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In the last few months, I have been working my way through the theory behind the solution to Hilbert’s fifth problem, as I (together with Emmanuel Breuillard, Ben Green, and Tom Sanders) have found this theory to be useful in obtaining noncommutative inverse sumset theorems in arbitrary groups; I hope to be able to report on this connection at some later point on this blog. Among other things, this theory achieves the remarkable feat of creating a smooth Lie group structure out of what is ostensibly a much weaker structure, namely the structure of a locally compact group. The ability of algebraic structure (in this case, group structure) to upgrade weak regularity (in this case, continuous structure) to strong regularity (in this case, smooth and even analytic structure) seems to be a recurring theme in mathematics, and an important part of what I like to call the “dichotomy between structure and randomness”.

The theory of Hilbert’s fifth problem sprawls across many subfields of mathematics: Lie theory, representation theory, group theory, nonabelian Fourier analysis, point-set topology, and even a little bit of group cohomology. The latter aspect of this theory is what I want to focus on today. The general question that comes into play here is the extension problem: given two (topological or Lie) groups ${H}$ and ${K}$, what is the structure of the possible groups ${G}$ that are formed by extending ${H}$ by ${K}$. In other words, given a short exact sequence

$\displaystyle 0 \rightarrow K \rightarrow G \rightarrow H \rightarrow 0,$

to what extent is the structure of ${G}$ determined by that of ${H}$ and ${K}$?

As an example of why understanding the extension problem would help in structural theory, let us consider the task of classifying the structure of a Lie group ${G}$. Firstly, we factor out the connected component ${G^\circ}$ of the identity as

$\displaystyle 0 \rightarrow G^\circ \rightarrow G \rightarrow G/G^\circ \rightarrow 0;$

as Lie groups are locally connected, ${G/G^\circ}$ is discrete. Thus, to understand general Lie groups, it suffices to understand the extensions of discrete groups by connected Lie groups.

Next, to study a connected Lie group ${G}$, we can consider the conjugation action ${g: X \mapsto gXg^{-1}}$ on the Lie algebra ${{\mathfrak g}}$, which gives the adjoint representation ${\hbox{Ad}: G \rightarrow GL({\mathfrak g})}$. The kernel of this representation consists of all the group elements ${g}$ that commute with all elements of the Lie algebra, and thus (by connectedness) is the center ${Z(G)}$ of ${G}$. The adjoint representation is then faithful on the quotient ${G/Z(G)}$. The short exact sequence

$\displaystyle 0 \rightarrow Z(G) \rightarrow G \rightarrow G/Z(G) \rightarrow 0$

then describes ${G}$ as a central extension (by the abelian Lie group ${Z(G)}$) of ${G/Z(G)}$, which is a connected Lie group with a faithful finite-dimensional linear representation.

This suggests a route to Hilbert’s fifth problem, at least in the case of connected groups ${G}$. Let ${G}$ be a connected locally compact group that we hope to demonstrate is isomorphic to a Lie group. As discussed in a previous post, we first form the space ${L(G)}$ of one-parameter subgroups of ${G}$ (which should, eventually, become the Lie algebra of ${G}$). Hopefully, ${L(G)}$ has the structure of a vector space. The group ${G}$ acts on ${L(G)}$ by conjugation; this action should be both continuous and linear, giving an “adjoint representation” ${\hbox{Ad}: G \rightarrow GL(L(G))}$. The kernel of this representation should then be the center ${Z(G)}$ of ${G}$. The quotient ${G/Z(G)}$ is locally compact and has a faithful linear representation, and is thus a Lie group by von Neumann’s version of Cartan’s theorem (discussed in this previous post). The group ${Z(G)}$ is locally compact abelian, and so it should be a relatively easy task to establish that it is also a Lie group. To finish the job, one needs the following result:

Theorem 1 (Central extensions of Lie are Lie) Let ${G}$ be a locally compact group which is a central extension of a Lie group ${H}$ by an abelian Lie group ${K}$. Then ${G}$ is also isomorphic to a Lie group.

