Growth of Quadratic Forms Under Anosov Subgroups

Source abstract

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