Further I reproduce, with small modifications, a comment to the post
by Terence Tao.
My motivation lies in the project described first time in public here. In fact, one of the reasons to start this blog is to have a place where I can leisurely explain stuff.
Background: The answer to the Hilbert fifth’s problem is: a connected locally compact group without small subgroups is a Lie group.
The key idea of the proof is to study the space of one parameter subgroups of the topological group. This space turns out to be a good model of the tangent space at the neutral element of the group (eventually) and the effort goes towards turning upside-down this fact, namely to prove that this space is a locally compact topological vector space and the “exponential map” gives a chart of (a neighbourhood of the neutral element of ) the group into this space.
Because I am a fan of differential structures (well, I think they are only the commutative, boring side of dilation structures or here or emergent algebras) I know a situation when one can prove the fact that a topological group is a Lie group without using the one parameter subgroups!
Here starts the original comment, slightly modified:
Contractive automorphisms may be as relevant as one-parameter subgroups for building a Lie group structure (or even more), as shown by the following result from E. Siebert, Contractive Automorphisms on Locally Compact Groups, Math. Z. 191, 73-90 (1986)
5.4. Proposition. For a locally compact group G the following assertions are equivalent:
(i) G admits a contractive automorphism group;
(ii) G is a simply connected Lie group whose Lie algebra g admits a positive graduation.
The corresponding result for local groups is proved in L. van den Dries, I. Goldbring, Locally Compact Contractive Local Groups, arXiv:0909.4565v2.
I used Siebert result for proving the Lie algebraic structure of the metric tangent space to a sub-riemannian manifold in M. Buliga, Infinitesimal affine geometry of metric spaces endowed with a dilatation structure, Houston Journal of Mathematics, 36, 1 (2010), 91-136. arXiv:0804.0135v2
(added here: see in Corollary 6.3 from “Infinitesimal affine …” paper, as well as Proposition 5.9 and Remark 5.10 from the paper A characterization of sub-riemannian spaces as length dilation structures constructed via coherent projections, Commun. Math. Anal. 11 (2011), No. 2, pp. 70-111 , arXiv:0810.5042v4 )
When saying that contractive automorphisms, or approximately contractive automorphisms [i.e. dilation structures], may be more relevant than one-parameter subgroups, I am thinking about sub-riemannian geometry again, where a one-parameter subgroup of a group, endowed with a left-invariant distribution and a corresponding Carnot-Caratheodory distance, is “smooth” (with respect to Pansu-type derivative) if and only if the generator is in the distribution. Metrically speaking, if the generator is not in the distribution then any trajectory of the one-parameter group has Hausdorff dimension greater than one. That means lots of problems with the definition of the exponential and any reasoning based on differentiating flows.