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Let ,
be additive groups (i.e., groups with an abelian addition group law). A map
is a homomorphism if one has
for all . A map
is an affine homomorphism if one has
for all additive quadruples in
, by which we mean that
and
. The two notions are closely related; it is easy to verify that
is an affine homomorphism if and only if
is the sum of a homomorphism and a constant.
Now suppose that also has a translation-invariant metric
. A map
is said to be a quasimorphism if one has
for all , where
denotes a quantity at a bounded distance from the origin. Similarly,
is an affine quasimorphism if
for all additive quadruples in
. Again, one can check that
is an affine quasimorphism if and only if it is the sum of a quasimorphism and a constant (with the implied constant of the quasimorphism controlled by the implied constant of the affine quasimorphism). (Since every constant is itself a quasimorphism, it is in fact the case that affine quasimorphisms are quasimorphisms, but now the implied constant in the latter is not controlled by the implied constant of the former.)
“Trivial” examples of quasimorphisms include the sum of a homomorphism and a bounded function. Are there others? In some cases, the answer is no. For instance, suppose we have a quasimorphism . Iterating (2), we see that
for any integer
and natural number
, which we can rewrite as
for non-zero
. Also,
is Lipschitz. Sending
, we can verify that
is a Cauchy sequence as
and thus tends to some limit
; we have
for
, hence
for positive
, and then one can use (2) one last time to obtain
for all
. Thus
is the sum of the homomorphism
and a bounded sequence.
In general, one can phrase this problem in the language of group cohomology (discussed in this previous post). Call a map a
-cocycle. A
-cocycle is a map
obeying the identity
for all . Given a
-cocycle
, one can form its derivative
by the formula
Such functions are called -coboundaries. It is easy to see that the abelian group of
-coboundaries is a subgroup of the abelian group of
-cocycles. The quotient of these two groups is the first group cohomology of
with coefficients in
, and is denoted
.
If a -cocycle is bounded then its derivative is a bounded
-coboundary. The quotient of the group of bounded
-cocycles by the derivatives of bounded
-cocycles is called the bounded first group cohomology of
with coefficients in
, and is denoted
. There is an obvious homomorphism
from
to
, formed by taking a coset of the space of derivatives of bounded
-cocycles, and enlarging it to a coset of the space of
-coboundaries. By chasing all the definitions, we see that all quasimorphism from
to
are the sum of a homomorphism and a bounded function if and only if this homomorphism
is injective; in fact the quotient of the space of quasimorphisms by the sum of homomorphisms and bounded functions is isomorphic to the kernel of
.
In additive combinatorics, one is often working with functions which only have additive structure a fraction of the time, thus for instance (1) or (3) might only hold “ of the time”. This makes it somewhat difficult to directly interpret the situation in terms of group cohomology. However, thanks to tools such as the Balog-Szemerédi-Gowers lemma, one can upgrade this sort of
-structure to
-structure – at the cost of restricting the domain to a smaller set. Here I record one such instance of this phenomenon, thus giving a tentative link between additive combinatorics and group cohomology. (I thank Yuval Wigderson for suggesting the problem of locating such a link.)
Theorem 1 Let
,
be additive groups with
, let
be a subset of
, let
, and let
be a function such that
for
additive quadruples
in
. Then there exists a subset
of
containing
with
, a subset
of
with
, and a function
such that
for all
(thus, the derivative
takes values in
on
), and such that for each
, one has
for
values of
.
Presumably the constants and
can be improved further, but we have not attempted to optimise these constants. We chose
as the domain on which one has a bounded derivative, as one can use the Bogulybov lemma (see e.g, Proposition 4.39 of my book with Van Vu) to find a large Bohr set inside
. In applications, the set
need not have bounded size, or even bounded doubling; for instance, in the inverse
theory over a small finite fields
, one would be interested in the situation where
is the group of
matrices with coefficients in
(for some large
, and
being the subset consisting of those matrices of rank bounded by some bound
.
Proof: By hypothesis, there are triples
such that
and
Thus, there is a set with
such that for all
, one has (6) for
pairs
with
; in particular, there exists
such that (6) holds for
values of
. Setting
, we conclude that for each
, one has
for values of
.
