On Relative Property (T) and Haagerup's Property

Abstract

We consider the following three properties for countable discrete groups $\Gamma$: (1) $\Gamma$ has an infinite subgroup with relative property (T), (2) the group von Neumann algebra $L\Gamma$ has a diffuse von Neumann subalgebra with relative property (T) and (3) $\Gamma$ does not have Haagerup's property. It is clear that (1) $⟹$ (2) $⟹$ (3). We prove that both of the converses are false.

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Cited by in corpus (7)

• Relative expanders
• Maximal subgroups and von Neumann subalgebras with the Haagerup property
• A characterization of relative Kazhdan Property T for semidirect products with abelian groups
• Proper actions of wreath products and generalizations
• Aspects de la géométrie des groupes
• Embeddability of generalized wreath products and box spaces
• Spaces with labelled partitions and isometric affine actions on Banach spaces