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
Contents
Formal definition
Let
- It is symmetric, that is if
g ∈ A theng − 1 ∈ A ; - There exists a subset
X ⊂ G of cardinality| X | ≤ K such thatA ⋅ A ⊂ X ⋅ 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
Examples of approximate subgroups which are not groups are given by symmetric intervals and more generally arithmetic progressions in the integers. Indeed, for all
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
The constants
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:
LetThe statement also gives some information on the characteristics (rank and step) of the nilpotent group
In the case where
The theorem applies for example to
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
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.