$2$-generator groups and parabolic class numbers

Abstract

It is shown that if $x,y$ are generators of the finite group $G$ such that ${x^p} = {y^q} = {(xy)^n} = 1$, where $p,q,n$ are integers $> 1,(p,q) = 1$, and $xy$ is of true order $n$, then the order $\mu = nt$ of $G$ satisfies $n \leqq pq{t^p}$. This result is used to show that if $F$ is a Fuchsian group of genus $0$ generated by 2 elliptic elements of coprime order and with 1 parabolic class, then $F$ possesses only finitely many normal subgroups having a given number of parabolic classes.