Measure rigidity for solvable group actions in the space of lattices

Abstract

We study invariant probability measures on the homogeneous space ${\mathrm{SL}}_n(\mathbb{R})/{\mathrm{SL}}_n(\mathbb{Z})$ for the action of subgroups of ${\mathrm{SL}}_n(\mathbb{R})$ of the form $SF$ where $F$ is generated by one parameter unipotent groups and $S$ is a one parameter $\mathbb{R}$-diagonalizable group normalizing $F$. Under the assumption that $S$ contains an element with only one eigenvalue less than one (counted with multiplicity) and others bigger than one we prove that all the $SF$ invariant and ergodic probability measures on ${\mathrm{SL}}_n(\mathbb{R})/{\mathrm{SL}}_n(\mathbb{Z})$ are homogeneous.

arXiv commentA few typos are fixed