Finite abelian subgroups of the mapping class group

Let S be a closed, smooth, orientable surface of genus 2. The mapping class group M of S (or MCG) is the group of isotopy classes of homeomorphisms of S . In this paper we shall investigate the conjugacy classes of finite elementary abelian subgroups, and more generally the finite abelian subgroups of M . While the general finite subgroup classification is important we focus on the case where G is an elementary abelian subgroup as a tractable case where complete classification methods by standard linear algebra is possible. We note that by finite elementary abelian, we mean G Š C v p Š F v p where Cp is a cyclic group of prime order p , and Fp is the finite field of the same cardinality (we use the latter notation when we want to emphasize the vector space structure of G ). We also consider the abelian case where positive steps can be made toward such a classification. The main result of our work is to describe methods which may be employed to completely classify all elementary abelian actions in a given genus and steps toward a classification of abelian actions. The classification is complex since it involves understanding the representation theory of certain subgroups of symmetric groups, but for a fixed low genus since the symmetric groups are small, we are able to produce very explicit results.