Hilbert space compression and exactness for discrete groups

Abstract

We show that the Hilbert space compression of any finite dimensional CAT(0) cube complex is 1 and deduce that any discrete group acting properly, co-compactly on a CAT(0) cube complex is exact. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)-T(4) small cancellation condition and all those word-hyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams.