Deformation theory and finite simple quotients of triangle groups II

TL;DR

This paper advances the study of finite simple quotients of triangle groups by employing deformation theory of representation varieties, exploring new deformation approaches beyond the principal homomorphism.

Contribution

It introduces novel deformation techniques using ${ m PGL}_2( R)$ and other representations, extending previous methods to analyze triangle groups.

Findings

Demonstrates new deformation methods for triangle groups

Identifies additional finite simple quotients via these methods

Extends the theoretical framework from Part I

Abstract

This paper is a continuation of our first paper [10] in which we showed how deformation theory of representation varieties can be used to study finite simple quotients of triangle groups. While in Part I, we mainly used deformations of the principal homomorphism from ${\rm SO}(3,\R)$, in this part we use ${\rm PGL}_2(\R)$ as well as deformations of representations which are very different from the principal homomorphism.