Growth of quadratic forms under Anosov subgroups

Abstract

Let $\rho:\Gamma\rightarrow PSL_d(\mathbb{K})$ 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 $\mathsf{S}^o$ be the corresponding geodesic copy of the Riemannian symmetric space of $PSO(o)$, inside the Riemannian symmetric space of $PSL_d(\mathbb{K})$. 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 $\mathsf{S}^o$ and $\rho\gamma\cdot\mathsf{S}^o$ 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.