Soluble quotients of triangle groups

TL;DR

This paper demonstrates that non-perfect hyperbolic triangle groups have finite soluble quotients of bounded derived length, explaining the frequent occurrence of soluble automorphism groups in regular maps.

Contribution

It establishes the existence of finite soluble quotients with bounded derived length for all non-perfect hyperbolic triangle groups, expanding understanding of their automorphism groups.

Findings

Every non-perfect hyperbolic triangle group has a finite soluble quotient of derived length at most 3.

Such groups have infinitely many soluble quotients of any derived length greater than the bound.

The results clarify the prevalence of soluble automorphism groups in regular maps of small genus.

Abstract

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 $d > c$.