Transversality in Flag Manifolds of SO(p,q)

par Roméo Troubat (Institut de Recherche Mathématique Avancée, Université de Strasbourg)

mercredi 20 nov. 2024, 14:00 → 15:15 Europe/Paris

Pierre Grisvard (IHP)

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 $F^{\prime }$ are transverse if for all $i$, $F_i^{\perp }$ and ${F_i^{\prime }}^{\perp }$ are in direct sum in $R^{p+q}$. The set $\mathrm{\Omega }(F^{\prime })$ of flags which are transverse to $F^{\prime }$ defines an affine chart of the flag manifold $F_{i_1,\dots ,i_k}$ which contains $F$.

Let $Hom(\mathrm{\Gamma },SO(p,q))/SO(p,q)$ be the set of representations from a hyperbolic group $\mathrm{\Gamma }$ 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 $Hom(\mathrm{\Gamma },SO(p,q))$ 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 $Hom(\mathrm{\Gamma },SO(p,q))$. One then only has to show that for a choice of a discrete group $\mathrm{\Gamma }$ and a flag manifold $F_{i_1,\dots ,i_k}$, the set of $F_{i_1,\dots ,i_k}$-Anosov representations is closed in $Hom(\mathrm{\Gamma },SO(p,q))$ 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 $\mathrm{\Omega }(F)\cap \mathrm{\Omega }(F^{\prime })$ of points transverse to $F$ in the affine chart $F^{\prime }$ 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.