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The classical foundations of probability theory (discussed for instance in this previous blog post) is founded on the notion of a probability space – a space (the sample space) equipped with a -algebra (the event space), together with a countably additive probability measure that assigns a real number in the interval to each event.

One can generalise the concept of a probability space to a *finitely additive* probability space, in which the event space is now only a Boolean algebra rather than a -algebra, and the measure is now only finitely additive instead of countably additive, thus when are disjoint events. By giving up countable additivity, one loses a fair amount of measure and integration theory, and in particular the notion of the expectation of a random variable becomes problematic (unless the random variable takes only finitely many values). Nevertheless, one can still perform a fair amount of probability theory in this weaker setting.

In this post I would like to describe a further weakening of probability theory, which I will call *qualitative probability theory*, in which one does not assign a precise numerical probability value to each event, but instead merely records whether this probability is zero, one, or something in between. Thus is now a function from to the set , where is a new symbol that replaces all the elements of the open interval . In this setting, one can no longer compute quantitative expressions, such as the mean or variance of a random variable; but one can still talk about whether an event holds almost surely, with positive probability, or with zero probability, and there are still usable notions of independence. (I will refer to classical probability theory as *quantitative* probability theory, to distinguish it from its qualitative counterpart.)

The main reason I want to introduce this weak notion of probability theory is that it becomes suited to talk about random variables living inside algebraic varieties, even if these varieties are defined over fields other than or . In algebraic geometry one often talks about a “generic” element of a variety defined over a field , which does not lie in any specified variety of lower dimension defined over . Once has positive dimension, such generic elements do not exist as classical, deterministic -points in , since of course any such point lies in the -dimensional subvariety of . There are of course several established ways to deal with this problem. One way (which one might call the “Weil” approach to generic points) is to extend the field to a sufficiently transcendental extension , in order to locate a sufficient number of generic points in . Another approach (which one might dub the “Zariski” approach to generic points) is to work scheme-theoretically, and interpret a generic point in as being associated to the zero ideal in the function ring of . However I want to discuss a third perspective, in which one interprets a generic point not as a deterministic object, but rather as a *random variable* taking values in , but which lies in any given lower-dimensional subvariety of with probability zero. This interpretation is intuitive, but difficult to implement in classical probability theory (except perhaps when considering varieties over or ) due to the lack of a natural probability measure to place on algebraic varieties; however it works just fine in qualitative probability theory. In particular, the algebraic geometry notion of being “generically true” can now be interpreted probabilistically as an assertion that something is “almost surely true”.

It turns out that just as qualitative random variables may be used to interpret the concept of a generic point, they can also be used to interpret the concept of a type in model theory; the type of a random variable is the set of all predicates that are almost surely obeyed by . In contrast, model theorists often adopt a Weil-type approach to types, in which one works with deterministic representatives of a type, which often do not occur in the original structure of interest, but only in a sufficiently saturated extension of that structure (this is the analogue of working in a sufficiently transcendental extension of the base field). However, it seems that (in some cases at least) one can equivalently view types in terms of (qualitative) random variables on the original structure, avoiding the need to extend that structure. (Instead, one reserves the right to extend the *sample space* of one’s probability theory whenever necessary, as part of the “probabilistic way of thinking” discussed in this previous blog post.) We illustrate this below the fold with two related theorems that I will interpret through the probabilistic lens: the “group chunk theorem” of Weil (and later developed by Hrushovski), and the “group configuration theorem” of Zilber (and again later developed by Hrushovski). For sake of concreteness we will only consider these theorems in the theory of algebraically closed fields, although the results are quite general and can be applied to many other theories studied in model theory.

In this previous post I recorded some (very standard) material on the structural theory of finite-dimensional complex Lie algebras (or *Lie algebras* for short), with a particular focus on those Lie algebras which were semisimple or simple. Among other things, these notes discussed the Weyl complete reducibility theorem (asserting that semisimple Lie algebras are the direct sum of simple Lie algebras) and the classification of simple Lie algebras (with all such Lie algebras being (up to isomorphism) of the form , , , , , , , , or ).

Among other things, the structural theory of Lie algebras can then be used to build analogous structures in nearby areas of mathematics, such as Lie groups and Lie algebras over more general fields than the complex field (leading in particular to the notion of a Chevalley group), as well as finite simple groups of Lie type, which form the bulk of the classification of finite simple groups (with the exception of the alternating groups and a finite number of sporadic groups).

In the case of complex Lie groups, it turns out that every simple Lie algebra is associated with a finite number of connected complex Lie groups, ranging from a “minimal” Lie group (the *adjoint form* of the Lie group) to a “maximal” Lie group (the *simply connected form* of the Lie group) that finitely covers , and occasionally also a number of intermediate forms which finitely cover , but are in turn finitely covered by . For instance, is associated with the projective special linear group as its adjoint form and the special linear group as its simply connected form, and intermediate groups can be created by quotienting out by some subgroup of its centre (which is isomorphic to the roots of unity). The minimal form is simple in the group-theoretic sense of having no normal subgroups, but the other forms of the Lie group are merely quasisimple, although traditionally all of the forms of a Lie group associated to a simple Lie algebra are known as *simple Lie groups*.