This result can be obtained by combining a result of Kuranishi with a result of Gleason; I am recording this argument below the fold. The point here is that while ${G}$ is initially only a topological group, the smooth structures of ${H}$ and ${K}$ can be combined (after a little bit of cohomology) to create the smooth structure on ${G}$ required to upgrade ${G}$ from a topological group to a Lie group. One of the main ideas here is to improve the behaviour of a cocycle by averaging it; this basic trick is helpful elsewhere in the theory, resolving a number of cohomological issues in topological group theory. The result can be generalised to show in fact that arbitrary (topological) extensions of Lie groups by Lie groups remain Lie; this was shown by Gleason. However, the above special case of this result is already sufficient (in conjunction with the rest of the theory, of course) to resolve Hilbert’s fifth problem.

Remark 1 We have shown in the above discussion that every connected Lie group is a central extension (by an abelian Lie group) of a Lie group with a faithful continuous linear representation. It is natural to ask whether this central extension is necessary. Unfortunately, not every connected Lie group admits a faithful continuous linear representation. An example (due to Birkhoff) is the Heisenberg-Weyl group

$\displaystyle G := \begin{pmatrix} 1 & {\bf R} & {{\bf R}/{\bf Z}} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix} / \begin{pmatrix} 1 & 0 & {\bf Z} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}.$

Indeed, if we consider the group elements

$\displaystyle A := \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

and

$\displaystyle B := \begin{pmatrix} 1 & 0 & 1/p \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

for some prime ${p}$, then one easily verifies that ${B}$ has order ${p}$ and is central, and that ${AB}$ is conjugate to ${A}$. If we have a faithful linear representation ${\rho: G \rightarrow GL_n({\bf C})}$ of ${G}$, then ${\rho(B)}$ must have at least one eigenvalue ${\alpha}$ that is a primitive ${p^{th}}$ root of unity. If ${V}$ is the eigenspace associated to ${\alpha}$, then ${\rho(A)}$ must preserve ${V}$, and be conjugate to ${\alpha \rho(A)}$ on this space. This forces ${\rho(A)}$ to have at least ${p}$ distinct eigenvalues on ${V}$, and hence ${V}$ (and thus ${{\bf C}^n}$) must have dimension at least ${p}$. Letting ${p \rightarrow \infty}$ we obtain a contradiction. (On the other hand, ${G}$ is certainly isomorphic to the extension of the linear group ${{\bf R}^2}$ by the abelian group ${{\bf R}/{\bf Z}}$.)

Vitaly Bergelson, Tamar Ziegler, and I have just uploaded to the arXiv our paper “An inverse theorem for the uniformity seminorms associated with the action of $F^\infty_p$“. This paper establishes the ergodic inverse theorems that are needed in our other recent paper to establish the inverse conjecture for the Gowers norms over finite fields in high characteristic (and to establish a partial result in low characteristic), as follows:

Theorem. Let ${\Bbb F}$ be a finite field of characteristic p.  Suppose that $X = (X,{\mathcal B},\mu)$ is a probability space with an ergodic measure-preserving action $(T_g)_{g \in {\Bbb F}^\omega}$ of ${\Bbb F}^\omega$.  Let $f \in L^\infty(X)$ be such that the Gowers-Host-Kra seminorm $\|f\|_{U^k(X)}$ (defined in a previous post) is non-zero.

1. In the high-characteristic case $p \geq k$, there exists a phase polynomial g of degree <k (as defined in the previous post) such that $|\int_X f \overline{g}\ d\mu| > 0$.
2. In general characteristic, there exists a phase polynomial of degree <C(k) for some C(k) depending only on k such that $|\int_X f \overline{g}\ d\mu| > 0$.

This theorem is closely analogous to a similar theorem of Host and Kra on ergodic actions of ${\Bbb Z}$, in which the role of phase polynomials is played by functions that arise from nilsystem factors of X.  Indeed, our arguments rely heavily on the machinery of Host and Kra.

