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

TL;DR

This paper investigates the dynamics of certain groups on the space of 2-lattices in 3-space, establishing uniqueness of stationary measures for Zariski dense actions and revealing a surprising algebraic embedding of the Poisson boundary.

Contribution

It proves the uniqueness of stationary measures for Zariski dense subgroups of SL(3,R) and uncovers an algebraic embedding of the Poisson boundary in the space of 2-lattices.

Findings

Unique stationary measure for Zariski dense SL(3,R) actions.

Dichotomy in stationary measures for SO(2,1)(R) actions.

Poisson boundary can be algebraically embedded in the space.

Abstract

We study the dynamics of $SL_3(\mathbb{R})$ 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 $SL_3(\mathbb{R})$ or $SO(2,1)(\mathbb{R})$. Using techniques of Benoist and Quint we prove that for a compactly supported probability measure $\mu$ on $SL_3(\mathbb{R})$ whose support generates a group which is Zariski dense in $SL_3(\mathbb{R})$, there exists a unique $\mu$-stationary probability measure on $X$. When the Zariski closure is $SO(2,1)(\mathbb{R})$ 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.