Thanks to the work of Chevalley, a very similar story holds for algebraic groups over arbitrary fields ; given any Dynkin diagram, one can define a simple Lie algebra with that diagram over that field, and also one can find a finite number of connected algebraic groups over (known as *Chevalley groups*) with that Lie algebra, ranging from an adjoint form to a universal form , with every form having an isogeny (the analogue of a finite cover for algebraic groups) to the adjoint form, and in turn receiving an isogeny from the universal form. Thus, for instance, one could construct the universal form of the algebraic group over a finite field of finite order.

When one restricts the Chevalley group construction to adjoint forms over a finite field (e.g. ), one usually obtains a finite simple group (with a finite number of exceptions when the rank and the field are very small, and in some cases one also has to pass to a bounded index subgroup, such as the derived group, first). One could also use other forms than the adjoint form, but one then recovers the same finite simple group as before if one quotients out by the centre. This construction was then extended by Steinberg, Suzuki, and Ree by taking a Chevalley group over a finite field and then restricting to the fixed points of a certain automorphism of that group; after some additional minor modifications such as passing to a bounded index subgroup or quotienting out a bounded centre, this gives some additional finite simple groups of Lie type, including classical examples such as the projective special unitary groups , as well as some more exotic examples such as the Suzuki groups or the Ree groups.

While I learned most of the classical structural theory of Lie algebras back when I was an undergraduate, and have interacted with Lie groups in many ways in the past (most recently in connection with Hilbert’s fifth problem, as discussed in this previous series of lectures), I have only recently had the need to understand more precisely the concepts of a Chevalley group and of a finite simple group of Lie type, as well as better understand the structural theory of simple complex Lie groups. As such, I am recording some notes here regarding these concepts, mainly for my own benefit, but perhaps they will also be of use to some other readers. The material here is standard, and was drawn from a number of sources, but primarily from Carter, Gorenstein-Lyons-Solomon, and Fulton-Harris, as well as the lecture notes on Chevalley groups by my colleague Robert Steinberg. The arrangement of material also reflects my own personal preferences; in particular, I tend to favour complex-variable or Riemannian geometry methods over algebraic ones, and this influenced a number of choices I had to make regarding how to prove certain key facts. The notes below are far from a comprehensive or fully detailed discussion of these topics, and I would refer interested readers to the references above for a properly thorough treatment.

A finite group is said to be a Frobenius group if there is a non-trivial subgroup of (known as the *Frobenius complement* of ) such that the conjugates of are “disjoint as possible” in the sense that whenever . This gives a decomposition

where the *Frobenius kernel* of is defined as the identity element together with all the non-identity elements that are not conjugate to any element of . Taking cardinalities, we conclude that

A remarkable theorem of Frobenius gives an unexpected amount of structure on and hence on :

Theorem 1 (Frobenius’ theorem)Let be a Frobenius group with Frobenius complement and Frobenius kernel . Then is a normal subgroup of , and hence (by (2) and the disjointness of and outside the identity) is the semidirect product of and .

I discussed Frobenius’ theorem and its proof in this recent blog post. This proof uses the theory of characters on a finite group , in particular relying on the fact that a character on a subgroup can induce a character on , which can then be decomposed into irreducible characters with *natural number* coefficients. Remarkably, even though a century has passed since Frobenius’ original argument, there is no proof known of this theorem which avoids character theory entirely; there are elementary proofs known when the complement has even order or when is solvable (we review both of these cases below the fold), which by the Feit-Thompson theorem does cover all the cases, but the proof of the Feit-Thompson theorem involves plenty of character theory (and also relies on Theorem 1). (The answers to this MathOverflow question give a good overview of the current state of affairs.)

I have been playing around recently with the problem of finding a character-free proof of Frobenius’ theorem. I didn’t succeed in obtaining a completely elementary proof, but I did find an argument which replaces character theory (which can be viewed as coming from the representation theory of the non-commutative group algebra ) with the Fourier analysis of class functions (i.e. the representation theory of the centre of the group algebra), thus replacing non-commutative representation theory by commutative representation theory. This is not a particularly radical depature from the existing proofs of Frobenius’ theorem, but it did seem to be a new proof which was technically “character-free” (even if it was not all that far from character-based in spirit), so I thought I would record it here.

The main ideas are as follows. The space of class functions can be viewed as a commutative algebra with respect to the convolution operation ; as the regular representation is unitary and faithful, this algebra contains no nilpotent elements. As such, (Gelfand-style) Fourier analysis suggests that one can analyse this algebra through the idempotents: class functions such that . In terms of characters, idempotents are nothing more than sums of the form for various collections of characters, but we can perform a fair amount of analysis on idempotents directly without recourse to characters. In particular, it turns out that idempotents enjoy some important integrality properties that can be established without invoking characters: for instance, by taking traces one can check that is a natural number, and more generally we will show that is a natural number whenever is a subgroup of (see Corollary 4 below). For instance, the quantity

is a natural number which we will call the *rank* of (as it is also the linear rank of the transformation on ).

