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arXiv · 2410.20329

Finite quotients of Fuchsian groups

Frankie Chan·Lindsey Styron
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Abstract

This work provides an effective algorithm for distinguishing finite quotients between two non-isomorphic finitely generated Fuchsian groups $Γ$ and $Λ$. 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 $Γ$ and $Λ$ which between them have differing degrees of smoothness. Regarding the order of a finite quotient that distinguishes between $Γ$ and $Λ$, we establish an upperbound as a function of the genera, the number of punctures, and the cone orders arising in $Γ$ and $Λ$.

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Frankie Chan, Lindsey Styron. 2024-10-27. Finite quotients of Fuchsian groups. https://arxiv.org/abs/2410.20329

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