Emmanuel Breuillard, Ben Green, and I have just uploaded to the arXiv our paper “The structure of approximate groups“, submitted to Pub. IHES. We had announced the main results of this paper in various forums (including this blog) for a few months now, but it had taken some time to fully write up the paper and put in various refinements and applications.

As announced previously, the main result of this paper is what is a (virtually, qualitatively) complete description of finite approximate groups in an arbitrary (local or global) group ${G}$. For simplicity let us work in the much more familiar setting of global groups, although our results also apply (but are a bit more technical to state) in the local group setting.

Recall that in a global group ${G = (G,\cdot)}$, a ${K}$-approximate group is a symmetric subset ${A}$ of ${G}$ containing the origin, with the property that the product set ${A \cdot A}$ is covered by ${K}$ left-translates of ${A}$. Examples of ${O(1)}$-approximate groups include genuine groups, convex bodies in a bounded dimensional vector space, small balls in a bounded dimensional Lie group, large balls in a discrete nilpotent group of bounded rank or step, or generalised arithmetic progressions (or more generally, coset progressions) of bounded rank in an abelian group. Specialising now to finite approximate groups, a key example of such a group is what we call a coset nilprogression: a set of the form ${\pi^{-1}(P)}$, where ${\pi: G' \rightarrow N}$ is a homomorphism with finite kernel from a subgroup ${G'}$ of ${G}$ to a nilpotent group ${N}$ of bounded step, and ${P = P(u_1,\ldots,u_r;N_1,\ldots,N_r)}$ is a nilprogression with a bounded number of generators ${u_1,\ldots,u_r}$ in ${N}$ and some lengths ${N_1,\ldots,N_r \gg 1}$, where ${P(u_1,\ldots,u_r;N_1,\ldots,N_r)}$ consists of all the words involving at most ${N_1}$ copies of ${u_1^{\pm 1}}$, ${N_2}$ copies of ${u_2^{\pm 1}}$, and so forth up to ${N_r}$ copies of ${u_r^{\pm 1}}$. One can show (by some nilpotent algebra) that all such coset nilprogressions are ${O(1)}$-approximate groups so long as the step and the rank ${r}$ are bounded (and if ${N_1,\ldots,N_r}$ are sufficiently large).

Our main theorem (which was essentially conjectured independently by Helfgott and by Lindenstrauss) asserts, roughly speaking, that coset nilprogressions are essentially the only examples of approximate groups.

Theorem 1 Let ${A}$ be a ${K}$-approximate group. Then ${A^4}$ contains a coset nilprogression ${P}$ of rank and step ${O_K(1)}$, such that ${A}$ can be covered by ${O_K(1)}$ left-translates of ${P}$.

In the torsion-free abelian case, this result is essentially Freiman’s theorem (with an alternate proof by Ruzsa); for general abelian case, it is due to Green and Ruzsa. Various partial results in this direction for some other groups (e.g. free groups, nilpotent groups, solvable groups, or simple groups of Lie type) are also known; see these previous blog posts for a summary of several of these results.

This result has a number of applications to geometric growth theory, and in particular to variants of Gromov’s theorem of groups of polynomial growth, which asserts that a finitely generated group is of polynomial growth if and only if it is virtually nilpotent. The connection lies in the fact that if the balls ${B_S(R)}$ associated to a finite set of generators ${S}$ has polynomial growth, then some simple volume-packing arguments combined with the pigeonhole principle will show that ${B_S(R)}$ will end up being a ${O(1)}$-approximate group for many radii ${R}$. In fact, since our theorem only needs a single approximate group to obtain virtually nilpotent structure, we are able to obtain some new strengthenings of Gromov’s theorem. For instance, if ${A}$ is any ${K}$-approximate group in a finitely generated group ${G}$ that contains ${B_S(R_0)}$ for some set of generators ${S}$ and some ${R_0}$ that is sufficiently large depending on ${K}$, our theorem implies that ${G}$ is virtually nilpotent, answering a question of Petrunin. Among other things, this gives an alternate proof of a recent result of Kapovitch and Wilking (see also this previous paper of Cheeger and Colding) that a compact manifold of bounded diameter and Ricci curvature at least ${-\epsilon}$ necessarily has a virtually nilpotent fundamental group if ${\epsilon}$ is sufficiently small (depending only on dimension). The main point here is that no lower bound on the injectivity radius is required. Another application is a “Margulis-type lemma”, which asserts that if a metric space ${X}$ has “bounded packing” (in the sense that any ball of radius (say) ${4}$ is covered by a bounded number of balls of radius ${1}$), and ${\Gamma}$ is a group of isometries on ${X}$ that acts discretely (i.e. every orbit has only finitely many elements (counting multiplicity) in each bounded set), then the near-stabiliser ${\{ \gamma \in \Gamma: d(\gamma x, x) \leq \epsilon \}}$ of a point ${x}$ is virtually nilpotent if ${\epsilon}$ is small enough depending on the packing constant.

