At the time of writing, the most recent blog post over at What's new by Terrence Tao is Cayley graphs and the geometry of groups, and that (excellent, as with most of Tao's writing) post most immediately inspired this question. Also, a disclaimer: Cayley graphs, and, indeed, abstract group theory generally, are not my area of expertise, so feel free to explain how my question is overly naive and/or needs revision.
It is a classical result that a connected topological group is generated by any open neighborhood of its identity element. (Proof: the subgroup generated by the open neighborhood is a union of opens, and hence open; but then so are all its cosets; so this subgroup is both open and closed; hence it is everything if the group is connected as a topological space.) So I am tempted to take some topological group $G$, and some very small neighborhood $S$ of the identity $e$, and construct the corresponding Cayley graph. My intuition says that as $S$ gets smaller and smaller, the Cayley graph better and better approximates the topological space $G$. Notice also that arbitrary small open neighborhoods $S\ni e$ know everything about the topology of $G$, because $G$ is homogeneous.
I don't know how to define a "formal neighborhood" of a topological space, unless it is actually a manifold or algebraic space or …. So maybe I should restrict my attention to Lie or Algebraic groups for this question. But anyway:
Is there a way to define the "infinitesimal generators" or "formal neighborhood" for a topological group in such a way that the corresponding "Cayley graph" is the group as a topological space (Edit: which of course doesn't make any sense; see the comments. I mean something like "so that the geometry/topology of the group comes immediately from the graph)? If not in this generality, does it at least work when the group is Lie? Algebraic? Or other regularity conditions on the topology?
Best Answer
Several people have been working on what might be described as geometric group theory for compactly generated, totally disconnected topological groups.
The main definitions as well as several nice results are in this paper:
Krön, Bernhard; Möller, Rögnvaldur G. Analogues of Cayley graphs for topological groups. Math. Z. 258 (2008), no. 3, 637--675.
and if you search for papers which reference this one you'll find some more recent work.