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Finite quotients of Fuchsian groups

Frankie Chan, Lindsey Styron

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 $Λ$.

Finite quotients of Fuchsian groups

Abstract

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

Paper Structure

This paper contains 18 sections, 30 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Profinite invariants
  3. Fuchsian groups
  4. Finite index subgroups of Fuchsian groups
  5. Abelian quotients
  6. First Betti numbers
  7. Euler characteristics
  8. Smooth and non-smooth quotients
  9. Smooth dihedral and PSL2 representations
  10. Degrees of smoothness of quotients
  11. Scrapes and smoothness
  12. Euler characteristics and homogeneous linear systems
  13. Proof of Theorem \ref{['thm:distinguishing_scrape_closures']}
  14. Badness and goodness
  15. Existence of maximally smooth maps with prescribed factors
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\Gamma$ and $\Lambda$ be finitely generated groups, then $\widehat{\Gamma}\cong\widehat{\Lambda}$ if and only if $\mathrm{FQ}(\Gamma)=\mathrm{FQ}(\Lambda)$.

Theorems & Definitions (62)

  • Definition
  • Theorem 1.1
  • Definition
  • Definition
  • Theorem 1.2: Riemann-Hurwitz formula
  • Definition
  • Definition
  • Proposition 2.1
  • Corollary 2.2
  • Definition
  • ...and 52 more