Composition of Permutation Representations of Triangle Groups

TL;DR

The paper investigates how permutation representations of triangle groups can be composed, revealing conditions for imprimitive representations and providing a detailed structure analysis in specific cases, advancing understanding of Fuchsian group images.

Contribution

It introduces a method for composing permutation representations of triangle groups and characterizes the structure of these compositions, especially for representations involving alternating groups.

Findings

Some compositions produce imprimitive representations.

Complete description of composition structure when using equivalent alternating group copies.

Application to understanding homomorphic images of Fuchsian groups.

Abstract

A triangle group is denoted by $\Delta(p,q,r)$ and has finite presentation $$ \Delta(p,q,r)=\langle x,y | x^p=y^q=(xy)^r=1 \rangle .$$ We examine a method for composition of permutation representations of a triangle group $\Delta(p,q,r)$ that was used in Everitt's proof of Higman's 1968 conjecture that every Fuchsian group has amongst its homomorphic images all but finitely many alternating groups. We see that some of these compositions must give imprimitive representations, and in particular situations, where the representations being composed are all equivalent copies of an alternating group in the same degree, we can give a complete description of the structure of the composition of the representations. This article contains the main results of the author's PhD thesis.