This is an addendum to last quarter’s course notes on Hilbert’s fifth problem, which I am in the process of reviewing in order to transcribe them into a book (as was done similarly for several other sets of lecture notes on this blog). When reviewing the zeroth set of notes in particular, I found that I had made a claim (Proposition 11 from those notes) which asserted, roughly speaking, that any sufficiently large nilprogression was an approximate group, and promised to prove it later in the course when we had developed the ability to calculate efficiently in nilpotent groups. As it turned out, I managed finish the course without the need to develop these calculations, and so the proposition remained unproven. In order to rectify this, I will use this post to lay out some of the basic algebra of nilpotent groups, and use it to prove the above proposition, which turns out to be a bit tricky. (In my paper with Breuillard and Green, we avoid the need for this proposition by restricting attention to a special type of nilprogression, which we call a nilprogression in -normal form, for which the computations are simpler.)
There are several ways to think about nilpotent groups; for instance one can use the model example of the Heisenberg group
over an arbitrary ring (which need not be commutative), or more generally any matrix group consisting of unipotent upper triangular matrices, and view a general nilpotent group as being an abstract generalisation of such concrete groups. (In the case of nilpotent Lie groups, at least, this is quite an accurate intuition, thanks to Engel’s theorem.) Or, one can adopt a Lie-theoretic viewpoint and try to think of nilpotent groups as somehow arising from nilpotent Lie algebras; this intuition is rigorous when working with nilpotent Lie groups (at least when the characteristic is large, in order to avoid issues coming from the denominators in the Baker-Campbell-Hausdorff formula), but also retains some conceptual value in the non-Lie setting. In particular, nilpotent groups (particularly finitely generated ones) can be viewed in some sense as “nilpotent Lie groups over “, even though Lie theory does not quite work perfectly when the underlying scalars merely form an integral domain instead of a field.
Another point of view, which arises naturally both in analysis and in algebraic geometry, is to view nilpotent groups as modeling “infinitesimal” perturbations of the identity, where the infinitesimals have a certain finite order. For instance, given a (not necessarily commutative) ring without identity (representing all the “small” elements of some larger ring or algebra), we can form the powers for , defined as the ring generated by -fold products of elements in ; this is an ideal of which represents the elements which are “ order” in some sense. If one then formally adjoins an identity onto the ring , then for any , the multiplicative group is a nilpotent group of step at most . For instance, if is the ring of strictly upper matrices (over some base ring), then vanishes and becomes the group of unipotent upper triangular matrices over the same ring, thus recovering the previous matrix-based example. In analysis applications, might be a ring of operators which are somehow of “order” or for some small parameter or , and one wishes to perform Taylor expansions up to order or , thus discarding (i.e. quotienting out) all errors in .
From a dynamical or group-theoretic perspective, one can also view nilpotent groups as towers of central extensions of a trivial group. Finitely generated nilpotent groups can also be profitably viewed as a special type of polycylic group; this is the perspective taken in this previous blog post. Last, but not least, one can view nilpotent groups from a combinatorial group theory perspective, as being words from some set of generators of various “degrees” subject to some commutation relations, with commutators of two low-degree generators being expressed in terms of higher degree objects, and all commutators of a sufficiently high degree vanishing. In particular, generators of a given degree can be moved freely around a word, as long as one is willing to generate commutator errors of higher degree.
With this last perspective, in particular, one can start computing in nilpotent groups by adopting the philosophy that the lowest order terms should be attended to first, without much initial concern for the higher order errors generated in the process of organising the lower order terms. Only after the lower order terms are in place should attention then turn to higher order terms, working successively up the hierarchy of degrees until all terms are dealt with. This turns out to be a relatively straightforward philosophy to implement in many cases (particularly if one is not interested in explicit expressions and constants, being content instead with qualitative expansions of controlled complexity), but the arguments are necessarily recursive in nature and as such can become a bit messy, and require a fair amount of notation to express precisely. So, unfortunately, the arguments here will be somewhat cumbersome and notation-heavy, even if the underlying methods of proof are relatively simple.
— 1. Some elementary group theory —
(Note that this convention for is not universal; for instance, the alternate convention also appears in the literature. The distinctions between the two conventions however are quite minor; the conventions here are optimised for pulling group elements to the right of a word, whereas other conventions may be slightly better for pulling group elements to the left of a word.)
