Wildness in the product groups

Non-abelian Polish groups arising as countable products of countable groups can be tame in arbitrarily complicated ways. This contrasts with some results of Solecki who revealed a very different picture in the abelian case. 0. Group trees. The class of all Polish (completely metrizable, separable) groups may be naturally divided into two classes. 0.1. Definition. A Polish group G is tame if whenever X is a Polish G-space (that is to say, G acts continuously on X) the orbit equivalence relation is Borel as a subset of X ×X. A Polish group that is not tame is wild. On the one hand the wild groups include almost all groups of reasonable topological complexity—for instance: S∞, the infinite permutation group in the topology of pointwise convergence; U∞, the unitary group of Hilbert space; RN (the infinite product of (R,+) in the product group structure and topology); c0; l2. The main examples of tame groups are the locally compact ones. These were not quite the only known examples—in [11] it is also shown that ⊕ p prime (Z(p∞)), the infinite product of the subgroups of R/Z generated by {p−n : n ∈ N} for p a prime, is tame. Thus there is a gap in the spectrum of examples. On the one hand we have the wild groups that can give rise to enormously complicated actions. On the other hand the principal examples of tame 2000 Mathematics Subject Classification: Primary 03A15, 03E15; Secondary 20B35, 03C15.