Consider the bipartite graph whose vertex sets are two copies of , and
and
connected by a (directed) edge if
and (7) holds. Then this graph has
edges. Applying (a slight modification of) the Balog-Szemerédi-Gowers theorem (for instance by modifying the proof of Corollary 5.19 of my book with Van Vu), we can then find a subset
of
with
with the property that for any
, there exist
triples
such that the edges
all lie in this bipartite graph. This implies that, for all
, there exist
septuples
obeying the constraints
and for
. These constraints imply in particular that
Also observe that
Thus, if and
are such that
, we see that
for octuples
in the hyperplane
By the pigeonhole principle, this implies that for any fixed , there can be at most
sets of the form
with
,
that are pairwise disjoint. Using a greedy algorithm, we conclude that there is a set
of cardinality
, such that each set
with
,
intersects
for some
, or in other words that
whenever . In particular,
This implies that there exists a subset of
with
, and an element
for each
, such that
for all . Note we may assume without loss of generality that
and
.
By construction of , and permuting labels, we can find
16-tuples
such that
and
for . We sum this to obtain
and hence by (8)
where . Since
we see that there are only possible values of
. By the pigeonhole principle, we conclude that at most
of the sets
can be disjoint. Arguing as before, we conclude that there exists a set
of cardinality
such that
whenever (10) holds.
For any , write
arbitrarily as
for some
(with
if
, and
if
) and then set
Then from (11) we have (4). For we have
, and (5) then follows from (9).
The von Neumann ergodic theorem (the Hilbert space version of the mean ergodic theorem) asserts that if is a unitary operator on a Hilbert space
, and
is a vector in that Hilbert space, then one has
in the strong topology, where is the
-invariant subspace of
, and
is the orthogonal projection to
. (See e.g. these previous lecture notes for a proof.) The same proof extends to more general amenable groups: if
is a countable amenable group acting on a Hilbert space
by unitary transformations
, and
is a vector in that Hilbert space, then one has
for any Følner sequence of
, where
is the
-invariant subspace. Thus one can interpret
as a certain average of elements of the orbit
of
.
I recently discovered that there is a simple variant of this ergodic theorem that holds even when the group is not amenable (or not discrete), using a more abstract notion of averaging:
Theorem 1 (Abstract ergodic theorem) Let
be an arbitrary group acting unitarily on a Hilbert space
, and let
be a vector in
. Then
is the element in the closed convex hull of
of minimal norm, and is also the unique element of
in this closed convex hull.
Proof: As the closed convex hull of is closed, convex, and non-empty in a Hilbert space, it is a classical fact (see e.g. Proposition 1 of this previous post) that it has a unique element
of minimal norm. If
for some
, then the midpoint of
and
would be in the closed convex hull and be of smaller norm, a contradiction; thus
is
-invariant. To finish the first claim, it suffices to show that
is orthogonal to every element
of
. But if this were not the case for some such
, we would have
for all
, and thus on taking convex hulls
, a contradiction.
Finally, since is orthogonal to
, the same is true for
for any
in the closed convex hull of
, and this gives the second claim.
This result is due to Alaoglu and Birkhoff. It implies the amenable ergodic theorem (1); indeed, given any , Theorem 1 implies that there is a finite convex combination
of shifts
of
which lies within
(in the
norm) to
. By the triangle inequality, all the averages
also lie within
of
, but by the Følner property this implies that the averages
are eventually within
(say) of
, giving the claim.
It turns out to be possible to use Theorem 1 as a substitute for the mean ergodic theorem in a number of contexts, thus removing the need for an amenability hypothesis. Here is a basic application:
Corollary 2 (Relative orthogonality) Let
be a group acting unitarily on a Hilbert space
, and let
be a
-invariant closed subspace of
. Then
and
are relatively orthogonal over their common subspace
, that is to say the restrictions of
and
to the orthogonal complement of
are orthogonal to each other.
Proof: By Theorem 1, we have for all
, and the claim follows. (Thanks to Gergely Harcos for this short argument.)
Now we give a more advanced application of Theorem 1, to establish some “Mackey theory” over arbitrary groups . Define a
-system
to be a probability space
together with a measure-preserving action
of
on
; this gives an action of
on
, which by abuse of notation we also call
:
(In this post we follow the usual convention of defining the spaces by quotienting out by almost everywhere equivalence.) We say that a
-system is ergodic if
consists only of the constants.
