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Rigidity for group actions on homogeneous spaces by affine transformations

Published online by Cambridge University Press:  22 March 2016

MOHAMED BOULJIHAD
MOHAMED BOULJIHAD*
Affiliation:
ENS Lyon – UMPA, 46 allée d’Italie, 69364 Lyon, France email [email protected] IRMAR, Campus de Beaulieu, bâtiments 22 et 23, 263 avenue du Général Leclerc, 35042 Rennes, France
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Abstract

We give a criterion for the rigidity of the action of a group of affine transformations of a homogeneous space of a real Lie group. Let $G$ be a real Lie group, $\unicode[STIX]{x1D6EC}$ a lattice in $G$ , and $\unicode[STIX]{x1D6E4}$ a subgroup of the affine group $\text{Aff}(G)$ stabilizing $\unicode[STIX]{x1D6EC}$ . Then the action of $\unicode[STIX]{x1D6E4}$ on $G/\unicode[STIX]{x1D6EC}$ has the rigidity property in the sense of Popa [On a class of type $\text{II}_{1}$ factors with Betti numbers invariants. Ann. of Math. (2) 163(3) (2006), 809–899] if and only if the induced action of $\unicode[STIX]{x1D6E4}$ on $\mathbb{P}(\mathfrak{g})$ admits no $\unicode[STIX]{x1D6E4}$ -invariant probability measure, where $\mathfrak{g}$ is the Lie algebra of $G$ . This generalizes results of Burger [Kazhdan constants for $\text{SL}(3,\mathbf{Z})$ . J. Reine Angew. Math. 413 (1991), 36–67] and Ioana and Shalom [Rigidity for equivalence relations on homogeneous spaces. Groups Geom. Dyn. 7(2) (2013), 403–417]. As an application, we establish rigidity for the action of a class of groups acting by automorphisms on nilmanifolds associated to two-step nilpotent Lie groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 7 , October 2017 , pp. 2060 - 2076
Copyright
© Cambridge University Press, 2016 

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