Analogy Help

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).

10 thoughts on “Analogy Help

  1. André Joyal once explained to me that “morally speaking”, the Lie group corresponding to the free Lie algebra on two generators is the group of paths in the plane which start at the origin. Think of the generators x and y as “infinitesimal steps” in the x and y directions on the plane, and you can probably figure out what he meant. The map from the free Lie algebra on two generators to the free abelian Lie algebra on two generators exponentiates to give the homomorphism mapping any path in the plane starting at the origin to its endpoint!

    He also explained how this lets one understand Fox’s “free differential calculus”. Alas, I don’t remember how that works, but this article might help.

    More recently, Misha Kapranov has made some of these ideas more precise, in his paper on noncommutative geometry and path integrals. The first two paragraphs give the game away, but also see page 2, where he writes: “The idea that the space of paths is related to the free group and to its various versions had been enunciated by K. T. Chen in the 1950s and can be traced throughout almost all of his work.”

  2. I’m talking about the group R*R, endowed with the distance inherited from the canonical distance in R. It seems that it is a classical object for people studying real trees and geometric group theory.

  3. R*R is a good candidate. One property that it has that I was not expecting in the final answer is that if you think of the elements of the group as paths that move in either the x or y direction for a certain amount of time, then each path is piecewise constant: you move in the x direction for a certain amount of time, then you switch to the y direction for a certain amount of time. If you base the analogy on the free Lie algebra, you can smoothly switch between the two directions, instead of jumping from one direction to the other.

    I initially thought that the free Lie algebra on two generators morally corresponded to paths in the plane, but I convinced myself that wasn’t quite right, and that paths in the plane is more like a group representation. Paths in the plane have the property that you can reach the same point using multiple paths. It could be that “endpoints” are just a bit of extra structure we can throw away, but the group should correspond to an abstract space of paths that has as its representations the set of all paths described by any two arbitrary vector fields on an arbitrary manifold. Maybe the space of paths in the plane is a faithful representation, and you just need to ignore the extra structure, but I’m not really clear if that’s the case.

  4. Walt writes:

    I initially thought that the free Lie algebra on two generators morally corresponded to paths in the plane, but I convinced myself that wasn’t quite right, and that paths in the plane is more like a group representation. Paths in the plane have the property that you can reach the same point using multiple paths.

    I don’t understand your worries here. I can’t tell if this is one thing you’re saying: general paths in the plane, with arbitrary endpoints, would certainly not be the right group. I agree with that.

    But, I repeat that Chen, Joyal, Kapranov and doubtless many other wise mathematicians agree on the “right” answer to your puzzle: namely, paths in the plane starting at the origin, with the obvious product (sticking one path at the end of the other), modulo a relation that makes the wannabe inverses into actual inverses.

    The question of whether to use continuous, piecewise smooth or whatever other sort of paths depends on what sort of analysis we enjoy. There are lots of choices.

    Perhaps the smallest group like this is the free product (= coproduct) R*R, which consisting of paths that consist of vertical and horizontal segments. R*R is dense in the group G coming from continuous paths starting at the origin, if we give the G the topology coming from uniform convergence. (As mentioned above, G is really a quotient space of the space of continuous paths starting at the origin. So, we really need to use the quotient topology.)

  5. I don’t object to the idea of paths in the plane as an analogue to the free group. This is exactly construction #1 that I suggested in my original post. (I glanced at the Kapranov paper, and the construction the outlines is exactly what I had in mind when I said “certain sets of paths on the plane”.)

    I’m quibbling at the idea that the Lie group associated with the free Lie algebra is morally the set of paths in the plane. Rather than try to explain my objection (which I’m not even sure is a true objection), I’ll ask a question. Let x be the vector field on the plane of translation to the right, and y be the vector field of translation up. Identify these with the two generators of the free Lie algebra. It’s clear what exp(x) and exp(y) are in the set of paths: translation to the right and translation up one unit. What’s exp([x,y])?

  6. One way of constructing the reals from the integers is via Gromov’s Asymptotic Cone. From this point of view, the natural analogy to Z*Z is the asymptotic cone of Z*Z. Very roughly speaking, the asymptotic cone of a group is the set of all bounded sequences of elements, modulo a relation that equates sequences that agree on all but finitely many terms.

    I expect this turns out to be the same as one of the other analogies suggested above, thought at this time of night I can’t tell for sure!

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