arXiv subjects

Mohamed Bouljihad

Publications and source records attributed to Mohamed Bouljihad.

2 recordsLinked to original sources

arXiv2016

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

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 $Γ$ considered by Fernos in \cite{Fer}, e.g. Zariski-dense subgroups in PSL$_n(\R)$, the product group $Γ\times \Z$ 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.

math.DS↗

arXiv2015

Rigidity for group actions on homogeneous spaces by affine transformations

We give a criterion for the rigidity of actions on homogeneous spaces. Let $G$ be a real Lie group, $Λ$ a lattice in $G$, and $Γ$ a subgroup of the affine group Aff$(G)$ stabilizing $Λ$. Then the action of $Γ$ on $G/Λ$ has the rigidity property in the sense of S. Popa, if and only if the induced action of $Γ$ on $\mathbb{P}(\frak{g})$ admits no $Γ$-invariant probability measure, where $\frak{g}$ is the Lie algebra of $G$. This generalizes results of M. Burger, and A. Ioana and Y. Shalom. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nilmanifolds associated to step 2 nilpotent Lie groups.

math.DS↗