Approximate groups, III
Let be a finitely generated free group. It is easy that for the standard generating system, . What about other generating systems?
Theorem 1 (Razborov, Safin) For an arbitrary generating set in ,
If a Product Theorem holds in a group, then . So this implies super-polynomial growth, but not exponential growth. Nevertheless, it might give an alternative proof of (a weak form) of Gromov’s polynomial growth theorem. Note that Product Theorems in this form are known only for certain finite simple groups (and even fail for alternating groups).
1.1. Proof of Gromov’s polynomial growth theorem
Indeed, the structure theorem for approximate groups (BGT) yields such a proof.
Assume that for all , . There are arbitrarily large integers such hat , i.e. is an approximate subgroup.
BGT states that where is virtually nilpotent and . For , this shows that has finite index.
Remark. We merely need one large value of such that . Also, the constant plays no role, merely the ratio . So that applies to generating sets of arbitrary size.
2. Margulis Lemma
Margulis Lemma states that in a compact negatively curved Riemannian manifold , the subgroup of generated by loops of length dimension,min sectional curvature is cyclic. In the non-compact case, cyclic needs be replaced with nilpotent.
Here is a generalization, which follows from BGT.
Lemma 2 Let be a metric space with -bounded geometry (every ball of radius 4 is covered by at mots balls of radius 1). Then, for every , any discrete group of isometries generated by elements that move at most away is virtually nilpotent.
Then is an approximate group, hence contained in with virtually nilpotent and , has finite index.
3. Proof of structure theorem
It goes by contradiction. Compactness plays a role, in the form of ultraproducts. The strategy was outlined by Hrushovski. We dug into the proof of the solution to Hilbert’s 5th problem.
Take any sequence of approximate subgroups in groups . Form ultraproducts . This is a non-standard approximate subgroup.
3.1. Step 1
Define a locally compact topology on .
A lemma due to independently to Sanders and Hrushovski says that in a finite -approximate group , there exists a subset of positive proportion such that . Such sets provide a basis for the required topology.
Then our non-standard maps to a locally compact group with image a compact neighborhood of .
3.2. Step 2
Use tools from the proof of the solution to Hilbert’s 5th problem: up to passing to an open subgroup, can mod out by a compact group and get a Lie group.
The point (Gleason Lemmas) is to show that one-parameter subgroups (obtained as limits of cyclic subgroups) can be multiplied. This produces a vectorspace, candidate to be the Lie algebra. These lemmas are quantitative, we can give approximative group versions.
Does this give a gap from polynomial growth? Not quite, because non effective. Kleiner’s proof is effective, Shalom and Tao managed to extract a gap from it, something like .