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

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 ${\mathbb{F}}_2\times \mathbb{Z}$. More generally, we show that for groups $\Gamma$ considered by Fernos in \cite{Fer}, e.g. Zariski-dense subgroups in PSL${}_n(\mathbb{R})$, the product group $\Gamma \times \mathbb{Z}$ 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.

arXiv admin note: text overlap with arXiv:1502.04339

References in corpus (2)

• Relative Property (T) for the Subequivalence Relations Induced by the Action of SL${}_2(\mathbb{Z})$ on ${\mathbb{T}}^2$
• Rigidity for group actions on homogeneous spaces by affine transformations

Cited by in corpus (1)

• Property (T) discrete quantum groups and subfactors with triangle presentations