(A technical point: the theory becomes slightly cleaner if we interpret our measure spaces abstractly (or “pointlessly“), removing the underlying space and quotienting
by the
-ideal of null sets, and considering maps such as
only on this quotient
-algebra (or on the associated von Neumann algebra
or Hilbert space
). However, we will stick with the more traditional setting of classical probability spaces here to keep the notation familiar, but with the understanding that many of the statements below should be understood modulo null sets.)
A factor of a
-system
is another
-system together with a factor map
which commutes with the
-action (thus
for all
) and respects the measure in the sense that
for all
. For instance, the
-invariant factor
, formed by restricting
to the invariant algebra
, is a factor of
. (This factor is the first factor in an important hierachy, the next element of which is the Kronecker factor
, but we will not discuss higher elements of this hierarchy further here.) If
is a factor of
, we refer to
as an extension of
.
From Corollary 2 we have
Corollary 3 (Relative independence) Let
be a
-system for a group
, and let
be a factor of
. Then
and
are relatively independent over their common factor
, in the sense that the spaces
and
are relatively orthogonal over
when all these spaces are embedded into
.
This has a simple consequence regarding the product of two
-systems
and
, in the case when the
action is trivial:
Lemma 4 If
are two
-systems, with the action of
on
trivial, then
is isomorphic to
in the obvious fashion.
This lemma is immediate for countable , since for a
-invariant function
, one can ensure that
holds simultaneously for all
outside of a null set, but is a little trickier for uncountable
.
Proof: It is clear that is a factor of
. To obtain the reverse inclusion, suppose that it fails, thus there is a non-zero
which is orthogonal to
. In particular, we have
orthogonal to
for any
. Since
lies in
, we conclude from Corollary 3 (viewing
as a factor of
) that
is also orthogonal to
. Since
is an arbitrary element of
, we conclude that
is orthogonal to
and in particular is orthogonal to itself, a contradiction. (Thanks to Gergely Harcos for this argument.)
Now we discuss the notion of a group extension.
Definition 5 (Group extension) Let
be an arbitrary group, let
be a
-system, and let
be a compact metrisable group. A
-extension of
is an extension
whose underlying space is
(with
the product of
and the Borel
-algebra on
), the factor map is
, and the shift maps
are given by
where for each
,
is a measurable map (known as the cocycle associated to the
-extension
).
An important special case of a -extension arises when the measure
is the product of
with the Haar measure
on
. In this case,
also has a
-action
that commutes with the
-action, making
a
-system. More generally,
could be the product of
with the Haar measure
of some closed subgroup
of
, with
taking values in
; then
is now a
system. In this latter case we will call
-uniform.
If is a
-extension of
and
is a measurable map, we can define the gauge transform
of
to be the
-extension of
whose measure
is the pushforward of
under the map
, and whose cocycles
for
are given by the formula
It is easy to see that is a
-extension that is isomorphic to
as a
-extension of
; we will refer to
and
as equivalent systems, and
as cohomologous to
. We then have the following fundamental result of Mackey and of Zimmer:
Theorem 6 (Mackey-Zimmer theorem) Let
be an arbitrary group, let
be an ergodic
-system, and let
be a compact metrisable group. Then every ergodic
-extension
of
is equivalent to an
-uniform extension of
for some closed subgroup
of
.
This theorem is usually stated for amenable groups , but by using Theorem 1 (or more precisely, Corollary 3) the result is in fact also valid for arbitrary groups; we give the proof below the fold. (In the usual formulations of the theorem,
and
are also required to be Lebesgue spaces, or at least standard Borel, but again with our abstract approach here, such hypotheses will be unnecessary.) Among other things, this theorem plays an important role in the Furstenberg-Zimmer structural theory of measure-preserving systems (as well as subsequent refinements of this theory by Host and Kra); see this previous blog post for some relevant discussion. One can obtain similar descriptions of non-ergodic extensions by working relative to the invariant factor (or via the ergodic decomposition, if one has enough separability hypotheses on the system), but the result becomes more complicated to state, and we will not do so here; see this paper of Austin for details.
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