Connected components of the space of triples of transverse partial flags in $\mathrm{SO}_0(p,q)$ and Anosov representations

Abstract: We count the number of connected components in the space of triples of transverse flags in any flag manifold of $\mathrm{SO}0(p,q)$. We compute the effect the involution of the unipotent radical has on those components and deduce that for certain parabolic subgroups $P{\Theta}$, any $P_{\Theta}$-Anosov subgroup is virtually isomorphic to either a surface group of a free group. We give examples of Anosov subgroups which are neither free nor surface groups for some sets of roots which do not fall under the previous results. As a consequence of the methods developed here, we get an explicit algorithm based on computation of minors to check if a unipotent matrix in $\mathrm{SO}0(p,q)$ belong to the $\Theta$-positive semigroup $U\Theta^{>0}$ when $p\neq q$.