Conjugation by a fixed element is an automorphism of , thus
for all . Conjugation is also an action, thus
An automorphism of the form is called an inner automorphism.
Conjugation is related to multiplication by the identity
thus one can pull to the right of at the cost of twisting (i.e. conjugating) it by . Commutation is related to multiplication by the identity
thus one can pull to the right of at the cost of adding an additional commutator factor to the right. Finally, commutation is related to conjugation by the identity
The commutator can be viewed as a nonlinear group-theoretic analogue of the Lie bracket. For instance, in a matrix group , we observe that the commutator of two elements and close to the identity is of the form , thus linking the group-theoretic commutator to the Lie bracket .
Because of this link, we expect the group-theoretic commutator to obey some nonlinear analogues of the basic Liebracket identities, and this is indeed the case. For instance, one easily observes that the commutator is antisymmetric in the sense that
for any . We also have the easily verified approximate bilinearity identities
A subgroup of is said to be normal if it is preserved by all inner automorphisms, thus for all (writing , of course), and characteristic if it is preserved by all automorphisms (not necessarily inner). Thus, all characteristic subgroups are normal, but the converse is not necessarily true. We write or if is a subgroup of , and or if is a normal subgroup of .
If is a normal subgroup of , we write for the quotient map from to , thus if . Given any other subgroup of , we write for the image of under the map, thus
Given two subgroups of a group , we define the commutator group to be the group generated by the commutators with . We say that and commute if is trivial, or equivalently if every element of commutes with every element of . Note that if are normal, then is also normal. In this case, one can view as the smallest subgroup of such that commute modulo (or equivalently, that and commute). Similarly, if are characteristic, then is also characteristic.
Observe from (3) that
for any subgroups . Also that if is a normal subgroup of , then (as is a homomorphism)
- (i) If are normal subgroups of generated by subsets , respectively, show that is the normal subgroup of generated (as a normal subgroup) by the commutators with , .
- (ii) If are normal subgroups of , show that lies in the normal subgroup generated by and . (Hint: use the Hall-Witt identity (6).)
Given an arbitrary group , we define the lower central series of by setting and for all . Observe that all of the groups in this series are characteristic and thus normal. A group is said to be nilpotent of step at most if is trivial; in particular, by (7), we see that is nilpotent of step at most for any group . We have a basic inclusion:
Exercise 2 We have for all . (Hint: use Exercise 1.)
The commutator structure can be clarified by passing to the “top order” components of this commutator, as follows. (Strictly speaking, this analysis is not needed to study nilprogressions, but it is still conceptually useful nonetheless.) Consider the subquotients of the group for . As contains , we see that each group is abelian. To emphasise this, we will write additively instead of multiplicatively, thus we shall denote the group operation on as .
Proof: We can quotient out by , and assume that is trivial. In particular, by Lemma 2, commutes with , and commutes with . As such, it is easy to see that if , then is unchanged if one multiplies (on the left or right) by an element of , and similarly for and . This defines the map as required.
We refer to the maps as quotiented commutator maps. The identity (8) asserts that these maps capture the “top order” nonlinear behaviour of the group . As the following exercise shows, these maps behave like (graded components of) a Lie bracket.
Exercise 3 Let be a group with lower central series , subquotients , and quotiented commutator maps . Let , let , , and . Establish the antisymmetry
the bihomomorphism properties
and the Jacobi identity
Remark 1 If one wished, one could combine all of these subquotients and quotiented commutator maps into a single graded object, namely the additive group consisting of tuples with (and all but finitely many of the trivial), with a bracket defined by
Then this bracket is an antisymmetric bihomomorphism obeying the Jacobi identity, thus is a Lie algebra over the integers . One could view this object as the “top order” or “Carnot” component of the “Lie algebra” of . We will however not need this object here.
We can iterate the “bilinear” commutator operations to create “multilinear” operations. Given a non-empty finite set of formal group elements , we can define a formal commutator word on to be a word that can be generated by the following rules:
- If is a formal group element, then is a formal commutator word on .
- If are disjoint finite non-empty sets of formal group elements, and are formal commutator words on respectively, then is a formal commutator word on .
We refer to as the length of the formal commutator word. Thus, for instance, is a formal commutator word on of length . Given a formal commutator word on and an assignment of group elements to each formal group element , we can define the group element by substituting for in for each . This gives a commutator word map . Given an assignment of a positive integer (or “degree”) to each formal group element , Lemma 1 and induction then gives a map which is a multihomomorphism (i.e. a homomorphism in each variable). From (8) one has that
where is shorthand for the assignment .
