REALIZABILITY OF TORSION FREE SUBGROUPS WITH PRESCRIBED SIGNATURES IN FUCHSIAN GROUPS

Abstract

The necessary and sufficient conditions for the existence of a subgroup of finite index in a Fuchsian group $\Gamma = \prod^{\ast n}_{j=1} {\bf Z}_{p_j}$, where each $p_j \geq 2$, are the Riemann-Hurwitz and diophantine conditions. For torsion free subgroups of index $d$ in $\Gamma$, the diophantine condition reduces to the one that $d$ is divisible by each $p_j$. The purpose of this paper is to study the realizability problem of torsion free subgroups of finite index with given signatures in $\Gamma$. In general, the Riemann-Hurwitz and diophantine conditions are not sufficient for our realizability problem if $n \geq 3$. An additional necessary end-condition for the existence of a subgroup $\Phi$ of index $d$ in $\Gamma$ is that the number $t$ of punctures of the Riemann surface ${\bf H^2} / \Phi$ is at most $d$. A major goal is to completely determine all possible $t \leq d$. Such signatures can always be realized under the Riemann-Hurwitz, diophantine and certain end conditions.