Connected components of the space of flags of SO0(p,q) transverse to a fixed pair and restrictions on Anosov subgroups

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 . 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 , any ‐Anosov subgroup is virtually isomorphic to either a surface group of a free group. We give examples of Anosov subgroups that are neither free nor surface groups for some sets of roots that do not fall under the previous results. As a consequence of the methods developed here, we get an explicit computation of some Plücker coordinates to check if a unipotent matrix in belongs to the ‐positive semigroup when .