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

TL;DR

This paper classifies connected components of flag spaces in $ ext{SO}_0(p,q)$, analyzes involutions on these components, and explores restrictions on Anosov subgroups, providing explicit criteria for positivity in unipotent matrices.

Contribution

It provides a detailed parametrization of flag components, studies involution effects, and characterizes certain Anosov subgroups, including examples beyond free and surface groups.

Findings

Connected components of flag spaces are classified and parametrized.

Involution effects on components are explicitly computed.

Restrictions on Anosov subgroups are established, with examples beyond classical types.

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 $p\neq q$.