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

Let n ∈ N n∈ N and let Θ ⊂ { 1 , … , n } Θ ⊂ \{1, ,n\} be a nonempty subset. We prove that if Θ Θ contains an odd integer, then any P Θ P_Θ -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 Θ does not contain any odd integers, then there exists a P Θ P_Θ -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.