Infinitely many Hypermaps of a given Type and Genus

Abstract

It is conjectured that given positive integers $l$, $m$, $n$ with $l^{-1}+m^{-1}+n^{-1} < 1$ and an integer $g \geq 0$, the triangle group $\Delta=\Delta (l,m,n)=\langle X, Y, Z | X^l=Y^m=Z^n=XYZ=1 \rangle $ contains infinitely many subgroups of finite index and of genus $g$. A slightly stronger version of this conjecture is as follows: given positive integers $l$, $m$, $n$ with $l^{-1}+m^{-1}+n^{-1} < 1$ and an integer $g \geq 0$, there are infinitely many nonisomorphic compact orientable hypermaps of type $(l,m,n)$ and genus $g$. We prove that these conjectures are true when two of the parameters $l$, $m$, $n$ are equal, by showing how to construct appropriate hypermaps.