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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 ${C}$-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

$\displaystyle H(R) :=\begin{pmatrix} 1 & R & R \\ 0 & 1 & R\\ 0 & 0 & 1 \end{pmatrix}$

over an arbitrary ring ${R}$ (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 ${{\bf Z}}$“, 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 ${R}$ without identity (representing all the “small” elements of some larger ring or algebra), we can form the powers ${R^j}$ for ${j=1,2,\ldots}$, defined as the ring generated by ${j}$-fold products ${r_1 \ldots r_j}$ of elements ${r_1,\ldots,r_j}$ in ${R}$; this is an ideal of ${R}$ which represents the elements which are “${j^{th}}$ order” in some sense. If one then formally adjoins an identity ${1}$ onto the ring ${R}$, then for any ${s \geq 1}$, the multiplicative group ${G := 1+R \hbox{ mod } R^{s+1}}$ is a nilpotent group of step at most ${s}$. For instance, if ${R}$ is the ring of strictly upper ${s \times s}$ matrices (over some base ring), then ${R^{s+1}}$ vanishes and ${G}$ becomes the group of unipotent upper triangular matrices over the same ring, thus recovering the previous matrix-based example. In analysis applications, ${R}$ might be a ring of operators which are somehow of “order” ${O(\epsilon)}$ or ${O(\hbar)}$ for some small parameter ${\epsilon}$ or ${\hbar}$, and one wishes to perform Taylor expansions up to order ${O(\epsilon^s)}$ or ${O(\hbar^s)}$, thus discarding (i.e. quotienting out) all errors in ${R^{s+1}}$.

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.