Lie groups acting on countable sets

Does every connected Lie group act faithfully on a countable set? In other words: is every Lie group a subgroup of $$\mathrm{Sym}(\mathbb{N})$$? This question is sometimes called Ulam’s problem and there is recent progress in a paper of Nicolas Monod. Monod proves that every nilpotent connected Lie group acts faithfully on a countable set. It … Continue reading "Lie groups acting on countable sets"