Computing Adapted Bases for Conformal Automorphism Groups of Riemann Surfaces

TL;DR

This paper extends the concept of adapted homology bases to arbitrary finite groups of conformal automorphisms of Riemann surfaces, providing a computational method to analyze their actions on homology.

Contribution

It introduces a method combining Schreier-Reidemeister rewriting to compute adapted homology bases for finite automorphism groups of Riemann surfaces.

Findings

Computed examples of adapted homology bases for specific automorphism groups.

Derived a simplified matrix form for the group action on homology.

Applications demonstrated in the context of the representation variety.

Abstract

The concept of an adapted homology basis for a prime order conformal automorphism of a compact Riemann surface extends to arbitrary finite groups of conformal automorphisms. Here we compute some examples of adapted homology bases for some groups of automorphisms. The method is to begin by apply the Schreier-Reidemeister rewriting process along with the Schreier-Reidemeister Theorem and then to eliminate generators and relations until there is one single large defining relation for the fundamental group in which every generator and its inverse occurs. We are then able to compute the action of the group on the homology image of these generators in the first homology group. The matrix of the action is in a simple form. This has applications to the representation variety.