I am interested in the amenability properties of infinite products of solvable groups. The following facts are well-known:


1. Any solvable group is amenable.
2. The class of solvable groups is closed under finite products and quotients (but not infinite products).
3. Infinite products of solvable groups of a fixed derived length are solvable.
4. Finite products and quotients of amenable groups are amenable (but not infinite products).


My specific question is whether an infinite product of solvable groups is still amenable?


The motivation arises from uniformity in multiple recurrence within ergodic theory. My coauthor and I wish to prove that the largeness of certain sets of multiple return times is independent of the underlying measure-preserving system *and* the acting group. Currently, we have achieved this for solvable groups of a fixed derived length, and more generally, for a class of uniformly amenable groups, where we use the device of ultraproducts as a key tool in our approach.



Is the infinite product of solvable groups amenable?
The free group $F\_2$ is residually nilpotent, meaning that the intersection of its lower central series is trivial, because the length of an element in the $k$th term of the lower central series is bounded below by $k$. It follows that $F\_2$ embeds in a direct product of nilpotent groups. So there is an infinite product of nilpotent groups that is not amenable.


