Some Notes on Property A

Introduction The coarse Baum-Connes conjecture states that a certain coarse assembly map µ : KX * (X) → K * (C * (X)) is an isomorphism (see for example [Roe96] or the piece by N. Higson and J. Roe in [FRR95]). It has many important consequences including one of the main motivations for this piece: a descent technique connects it to injectivity of the ('ordinary') Baum-Connes assembly map µ : K * (EG) → K * (C * r (G)) when X is the underlying coarse space of a countable discrete group G. In-jectivity of this map is called the strong Novikov conjecture and (as one might guess) implies the 'usual' Novikov conjecture on higher signatures in the case G is the fundamental group of a compact manifold. Our focus in these notes is on property A, which was introduced by G. Yu in [Yu00] as a weak form of amenability. He proved that a property A metric space X coarsely embeds into (looks like a subspace of, on the large scale) Hilbert space, that such coarse embeddability implies coarse Baum-Connes, and from here the strong Novikov conjecture if X is the underlying coarse space of a countable discrete group G (two finiteness restrictions were later removed in [Hig00] by Higson, and in more generality in [STY02] by G. Skandalis, J-L. Tu and Yu). The above is background and motivation. We will focus on rather less complicated aspects of property A, which is a relatively elementary 'large-scale' (or coarse) invariant of metric spaces. It is part of the appeal of A that such a seemingly simple property really does have a great many connections with, and implications for, other areas of mathematics. These include amenability, group representations and various coarse embeddability problems, all of which this piece starts to explore. The classical theory of kernels touches most of these areas, and helps to unify a lot of the material. The two major omissions we make are the relatively sophisticated K-theoretic consequences mentioned above, and some finer invariants associated with prop-1 erty A. The latter unfortunately meant leaving out many of the results showing that particular classes of groups have property A, as well as certain quasi-isometry invariants, for example those studied in [GK04], [Now07a] and [Tes07], all of which is interesting and reasonably accessible; references are included in the bibliography. Our approach does have the benefit that almost everything discussed …