Subgroups of Fuchsian Groups and Finite Permutation Groups

TLDR: In this article, the authors consider the problem of determining necessary and sufficient conditions for a subgroup of finite index in F to have some given signature, where the generators are the elements which map the region onto a full neighbour.

Abstract: We then say F has signature (g;mi,m2,...,mr;s;t) (1) The integers mu m2,... mr are called the periods of F. We note the following facts about this presentation. Every elliptic element of F is conjugate to a power of one of the Xj{\ < j ^ r), every parabolic element of T is conjugate to a power of one of the pk(l < k < s) and every hyperbolic boundary element of T is conjugate to a power of one of the ht(l ^ / ^ t). Moreover no nontrivial power of one of the generators can be conjugate to a power of another generator. These well-known facts follow from the existence of a fundamental region for T with the property that the generators are the elements which map the region onto a full neighbour [1, 2]. The problem we consider here is to determine necessary and sufficient conditions for a subgroup of finite index in F to have some given signature. In §3 we give an application to finite permutation groups.