RESTRICTIONS ON ANOSOV SUBGROUPS OF Sp (2n, n, R)

Let n is an element of N and let Theta subset of {1, 1 , ... , n } be a nonempty subset. We prove that if Theta contains an odd integer, then any P Theta-Anosov subgroup of Sp(2n, n, R) is virtually isomorphic to a free group or a surface group. In particular, any Borel Anosov subgroup of Sp(2n, n, R) is virtually isomorphic to a free or surface group. On the other hand, if Theta does not contain any odd integers, then there exists a P Theta-Anosov subgroup of Sp(2n, n, R) which is not virtually isomorphic to a free or surface group. We also exhibit new examples of maximally antipodal subsets of certain flag manifolds; these arise as limit sets of rank 1 subgroups.