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

Abstract

We show that a group G G contains a subgroup K K with e ( G , K ) > 1 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 G with such a subgroup does not satisfy Kazhdan’s property (T).