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

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

2007-01-11

arxiv.org/abs/math/0701330v2

Mathematics

Geometric Topology

24 pages; change in title; typos corrected; some theorems revised; computational complexity added; final version of paper to a

Scientific paper

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.

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