Equidistribution of lattice orbits in the space of homothety classes of rank $2$ sublattices in $\mathbb R^3$

Abstract

We study the distribution of orbits of a lattice $\Gamma\leq\text{SL}(3,\mathbb R)$ in the moduli space $X_{2,3}$ of covolume one rank-two discrete subgroups in $\mathbb R^3$. Each orbit is dense, and our main result is the limiting distribution of these orbits with respect to norm balls, where the norm is given by the sum of squares. Specifically, we consider $\Gamma_T=\{\gamma\in\Gamma:\|\gamma\|\leq T\}$ and show that, for any fixed $x_0\in X_{2,3}$ and $\varphi\in C_c(X_{2,3})$, $$\lim_{T\to\infty}\frac{1}{\#\Gamma_T}\sum_{\gamma\in\Gamma_T}\varphi(x_0\cdot\gamma)=\int_{X_{2,3}}\varphi(x)d \tilde\nu_{x_0}(x),$$ where $\tilde\nu_{x_0}$ is an explicit probability measure on $X_{2,3}$ depending on $x_0$. To prove our result, we use the duality principle developed by Gorodnik and Weiss which recasts the above problem into the problem of computation of certain volume estimates of growing skewed balls in $H$ and proving ergodic theorems of the left action of the skewed balls on $\text{SL}(3,\mathbb{R})/\Gamma$. The ergodic theorems are proven by applying theorems of Shah building on the linearisation technique. The main contribution of the paper is the application of the duality principle in the case where $H$ has infinitely many non-compact connected components.