Here’s an analogy that I try to complete from time to time.
Integers:Reals::Free Group on Two Generators:?
(Under addition, the integers are the free group on one generator.) I’m not precisely sure what properties of the construction of the reals I’m trying to generalize to the right-hand side of the analogy, beyond the fact that the answer should be a group that is a path-connected topological space.
Two candidates I’ve considered are: 1) certain sets of paths on the plane (which is naturally a groupoid, but you can bully it into being a group) or 2) the Lie group corresponding to the free Lie algebra on two generators (I don’t know in what sense, if any, such an object exists).