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 ${\Delta }^{+}(p,q,r)=〈x,y|x^p=y^q={\left(xy\right)}^r=1〉$ 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 $d>c$.