Existence of rigid actions for finitely-generated non-amenable linear groups

TL;DR

This paper proves that all finitely-generated non-amenable linear groups over characteristic zero fields have ergodic actions that are rigid, with some being free, and explores how rigidity behaves under certain group operations.

Contribution

It establishes the existence of ergodic rigid actions for broad classes of non-amenable linear groups, including free actions for specific groups, and analyzes rigidity transfer properties.

Findings

Existence of ergodic rigid actions for all such groups.

Construction of free ergodic rigid actions for specific groups like  imes \u211d.

Rigidity passes to co-amenable subgroups.

Abstract

We show that every finitely-generated non-amenable linear group over a field of characteristic zero admits an ergodic action which is rigid in the sense of Popa. If this group has trivial solvable radical, we prove that these actions can be chosen to be free. Moreover, we give a positive answer to a question raised by Ioana and Shalom concerning the existence of such a free action for $\mathbb{F}_2\times\Z$. More generally, we show that for groups $\Gamma$ considered by Fernos in \cite{Fer}, e.g. Zariski-dense subgroups in PSL$_n(\R)$, the product group $\Gamma\times \Z$ admit a free ergodic rigid action. Also, we investigate how rigidity of an action behaves under restriction and co-induction. In particular, we show that rigidity passes to co-amenable subgroups.