Let $Q$ be the canonical real quadratic form of signature $(p,q)$ on $R^{p+q}$ and $F_{i_1,\dots,i_k}$ the set of flags $F=(F_{i_1},\dots,F_{i_k})$ where each $F_i$ is isotropic. We'll say that $F$ and $F'$ are transverse if for all $i$, $F_i^{\perp}$ and ${F'_i}^{\perp}$ are in direct sum in $R^{p+q}$. The set $\Omega(F')$ of flags which are transverse to $F'$ defines an affine chart of the flag manifold $F_{i_1,\dots,i_k}$ which contains $F$.
Let $Hom(\Gamma,SO(p,q))/SO(p,q)$ be the set of representations from a hyperbolic group $\Gamma$ to $SO(p,q)$ up to conjugation endowed with the topology inherited from $SO(p,q)$. One of the goal of the modern theory of deformation of discrete subgroups is to find connected components of $Hom(\Gamma,SO(p,q))$ which only contain faithful and discrete representations, in the hope that they will correspond to holonomies of interesting geometric structures. A property on representations which often yield such components is that of $F_{i_1,\dots,i_k}$-Anosov representations. Those are automatically faithful and discrete, and the set of $F$-Anosov representations is always open in $Hom(\Gamma, SO(p,q))$. One then only has to show that for a choice of a discrete group $\Gamma$ and a flag manifold $F_{i_1,\dots,i_k}$, the set of $F_{i_1,\dots,i_k}$-Anosov representations is closed in $Hom(\Gamma,SO(p,q))$ to show that they form a union of connected components, thus making Anosov representation a class of representations of high interest for experts in the field.
Our goal will be to count the number of connected component of the set $\Omega(F)\cap\Omega(F')$ of points transverse to $F$ in the affine chart $F'$ and to deduce from this count that for some choice of $q$, any $F_q$-Anosov subgroup of either $SO(q+1,q)$ or $SO(q,q)$ is virtually isomorphic to a free group or a surface group.