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

2016/06/19 by Mohamed Bouljihad, Bouljihad, Mohamed
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1606.05911

openalex publication_date 2016/06/19 · openalex created_date 2016/07/22 · openalex updated_date 2026/07/28

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 \mathbbF₂×\Z. More generally, we show that for groups Γ considered by Fernos in \citeFer, e.g. Zariski-dense subgroups in PSL_n(\R), the product group Γ× \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.

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