Canonical Symplectic Representations for Prime Order Conjugacy Classes of the Mapping-class Group

In this paper we find a unique normal form for the symplectic matrix representation of the conjugacy class of a prime order element of the mapping-class group. We find a set of generators for the fundamental group of a surface with a conformal automorphism of prime order which reflects the action the automorphism in an optimal way. This is called an {\sl adapted} homotopy basis and there is a corresponding {\sl adapted presentation}. We also give a necessary and sufficient condition for a prime order symplectic matrix to be the image of a prime order element in the mapping-class group.Comment: 24 pages; change in title; typos corrected; some theorems revised; computational complexity added; final version of paper to appear in Journal of Algebr

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 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.Comment: Typos and spelling error fixed; 17 pages; to appear AMS Conn Math; Proc. Linkopoing Conference, 201