Approximate group

In mathematics, an approximate group is a subset of a group which behaves like a subgroup "up to a constant error", in a precise quantitative sense (so the term approximate subgroup may be more correct). For example, it is required that the set of products of elements in the subset be not much bigger than the subset itself (while for a subgroup it is required that they be equal). The notion was introduced in the 2010s but can be traced to older sources in additive combinatorics.

Formal definition

Let ${\displaystyle G}$ be a group and ${\displaystyle K\geq 1}$; for two subsets ${\displaystyle X,Y\subset G}$ we denote by ${\displaystyle X\cdot Y}$ the set of all products ${\displaystyle xy,x\in X,y\in Y}$. A non-empty subset ${\displaystyle A\subset G}$ is a ${\displaystyle K}$-approximate subgroup of ${\displaystyle G}$ if: ^[1]

1. It is symmetric, that is if ${\displaystyle g\in A}$ then ${\displaystyle g^{-1}\in A}$;
2. There exists a subset ${\displaystyle X\subset G}$ of cardinality ${\displaystyle |X|\leq K}$ such that ${\displaystyle A\cdot A\subset X\cdot A}$.

It is immediately verified that a 1-approximate subgroup is the same thing as a genuine subgroup. Of course this definition is only interesting when ${\displaystyle K}$ is small compared to ${\displaystyle |A|}$ (in particular, any subset ${\displaystyle B\subset G}$ is a ${\displaystyle |B|}$-approximate subgroup). In applications it is often used with ${\displaystyle K}$ being fixed and ${\displaystyle |A|}$ going to infinity.

Examples of approximate subgroups which are not groups are given by symmetric intervals and more generally arithmetic progressions in the integers. Indeed, for all ${\displaystyle n\geq 1}$ the subset ${\displaystyle X=[-N,N]\subset \mathbb {Z} }$ is a 2-approximate subgroup: the set ${\displaystyle X+X=[-2N,2N]}$ is contained in the union of the two translates ${\displaystyle X+N}$ and ${\displaystyle X-N}$ of ${\displaystyle X}$. A generalised arithmetic progression in ${\displaystyle \mathbb {Z} }$ is a subset in ${\displaystyle \mathbb {Z} }$ of the form ${\displaystyle \{n_{1}x_{1}+\cdots +n_{d}x_{d}:|n_{i}|\leq N_{i}\}}$, and it is a ${\displaystyle 2^{d}}$-approximate subgroup.

A more general example is given by balls in the word metric in finitely generated nilpotent groups.

Classification of approximate subgroups

Approximate subgroups of the integer group ${\displaystyle \mathbb {Z} }$ were completely classified by Imre Z. Ruzsa and Freiman. ^[2] The result is stated as follows:

For any ${\displaystyle K\geq 1}$ there are ${\displaystyle C_{K},c_{K}>0}$ such that for any ${\displaystyle K}$-approximate subgroup ${\displaystyle A\subset \mathbb {Z} }$ there exists a generalised arithmetic progression ${\displaystyle P}$ generated by at most ${\displaystyle C_{K}}$ integers and containing at least ${\displaystyle c_{K}|A|}$ elements, such that ${\displaystyle A+A+A+A\supset P}$.

The constants ${\displaystyle C_{K},C_{K}',c_{K}}$ can be estimated sharply. ^[3] In particular ${\displaystyle A}$ is contained in at most ${\displaystyle C_{K}'}$translates of ${\displaystyle P}$: this means that approximate subgroups of ${\displaystyle \mathbb {Z} }$ are "almost" generalised arithmetic progressions.

