Finite quotients of Fuchsian groups

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