Finite quotients of Fuchsian groups

TL;DR

This paper introduces an effective algorithm to distinguish finite quotients of non-isomorphic finitely generated Fuchsian groups, focusing on specific types of quotients and establishing bounds based on geometric properties.

Contribution

It develops a method for identifying finite quotients of Fuchsian groups and provides bounds related to the groups' geometric features, advancing understanding of their finite representations.

Findings

Algorithm for distinguishing finite quotients of Fuchsian groups

Finite quotients can be abelian, dihedral, or subgroups of PSL(2, F_q)

Upper bounds on quotient order based on geometric parameters

Abstract

This work provides an effective algorithm for distinguishing finite quotients between two non-isomorphic finitely generated Fuchsian groups $\Gamma$ and $\Lambda$. It will suffice to take a finite quotient which is abelian, dihedral, a subgroup of $\mathrm{PSL}(2,\mathbf{F}_q)$, or an abelian extension of one of these 3. We will develop an approach for creating group extensions upon a shared finite quotient of $\Gamma$ and $\Lambda$ which between them have differing degrees of smoothness. Regarding the order of a finite quotient that distinguishes between $\Gamma$ and $\Lambda$, we establish an upperbound as a function of the genera, the number of punctures, and the cone orders arising in $\Gamma$ and $\Lambda$.