Connected components of the space of triples of transverse partial flags in SO(p,q) and Anosov representations
Résumé
We count the number of connected components in the space of triples of transverse flags in any flag manifold of 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_Θ , any P_Θ -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 SO_0 (p, q) belong to the Θ-positive semigroup U_0 Θ when p ̸ = q.
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