Restrictions on Anosov subgroups of 𝑆𝑝(2𝑛,ℝ)

Let n ∈ N n N and let Θ ⊂ 1, …, n \1, , n\ be a nonempty subset. We prove that if Θ contains an odd integer, then any P Θ P_ -Anosov subgroup of Sp ⁡ (2 n, R) Sp (2n, R) is virtually isomorphic to a free group or a surface group. In particular, any Borel Anosov subgroup of Sp ⁡ (2 n, R) Sp (2n, R) is virtually isomorphic to a free or surface group. On the other hand, if Θ does not contain any odd integers, then there exists a P Θ P_ -Anosov subgroup of Sp ⁡ (2 n, R) Sp (2n, 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 1 subgroups.