Infinitely many Hypermaps of a given Type and Genus

It is conjectured that given positive integers l, m, n with l−1 + m−1 + n−1 < 1 and an integer g > 0, the triangle group ∆ = ∆(l,m, n) = 〈X,Y,Z|X l = Y m = Z = XY Z = 1〉 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 > 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.