Euler Characteristics for Orbifold Generalised Triangle Groups

A generalized triangle group is a group with a presentation GT(l, m, n; w) = [left angle bracket]a, b | al = bm = wn = 1[right angle bracket]; where l, m, n [gt-or-equal, slanted] 2, and w is a cyclically reduced word in the free product on {a, b}. In the setting of one-relator products of cyclic groups, Fine et al. calculated their Euler characteristics for n [gt-or-equal, slanted] 3 under certain conditions on the defining word w.