On relative property (T) and Haagerup’s property

Chifan, Ionut; Ioana, Adrian

doi:10.1090/S0002-9947-2011-05259-1

On relative property (T) and Haagerup's property

Authors: Ionut Chifan and Adrian Ioana
Journal: Trans. Amer. Math. Soc. 363 (2011), 6407-6420
MSC (2010): Primary 20F69; Secondary 46L10
DOI: https://doi.org/10.1090/S0002-9947-2011-05259-1
Published electronically: July 14, 2011
MathSciNet review: 2833560
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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) $\Longrightarrow$ (2) $\Longrightarrow$ (3). We prove that both of the converses are false.

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