Dynamics on the space of 2-lattices in 3-space

Abstract

We study the dynamics of $${{\rm SL_3}(\mathbb{R})}$$ and its subgroups on the homogeneous space X consisting of homothety classes of rank-2 discrete subgroups of $${\mathbb{R}^3}$$ . We focus on the case where the acting group is Zariski dense in either $${{\rm SL_3}(\mathbb{R})}$$ or $${{\rm SO(2,1)}(\mathbb{R})}$$ . Using techniques of Benoist and Quint we prove that for a compactly supported probability measure $${\mu}$$ on $${{\rm SL_3}(\mathbb{R})}$$ whose support generates a group which is Zariski dense in $${{\rm SL_3}(\mathbb{R})}$$ , there exists a unique $${\mu}$$ -stationary probability measure on X. When the Zariski closure is $${{\rm SO(2,1)}(\mathbb{R})}$$ we establish a certain dichotomy regarding stationary measures and discover a surprising phenomenon: The Poisson boundary can be embedded in X. The embedding is of algebraic nature and raises many natural open problems. Furthermore, motivating applications to questions in the geometry of numbers are discussed.