There are also some variants and refinements to the main theorem proved in the paper, such as an extension to local groups, and also an improvement on the bound on the rank and step from ${O_K(1)}$ to ${O(\log K)}$ (but at the cost of replacing ${A^4}$ in the theorem with ${A^{O(1)}}$).

I’ll be discussing the proof of the main theorem in detail in the next few lecture notes of my current graduate course. The full proof is somewhat lengthy (occupying about 50 pages of the 90-page paper), but can be summarised in the following steps:

1. (Hrushovski) Take an arbitrary sequence ${A_n}$ of finite ${K}$-approximate groups, and show that an appropriate limit ${A}$ of such groups can be “modeled” in some sense by an open bounded subset of a locally compact group. (The precise definition of “model” is technical, but “macroscopically faithful representation” is a good first approximation.) As discussed in the previous lecture notes, we use an ultralimit for this purpose; the paper of Hrushovski where this strategy was first employed also considered more sophisticated model-theoretic limits. To build a locally compact topology, Hrushovski used some tools from definability theory; in our paper, we instead use a combinatorial lemma of Sanders (closely related to a similar result of Croot and Sisask.)
2. (Gleason-Yamabe) The locally compact group can in turn be “modeled” by a Lie group (possibly after shrinking the group, and thus the ultralimit ${A}$, slightly). (This result arose from the solution to Hilbert’s fifth problem, as discussed here. For our extension to local groups, we use a recent local version of the Gleason-Yamabe theorem, due to Goldbring.)
3. (Gleason) Using the escape properties of the Lie model, construct a norm ${\| \|}$ (and thus a left-invariant metric ${d}$) on the ultralimit approximate group ${A}$ (and also on the finitary groups ${A_n}$) that obeys a number of good properties, such as a commutator estimate ${\| [g,h]\| \ll \|g\| \|h\|}$. (This is modeled on an analogous construction used in the theory of Hilbert’s fifth problem, as discussed in this previous set of lecture notes.) This norm is essentially an escape norm associated to (a slight modification) of ${A}$ or ${A_n}$.
4. (Jordan-Bieberbach-Frobenius) We now take advantage of the finite nature of the ${A_n}$ by locating the non-trivial element ${e}$ of ${A_n}$ with minimal escape norm (but one has to first quotient out the elements of zero escape norm first). The commutator estimate mentioned previously ensures that this element is essentially “central” in ${A_n}$. One can then quotient out a progression ${P(e;N)}$ generated by this central element (reducing the dimension of the Lie model by one in the process) and iterates the process until the dimension of the model drops to zero. Reversing the process, this constructs a coset nilprogression inside ${A_n^4}$. This argument is based on the classic proof of Jordan’s theorem due to Bieberbach and Frobenius, as discussed in this blog post.

One quirk of the argument is that it requires one to work in the category of local groups rather than global groups. (This is somewhat analogous to how, in the standard proofs of Freiman’s theorem, one needs to work with the category of Freiman homomorphisms, rather than group homomorphisms.) The reason for this arises when performing the quotienting step in the Jordan-Bieberbach-Frobenius leg of the argument. The obvious way to perform this step (and the thing that we tried first) would be to quotient out by the entire cyclic group ${\langle e \rangle}$ generated by the element ${e}$ of minimal escape norm. However, it turns out that this doesn’t work too well, because the group quotiented out is so “large” that it can create a lot of torsion in the quotient. In particular, elements which used to have positive escape norm, can now become trapped in the quotient of ${A_n}$, thus sending their escape norm to zero. This leads to an inferior conclusion (in which a coset nilprogression is replaced by a more complicated tower of alternating extensions between central progressions and finite groups, similar to the towers encountered in my previous paper on this topic). To prevent this unwanted creation of torsion, one has to truncate the cyclic group ${\langle e \rangle}$ before it escapes ${A_n}$, so that one quotients out by a geometric progression ${P(e;N)}$ rather than the cyclic group. But the operation of quotienting out by a ${P(e;N)}$, which is a local group rather than a global one, cannot be formalised in the category of global groups, but only in the category of local groups. Because of this, we were forced to carry out the entire argument using the language of local groups. As it turns out, the arguments are ultimately more natural in this setting, although there is an initial investment of notation required, given that global group theory is much more familiar and well-developed than local group theory.

One interesting feature of the argument is that it does not use much of the existing theory of Freiman-type theorems, instead building the coset nilprogression directly from the geometric properties of the approximate group. In particular, our argument gives a new proof of Freiman’s theorem in the abelian case, which largely avoids Fourier analysis (except through the use of the theory of Hilbert’s fifth problem, which uses the Peter-Weyl theorem (or, in the abelian case, Pontryagin duality), which is basically a version of Fourier analysis).