Matrix-valued random walks and variations on property AT

We reformulate matrix-valued random walks and their associated group actions in terms of dimension groups, suitably modified to deal with measure-theoretic classification. This leads naturally to a notion of rank denoted AT(n), for integers n (approximately transitive, that is, AT, actions constitute the rank one situation). This yields wide classes of examples, and easily verified criteria for basic properties (such as ergodicity) are established. We present an (ergodic) AT(2) action of the integers (from an involution) that is not AT, effectively answering an old question of Thouvenot, but on the other hand, give criteria for matrix-valued random walks to be AT. One of the criteria resembles a result of Mineka on mass cancellation. What are known as bounded AT actions in the literature are shown to be exactly the AT actions for which the corresponding random walk comes from a sequence of Poisson distributions, and we show that the natural involutions on bounded AT actions have orbit space that is AT (unlike more general AT actions), and generically are bounded. We also present a rather unusual ergodic action of the free group on two generators which is AT(2) (and not AT), but is given by a constant sequence (for an amenable group, a constant sequence would yield an uninteresting action).