Conformal Automorphism Groups, Adapted Generating Sets and Bases

TL;DR

This paper introduces a method to construct adapted generating sets and bases for the first homology group of compact Riemann surfaces with conformal automorphism groups, simplifying the representation of group actions.

Contribution

It defines and proves the existence of adapted bases for any conformal automorphism group on such surfaces, extending previous results to more general groups.

Findings

Constructed adapted bases for automorphism groups of arbitrary order

Derived matrices representing group actions on homology

Extended prior results to broader classes of automorphism groups

Abstract

Let S be a compact Riemann surfaces of genus g >= 2 and G a conformal automoprhism group of order n acting on S. In this paper we give the definition of an adapted generating set and an adapted basis for the first homology group of such a compact Riemann surface. This generating set and basis reflect the action of G in as simple manner as possible. This can be seen in the matrix of the action of G which we obtain. We prove the existence of such a generating set and basis for any conformal group acting on such a surface and find the matrix. This extends our earlier results on adapted bases and matrices for automorphism groups of prime orders and other specific groups.