A study of exactness for discrete groups
We recall the concepts of exactness for both C*-algebras and groups. We explore some new properties linked or equivalent to exactness, including Property A, a second property we term Property O, and Hilbert space compression [GK2, O, Yu]. We use geometric methods to show that a variety of groups satisfy these properties. We then deduce that those groups are exact. In particular we show that Properties O and A are equivalent. We show that the integers, groups of subexponential growth, amenable groups and free groups satisfy Property O by constructing a family of Ozawa kernels for each case. To construct these families we exploit growth properties of the integers and groups of subexponential growth, Følner’s criterion for amenable groups and geometric properties of the Cayley graph for free groups. For each of these groups we deduce that they are exact and have Property A. Finally we turn to Hilbert space compression to prove our main theorem that groups acting properly and cocompactly on CAT(0) cube complexes are exact and have Property A.