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We study the distribution of orbits of a lattice$\Gamma\leq\text{SL}(3,\mathbb R)$ in the moduli space $X_{2,3}$ of covolumeone rank-two discrete subgroups in $\mathbb R^3$. Each orbit is dense, and ourmain result is the limiting distribution of these orbits with respect to normballs, 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 probabilitymeasure on $X_{2,3}$ depending on $x_0$. To prove our result, we use theduality principle developed by Gorodnik and Weiss which recasts the aboveproblem into the problem of computation of certain volume estimates of growingskewed balls in $H$ and proving ergodic theorems of the left action of theskewed balls on $\text{SL}(3,\mathbb{R})/\Gamma$. The ergodic theorems areproven by applying theorems of Shah building on the linearisation technique.The main contribution of the paper is the application of the duality principlein the case where $H$ has infinitely many non-compact connected components.

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