The work of Breuillard–Green–Tao (the culmination of an effort started a few years earlier by various other people) is a vast generalisation of this result. In a very general form its statement is the following: ^[4]

Let ${\displaystyle K\geq 1}$; there exists ${\displaystyle C_{K}}$ such that the following holds. Let ${\displaystyle G}$ be a group and ${\displaystyle A}$ a ${\displaystyle K}$-approximate subgroup in ${\displaystyle G}$. There exists subgroups ${\displaystyle H\triangleleft G_{0}\leq G}$ with ${\displaystyle H}$ finite and ${\displaystyle G_{0}/H}$ nilpotent such that ${\displaystyle A^{4}\supset H}$, the subgroup generated by ${\displaystyle A^{4}}$ contains ${\displaystyle G_{0}}$, and ${\displaystyle A\subset X\cdot G_{0}}$ with ${\displaystyle |X|\leq C_{K}}$.

The statement also gives some information on the characteristics (rank and step) of the nilpotent group ${\displaystyle G_{0}/H}$.

In the case where ${\displaystyle G}$ is a finite matrix group the results can be made more precise, for instance: ^[5]

Let ${\displaystyle K\geq 1}$. For any ${\displaystyle d}$ there is a constant ${\displaystyle C_{d}}$ such that for any finite field ${\displaystyle \mathbb {F} _{q}}$, any simple subgroup ${\displaystyle G\subset \mathrm {GL} _{d}(\mathbb {F} _{q})}$ and any ${\displaystyle K}$-approximate subgroup ${\displaystyle A\subset G}$ then either ${\displaystyle A}$ is contained in a proper subgroup of ${\displaystyle G}$, or ${\displaystyle |A|\leq K^{C_{d}}}$, or ${\displaystyle |A|\geq |G|/K^{C_{d}}}$.

The theorem applies for example to ${\displaystyle \mathrm {SL} _{d}(\mathbb {F} _{q})}$; the point is that the constant does not depend on the cardinality ${\displaystyle q}$ of the field. In some sense this says that there are no interesting approximate subgroups (besides genuine subgroups) in finite simple linear groups (they are either "trivial", that is very small, or "not proper", that is almost equal to the whole group).

Applications

The Breuillard–Green–Tao theorem on classification of approximate groups can be used to give a new proof of Gromov's theorem on groups of polynomial growth. The result obtained is actually a bit stronger since it establishes that there exists a "growth gap" between virtually nilpotent groups (of polynomial growth) and other groups; that is, there exists a (superpolynomial) function ${\displaystyle f}$ such that any group with growth function bounded by a multiple of ${\displaystyle f}$ is virtually nilpotent. ^[6]

Other applications are to the construction of expander graphs from the Cayley graphs of finite simple groups, and to the related topic of superstrong approximation. ^{[7] [8]}

Notes

1. Green 2012.
2. Ruzsa, I. Z. (1994). "Generalized arithmetical progressions and sumsets". Acta Mathematica Hungarica . 65 (4): 379–388. doi:10.1007/bf01876039. S2CID   121469006.
3. Breuillard, Tao & Green 2012, Theorem 2.1.
4. Breuillard, Tao & Green 2012, Theorem 1.6.
5. Breuillard 2012, Theorem 4.8.
6. Breuillard, Tao & Green 2012, Theorem 1.11.
7. Breuillard 2012.
8. Helfgott, Harald; Seress, Ákos; Zuk, Andrzej (2015). "Expansion in the symmetric groups". Journal of Algebra . 421: 349–368. arXiv: 1311.6742 . doi: 10.1016/j.jalgebra.2014.08.033 . S2CID   119315830.

References

• Breuillard, Emmanuel (2012). "Expander graphs, property (τ) and approximate groups". In Bestvina, Mladen; Sageev, Michah; Vogtmann, Karen (eds.). Geometric group theory (PDF). IAS/Park City mathematics series. Vol. 21. American Math. Soc. pp. 325–378.
• Breuillard, Emmanuel; Tao, Terence; Green, Ben (2012). "The structure of approximate groups". Publications Mathématiques de l'IHÉS . 116: 115–221. arXiv: 1110.5008 . doi: 10.1007/s10240-012-0043-9 . S2CID   119603959.
• Green, Ben (May 2012). "What is... an approximate group?" (PDF). Notices of the AMS. 59 (5).