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 ${\mathrm{\Delta }}^{+}(p,q,r)=⟨x,y\mathrm{\mid }{x}^p=y^q=(xy{)}^r=1⟩$ 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 $d>c$.