Infinitely many hypermaps of a given type and genus (Q612931)

Summary: 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 \[ \Delta= \Delta(l,m,n)= \langle X,Y,Z\mid X^l= Y^m= Z^n= XYZ=1\rangle \] 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 typ \((l,m,n)\) 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.