Computing Adapted Bases for Conformal Automorphism Groups of Riemann Surfaces

The concept of an adapted homology basis for a prime order conformal automorphism of a compact Riemann surface originated in [6, 7, 8, 9] and is extended to arbitrary finite groups of conformal automorphisms in [12]. 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 and the Schreier-Reidemeister Theorem from [25] 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.