I think both.
Finally Terence Tao presented a sketch of his project relating Hilbert’ fifth problem and approximate groups. For me the most interesting part is his Theorem 12 and its relation with Gromov’ theorem on finitely generated groups with polynomial growth.
A bit dissapointing (for me) is that he seems to choose to rely on “commutative” techniques and, not surprising, he is bound to get results valid only in riemannian geometry (or spaces of Alexander type) AND also to regard the apparition of nilpotent structures as qualitatively different from smooth structures.
For clarity reasons, I copy-paste his two “broad principles”
The above three families of results exemplify two broad principles (part of what I like to call “the dichotomy between structure and randomness“):
- (Rigidity) If a group-like object exhibits a weak amount of regularity, then it (or a large portion thereof) often automatically exhibits a strong amount of regularity as well;
- (Structure) This strong regularity manifests itself either as Lie type structure (in continuous settings) or nilpotent type structure (in discrete settings). (In some cases, “nilpotent” should be replaced by sister properties such as “abelian“, “solvable“, or “polycyclic“.)
Let me contrast with my
(from this previous post).
While his “dichotomy between structure and randomness“ principle is a very profound and mysterious one, the second part (Structure) is only an illusion created by psychological choices (I guess). Indeed, both (Lie) smooth structure and nilpotence are just “noncommutative (local) linearity”, as explained previously. Both smooth structure and “conical algebraic” structure (nilpotent in particular) stem from the same underlying dilation structure. What is most mysterious, in my opinion, be it in the “exact” or “approximate” group structure, is HOW a (variant of) dilation structure appears from non-randomness (presumably as a manifestation of Tao randomness/structure principle), i.e. HOW just a little bit of approximate self-similarity bootstraps to a dilation structure, or emergent algebra, which then leads to various flavors of “smoothness” and “nilpotence”.