The paper is rather technical (60+ pages!) and difficult to describe in detail here, but I will try to sketch out (in very broad brush strokes) what the key steps in the proof of part 2 of the theorem are.  (Part 1 is similar but requires a more delicate analysis at various stages, keeping more careful track of the degrees of various polynomials.)

A dynamical system is a space X, together with an action $(g,x) \mapsto gx$ of some group $G = (G,\cdot)$.  [In practice, one often places topological or measure-theoretic structure on X or G, but this will not be relevant for the current discussion.  In most applications, G is an abelian (additive) group such as the integers ${\Bbb Z}$ or the reals ${\Bbb R}$, but I prefer to use multiplicative notation here.]  A useful notion in the subject is that of an (abelian) cocycle; this is a function $\rho: G \times X \to U$ taking values in an abelian group $U = (U,+)$ that obeys the cocycle equation

$\rho(gh, x) = \rho(h,x) + \rho(g,hx)$ (1)

for all $g,h \in G$ and $x \in X$.  [Again, if one is placing topological or measure-theoretic structure on the system, one would want $\rho$ to be continuous or measurable, but we will ignore these issues.] The significance of cocycles in the subject is that they allow one to construct (abelian) extensions or skew products $X \times_\rho U$ of the original dynamical system X, defined as the Cartesian product $\{ (x,u): x \in X, u \in U \}$ with the group action $g(x,u) := (gx,u + \rho(g,x))$.  (The cocycle equation (1) is needed to ensure that one indeed has a group action, and in particular that $(gh)(x,u) = g(h(x,u))$.)  This turns out to be a useful means to build complex dynamical systems out of simpler ones.  (For instance, one can build nilsystems by starting with a point and taking a finite number of abelian extensions of that point by a certain type of cocycle.)

A special type of cocycle is a coboundary; this is a cocycle $\rho: G \times X \to U$ that takes the form $\rho(g,x) := F(gx) - F(x)$ for some function $F: X \to U$.  (Note that the cocycle equation (1) is automaticaly satisfied if $\rho$ is of this form.)  An extension $X \times_\rho U$ of a dynamical system by a coboundary $\rho(g,x) := F(gx) - F(x)$ can be conjugated to the trivial extension $X \times_0 U$ by the change of variables $(x,u) \mapsto (x,u-F(x))$.

While every coboundary is a cocycle, the converse is not always true.  (For instance, if X is a point, the only coboundary is the zero function, whereas a cocycle is essentially the same thing as a homomorphism from G to U, so in many cases there will be more cocycles than coboundaries.  For a contrasting example, if X and G are finite (for simplicity) and G acts freely on X, it is not difficult to see that every cocycle is a coboundary.)  One can measure the extent to which this converse fails by introducing the first cohomology group $H^1(G,X,U) := Z^1(G,X,U) / B^1(G,X,U)$, where $Z^1(G,X,U)$ is the space of cocycles $\rho: G \times X \to U$ and $B^1(G,X,U)$ is the space of coboundaries (note that both spaces are abelian groups).  In my forthcoming paper with Vitaly Bergelson and Tamar Ziegler on the ergodic inverse Gowers conjecture (which should be available shortly), we make substantial use of some basic facts about this cohomology group (in the category of measure-preserving systems) that were established in a paper of Host and Kra.

The above terminology of cocycles, coboundaries, and cohomology groups of course comes from the theory of cohomology in algebraic topology.  Comparing the formal definitions of cohomology groups in that theory with the ones given above, there is certainly quite a bit of similarity, but in the dynamical systems literature the precise connection does not seem to be heavily emphasised.   The purpose of this post is to record the precise fashion in which dynamical systems cohomology is a special case of cochain complex cohomology from algebraic topology, and more specifically is analogous to singular cohomology (and can also be viewed as the group cohomology of the space of scalar-valued functions on X, when viewed as a G-module); this is not particularly difficult, but I found it an instructive exercise (especially given that my algebraic topology is extremely rusty), though perhaps this post is more for my own benefit that for anyone else.