Subgroups of products of surface groups

Abstract

The subgroup structure of a direct product of two Fuchsian groups is very complicated; for example, Baumslag and Roseblade [1] have shown that a direct product F_m1×F_m2 of finitely generated free groups contains continuously many distinct isomorphism classes of finitely generated subgroups. The situation is much simpler, however, if attention is restricted to finitely generated normal subgroups of F_m1×F_m2; then on general grounds one cannot expect to get more than a countable infinity, but, in fact, the situation is almost finite; we showed, in [4], that F_m1×F_m2 contains precisely 1+min{m₁, m₂} orbits of maximal normal subdirect products under the natural action of its automorphism group.

In this paper we study the situation which arises if the free groups F_mi are replaced by fundamental groups of closed surfaces. This question was previously considered by Nigel Carr in the final chapter of his (unpublished) thesis [2]. By appealing to the relative invariant theory of pairs of skew bilinear forms, Carr was able to show that in a direct product Σ_g1×Σ_g2 of orientable surface groups of genus [ges ]2 the number of orbits of maximal normal subdirect products is always infinite. Here we refine and extend Carr's approach to study the manner in which the finiteness result of [4] breaks down on passing to more general Fuchsian groups.