Growth of Quadratic Forms Under Anosov Subgroups

Abstract Let $\rho :\Gamma \rightarrow \textrm{PSL}_d({\mathbb{K}})$ be a Zariski dense Borel–Anosov representation for ${\mathbb{K}}$ equal to ${\mathbb{R}}$ or ${\mathbb{C}}$. $o$ form of signature $(p,d-p)$ on ${\mathbb{K}}^d$ (where $0<p<d)$. $\textsf{S}^o$ the corresponding geodesic copy Riemannian symmetric space $\textrm{PSO}(o)$ inside $\textrm{PSL}_d({\mathbb{K}})$. For certain choices and every $t$ large enough, we show exponential bounds number $\gamma \in \Gamma $ which distance between \gamma \cdot \textsf{S}^o$ is smaller than $t$. Under an extra assumption, satisfied instance when boundary $\Gamma connected, asymptotic as $t\rightarrow \infty counting function relative functional in interior dual limit cone.