Soluble quotients of triangle groups

This paper helps explain the prevalence of soluble groups among the automorphism groups of regular maps (at least for `small' genus), by showing that every non-perfect hyperbolic ordinary triangle group $\Delta^+(p,q,r) = \langle\, x,y \ | \ x^p = y^q = (xy)^r = 1 \,\rangle$ has a smooth finite soluble quotient of derived length $c$ for some $c \le 3$, and infinitely many such quotients of derived length $d$ for every $d > c$.