Groups acting on cubes and Kazhdan’s property (T)

Niblo, Graham; Roller, Martin

doi:10.1090/S0002-9939-98-04463-3

Groups acting on cubes and Kazhdan’s property (T)
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by Graham A. Niblo and Martin A. Roller PDF

Proc. Amer. Math. Soc. 126 (1998), 693-699 Request permission

Abstract:

We show that a group $G$ contains a subgroup $K$ with $e(G,K) > 1$ if and only if it admits an action on a connected cube that is transitive on the hyperplanes and has no fixed point. As a corollary we deduce that a countable group $G$ with such a subgroup does not satisfy Kazhdan’s property (T).

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