Boundary actions by higher-rank lattices: Classification and embedding in low dimensions, local rigidity, smooth factors

TL;DR

This paper classifies and analyzes boundary actions of higher-rank lattices on compact manifolds, establishing rigidity results, classifying smooth factors, and detecting blow-ups through measure invariants.

Contribution

It provides new classification and rigidity results for boundary actions of higher-rank lattices, especially in low dimensions, and introduces methods to detect blow-ups via measure invariants.

Findings

All low-dimensional actions with infinite image are conjugate to boundary actions.

Standard boundary actions are locally rigid.

A mechanism to detect blow-ups using measure invariants is developed.

Abstract

We study actions by lattices in higher-rank (semi)simple Lie groups on compact manifolds. By classifying certain measures invariant under a related higher-rank abelian action (the diagonal action on the suspension space) we deduce a number of new rigidity results related to standard projective actions (i.e. boundary actions) by such groups. Specifically, in low dimensions we show all actions (with infinite image) are conjugate to boundary actions. We also show standard boundary actions (e.g. projective actions on generalized flag varieties) are local rigid and classify all smooth actions that are topological factors of such actions. Finally, for volume-preserving actions in low dimensions (with infinite image) we provide a mechanism to detect the presence of "blow-ups" for the action by studying measures that are $P$-invariant but not $G$-invariant for the suspension action.