Wm Groups and Ramsey Theory

Our goal in this paper is to exhibit a connection between two seemingly disparate areas: Ramsey theory and the theory of unitary representations of a class of locally compact groups. The class of groups that we are interested in consists of the so-called ”minimally periodic groups” introduced by von Neumann in [N]; namely, groups having the property that they do not admit non-trivial almost periodic functions. This, in turn, is equivalent to the property of not having non-trivial finite dimensional unitary representations. One can show that yet another equivalent form of the above condition is that any ergodic measure preserving action of such a group on a finite measure space is actually weakly mixing. It is this aspect that will interest us and so we call these groups WM groups, and the class of these is denoted WM. As we shall see WM groups that are also amenable have unexpected Ramsey-theoretical properties. Here are some examples of WM groups: (i) SL(2,R), or more generally, any simple non-compact Lie group with finite center. (ii) The group Alt(N) of even permutations of N, or more generally, any group which is a direct limit of compact simple groups. Note that the examples in (i) are non-amenable and those in (ii) are amenable. Amenability of a locally compact group can be defined in two ways. One of these is in terms of an invariant mean, that is, a functional on either bounded continuous functions or bounded Borel measurable functions having the same value on a function as on its translate: m(f) = m(fu), where fu(g) = f(ug). An alternative characterization is in terms of Følner sequences. A sequence of compact sets (Fn) in G is called (left) Følner sequence, if for any g ∈ G one has |Fn ∩ gFn| |Fn| → 1 as n→∞,