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

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