# Measure rigidity for solvable group actions in the space of lattices

## TL;DR

This paper proves that under certain conditions, all invariant ergodic probability measures for specific solvable group actions on the space of lattices are homogeneous, extending measure rigidity results in homogeneous dynamics.

## Contribution

It establishes measure rigidity for a class of solvable group actions on the space of lattices, generalizing previous results to new group configurations.

## Key findings

- All invariant ergodic measures are homogeneous under the given conditions.
- The result applies to groups generated by unipotent and diagonalizable elements with specific eigenvalue properties.
- The proof extends measure classification techniques in homogeneous dynamics.

## 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.