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

Let $n\in\mathbb{N}$ and let $\Theta \subset \{1,\dots,n\}$ be a non-empty subset. We prove that if $\Theta$ contains an odd integer, then any $P_\Theta$-Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ is virtually isomorphic to a free group or a surface group. In particular, any Borel Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ 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 ${\rm Sp}(2n,\mathbb{R})$ 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 $1$ subgroups.