Ars Mathematica

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).

I’ve gotten complaints recently about Ars Math’s comment registration system, so as an experiment, I’ve enabled users to post comments without registering. Here’s hoping that I get to spent part of the next 24 hours not deleting spam…