In particular, using the degree map , we obtain a multihomomorphism
for any commutator word of length , such that
for any and integers . A variant of this approximate identity will be key in understanding nilprogressions.
One can use commutator words to generate a nilpotent group and its lower central series:
Exercise 4 Let be a nilpotent group that is generated by a set of generators.
- (i) Show that for every positive integer , is generated by the words , where ranges over formal commutator words of length at least , and is a collection of generators from (possibly with repetition).
- (ii) Suppose further that is finite (so that there are only finitely many possible choices for for any given length , and let be all the values of as varies over formal commutator words of length at most , and ranges over collections of generators from , arranged in non-decreasing length of (this arrangement is not unique, and may contain repeated values). Show that for every positive integer , any element in can be expressed in the form , where are the elements of arising from words of length at least , and are integers.
— 2. Nilprogressions —
Recall from Notes 0 that a noncommutative progression in a group with generators and dimensions is the set of all words with alphabet , such that for each , the symbols are used a total of at most times. A nilprogression is a noncommutative progression in a nilpotent group. The objective of this section is to prove Proposition 11 from Notes 0, restated here:
Recall that a subset of a group is a -approximate group if it is symmetric, contains the origin, and if can be covered by left-translates (or equivalently, right-translates) of . The first two properties are clear, so it suffices to show that can be covered by right-translates of .
We allow all implied constants to depend on . We pick a small constant (depending on ). It will suffice to show the inclusion
for some set of cardinality .
where consists of all group elements of the form where is a formal commutator word of length , , and each is associated to the dimension (note that we allow some of the to be equal, or to degenerate to the identity). Thus the are non-increasing in and become trivial for . We will usually abbreviate as . Trivially we also have
Observe also that the are symmetric, contain the identity, and .
Next, we need the following variant of (9).
Lemma 3 Let be a formal commutator word of length , and let . For each , let be an integer with . Then
Proof: When , the expressions involving collapse to the identity, and the claim follows, so we may assume that . We induct on . The claim is trivial, so suppose that and that the claim has been proven for smaller values of .
We can write . It is then not difficult to see (and we leave as an exercise to the reader) that the claim will follow if we can show
Thus we have effectively reduced the problem to the case . A similar argument allows us to also obtain the additional reduction , at which point the claim is trivial. This completes the treatment of the case.
Now we handle the general case. After some relabeling, we may write , where are words of length respectively for some adding up to , with
By induction hypothesis, one has
In particular we have
Similarly we have
Using (5) and the approximate homomorphism properties of commutators, we conclude that
but from (11) we have
The claim follows.
Exercise 6 For positive integers , show that
Next, we need the following elementary number-theoretic lemma:
Exercise 7 Let , and let be an integer with . Show that can be expressed as the sum of terms of the form , where the are integers with for each . (Hint: Despite the superficial similarity here with non-trivial number-theoretic questions such as the Waring problem, this is actually a very elementary fact which can be proven by induction on .)
Exercise 8 Let be a formal commutator word of length , and let . Let . Show that
whenever , and on iteration conclude that
A similar argument shows that for a sufficiently small depending only on , assuming that are sufficiently large depending on . Since , it thus suffices to show that for any , that can be covered by right-translates of . But this then follows by iterating the following exercise:
Exercise 9 Let , , and . Show that can be covered by right-translates of . (Hint: this is similar to Exercise 5. Factor an element of into words of length , together with words of higher length. Gather all the words of length into monomials with , times a factor in . Split these monomials into a monomial with exponent , times a monomial which can take at most possible values. Then push the latter monomials (and the factor) to the right.)
Exercise 10 (Polynomial growth) Suppose that generate a nilpotent group of step , and suppose that . Show that
Exercise 11 Suppose that generate a nilpotent group of step , and suppose that are sufficiently large depending on . Write . Let be as in (10), arranged in non-decreasing order of the length of the associated formal commutator words.
- (i) Show that every element of can be represented in the form
for some integers . (We do not claim however that this representation is unique, and indeed the generators are likely to contain quite a lot of redundancy.)
- (ii) Conversely, show that there exists an depending only on such that any expression of the form
with integers with , lies in .