Canonical symplectic representations for prime order conjugacy classes of the mapping-class group

Abstract

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 of the automorphism in an optimal way. This is called an adapted homotopy basis and there is a corresponding 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.