce of a subgroup \(\varPhi\) of index \(d\) in \(\Gamma\) is that the number \(t\) of punctures of the Riemann surface \(\mathbb{H}^2/\varPhi\) is at most \(d\). A major goal is to determine completely all possible \(t\leq d\). In fact, such signatures can always be realized under the Riemann-Hurwitz, diophantine and certain end-conditions.