Product groups acting on manifolds

TL;DR

This paper establishes a superrigidity theorem for volume-preserving actions of product groups on manifolds, leading to a dichotomy between linear and isometric actions, with implications for manifold dimension and group non-linearity.

Contribution

It introduces a new superrigidity result for ergodic cocycles of product groups with minimal assumptions, and applies it to classify actions on manifolds as either linear or isometric.

Findings

Actions are either infinitesimally linear or measurably isometric.

Manifold dimension bounds depend on the number of group factors.

Restrictions are derived for actions of non-linear groups.

Abstract

We analyse volume-preserving actions of product groups on Riemannian manifolds. To this end, we establish a new superrigidity theorem for ergodic cocycles of product groups ranging in linear groups. There are no a priori assumptions on the acting groups, except a spectral gap assumption on their action. Our main application to manifolds concerns irreducible actions of Kazhdan product groups. We prove the following dichotomy: Either the action is infinitesimally linear, which means that the derivative cocycle arises from unbounded linear representations of all factors. Otherwise, the action is measurably isometric, in which case there are at most two factors in the product group. As a first application, this provides lower bounds on the dimension of the manifold in terms of the number of factors in the acting group. Another application is a strong restriction for actions of non-linear groups.