Partially hyperbolic lattice actions on 2-step nilmanifolds

TL;DR

This paper establishes global rigidity for higher rank lattice actions on 2-step nilmanifolds with partially hyperbolic elements, showing such actions are affine under certain conditions, extending previous results to non-Anosov cases.

Contribution

It proves that under specific assumptions, higher rank lattice actions with partially hyperbolic elements on nilmanifolds are necessarily affine, broadening the scope of rigidity results.

Findings

Actions are affine under the given conditions.

Extends rigidity results to non-Anosov lattice actions.

Applicable to 2-step nilmanifolds with 1-dimensional center.

Abstract

We prove global rigidity results for actions of higher rank lattices on nilmanifolds containing a partially hyperbolic element. We consider actions of higher rank lattices on tori or on $2-$step nilpotent nilmanifolds, such that the actions contain a partially hyperbolic element with $1-$dimensional center. In this setting we prove, under a technical assumption on the partially hyperbolic element, that any such action must be by affine maps. This extends results by Brown, Rodriguez Hertz, and Wang to certain lattice actions that are not Anosov.