Dynamics on the space of 2-lattices in 3-space (Q1999363)

This work is motivated by a conjecture about the distribution of `shapes' of lattices in homogeneous spaces. To formulate this conjecture, let \(X_2\) be the space of lattices in \(\mathbb{R}^2\) identified up to scaling, so that the quotient \(\mathrm{O}_2(\mathbb{R})\backslash X_2\) by the natural action of the orthogonal group may be thought of as the space of `shapes of lattices' in \(\mathbb{R}^2\) (with the `size' of the lattice factored out by definition of \(X_2\) and the `position' factored out via the action of \(\mathrm{O}_2(\mathbb{R})\)). For a discrete subgroup \(\Lambda\) of rank \(2\) in \(\mathbb{R}^3\), denote by \(s(\Lambda)\) the point in \(\mathrm{O}_2(\mathbb{R})\backslash X_2\) that corresponds to an image of \(\Lambda\) in \(X_2\) obtained by choosing an isometry from the plane spanned by \(\Lambda\) to \(\mathbb{R}^2\). For the quadratic form \(Q(v_1,v_2,v_3)=2v_1v_3-v_2^2\) and the associated level set variety \(V=\{v\in\mathbb{R}^3\mid Q(v)=1\}\), the conjecture asks if the collection of orthogonal shapes of the integer points on \(V\), namely \(\{s(\mathbb{Z}^3\cap v^{\perp})\mid v\in V\cap\mathbb{Z}^3\}\), is a dense subset of \(\mathrm{O}_2(\mathbb{R})\backslash X_2\). While there is some numerical evidence in support of this conjecture, little seems to be known. Here the dynamics of subgroups of \(\mathrm{SL}_3(\mathbb{R})\) on the space of homothety classes of discrete rank \(2\) subgroups of \(\mathbb{R}^3\), particularly under the assumption that the subgroup is Zariski dense in \(\mathrm{SL}_3(\mathbb{R})\) or \(\mathrm{SO}(2,1)(\mathbb{R})\) is studied. A consequence of one of the main results relevant to the conjecture formulated above is that if \(\Gamma\) is a compactly generated Zariski dense subgroup of \(\mathrm{SL}_3(\mathbb{R})\) and \(\Lambda\) is a rank \(2\) discrete subgroup of \(\mathbb{R}^3\), then the collection of shapes \(\{s(g\Lambda)\mid g\in\Gamma\}\) is dense in \(\mathrm{O}_2(\mathbb{R})\backslash X_2\). The approach adopted in this substantial work involves the study of random walks on the space of \(2\)-dimensional lattices, building on work of \textit{Y. Benoist} and \textit{J.-F. Quint} [Ann. Math. (2) 178, No. 3, 1017--1059 (2013; Zbl 1279.22013)] and references therein, and leading to classification results for stationary measures under the group action. Applications of some of these ideas and results to problems in the geometry of numbers are discussed.