An action of subgroups of mapping class groups on polynomial algebras

We study a certain subgroup of the mapping class group of a surface with boundary by obtaining an action of this subgroup on a polynomial algebra. This action comes from a lift of the action of the subgroup on a trace algebra, Magnus having done a similar thing for the braid groups. We then investigate the invariant theory for this action and various representations that this action affords. In particular, we obtain finite permutation representations and infinite linear representations of these subgroups that are non-trivial on subgroups of the Torelli group.