Connected components of the space of flags of $\mathrm{SO}_0(p,q)$ transverse to a fixed pair and restrictions on Anosov subgroups

Abstract

We count and give a parametrization of connected components in the space of flags transverse to a given transverse pair in every flag varieties of $\mathrm{SO}_0(p,q)$. We compute the effect the involution of the unipotent radical has on those components and, using methods of Dey--Greenberg--Riestenberg, we show 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 computation of some Pl\"ucker coordinates to check if a unipotent matrix in $\mathrm{SO}_0(p,q)$ belong to the $\Theta$-positive semigroup $U_\Theta^{>0}$ when $p\neq q$.