In the case that is a Frobenius group with kernel , the above integrality properties can be used after some elementary manipulations to establish that for any idempotent , the quantity

is an integer. On the other hand, one can also show by elementary means that this quantity lies between and . These two facts are not strong enough on their own to impose much further structure on , unless one restricts attention to *minimal* idempotents . In this case spectral theory (or Gelfand theory, or the fundamental theorem of algebra) tells us that has rank one, and then the *integrality gap* comes into play and forces the quantity (3) to always be either zero or one. This can be used to imply that the convolution action of every minimal idempotent either preserves or annihilates it, which makes itself an idempotent, which makes normal.

Suppose that is a finite group of even order, thus is a multiple of two. By Cauchy’s theorem, this implies that contains an involution: an element in of order two. (Indeed, if no such involution existed, then would be partitioned into doubletons together with the identity, so that would be odd, a contradiction.) Of course, groups of odd order have no involutions , thanks to Lagrange’s theorem (since cannot split into doubletons ).

The classical Brauer-Fowler theorem asserts that if a group has many involutions, then it must have a large non-trivial subgroup:

Theorem 1 (Brauer-Fowler theorem)Let be a finite group with at least involutions for some . Then contains a proper subgroup of index at most .

This theorem (which is Theorem 2F in the original paper of Brauer and Fowler, who in fact manage to sharpen slightly to ) has a number of quick corollaries which are also referred to as “the” Brauer-Fowler theorem. For instance, if is a an involution of a group , and the centraliser has order , then clearly (as contains and ) and the conjugacy class has order (since the map has preimages that are cosets of ). Every conjugate of an involution is again an involution, so by the Brauer-Fowler theorem contains a subgroup of order at least . In particular, we can conclude that every group of even order contains a proper subgroup of order at least .

Another corollary is that the size of a simple group of even order can be controlled by the size of a centraliser of one of its involutions:

Corollary 2 (Brauer-Fowler theorem)Let be a finite simple group with an involution , and suppose that has order . Then has order at most .

Indeed, by the previous discussion has a proper subgroup of index less than , which then gives a non-trivial permutation action of on the coset space . The kernel of this action is a proper normal subgroup of and is thus trivial, so the action is faithful, and the claim follows.

If one assumes the Feit-Thompson theorem that all groups of odd order are solvable, then Corollary 2 suggests a strategy (first proposed by Brauer himself in 1954) to prove the classification of finite simple groups (CFSG) by induction on the order of the group. Namely, assume for contradiction that the CFSG failed, so that there is a counterexample of minimal order to the classification. This is a non-abelian finite simple group; by the Feit-Thompson theorem, it has even order and thus has at least one involution . Take such an involution and consider its centraliser ; this is a proper subgroup of of some order . As is a minimal counterexample to the classification, one can in principle describe in terms of the CFSG by factoring the group into simple components (via a composition series) and applying the CFSG to each such component. Now, the “only” thing left to do is to verify, for each isomorphism class of , that all the possible simple groups that could have this type of group as a centraliser of an involution obey the CFSG; Corollary 2 tells us that for each such isomorphism class for , there are only finitely many that could generate this class for one of its centralisers, so this task should be doable *in principle* for any given isomorphism class for . That’s all one needs to do to prove the classification of finite simple groups!

Needless to say, this program turns out to be far more difficult than the above summary suggests, and the actual proof of the CFSG does not quite proceed along these lines. However, a significant portion of the argument *is* based on a generalisation of this strategy, in which the concept of a centraliser of an involution is replaced by the more general notion of a normaliser of a -group, and one studies not just a single normaliser but rather the entire family of such normalisers and how they interact with each other (and in particular, which normalisers of -groups commute with each other), motivated in part by the theory of Tits buildings for Lie groups which dictates a very specific type of interaction structure between these -groups in the key case when is a (sufficiently high rank) finite simple group of Lie type over a field of characteristic . See the text of Aschbacher, Lyons, Smith, and Solomon for a more detailed description of this strategy.

The Brauer-Fowler theorem can be proven by a nice application of character theory, of the type discussed in this recent blog post, ultimately based on analysing the alternating tensor power of representations; I reproduce a version of this argument (taken from this text of Isaacs) below the fold. (The original argument of Brauer and Fowler is more combinatorial in nature.) However, I wanted to record a variant of the argument that relies not on the fine properties of characters, but on the cruder theory of *quasirandomness* for groups, the modern study of which was initiated by Gowers, and is discussed for instance in this previous post. It gives the following slightly weaker version of Corollary 2:

Corollary 3 (Weak Brauer-Fowler theorem)Let be a finite simple group with an involution , and suppose that has order . Then can be identified with a subgroup of the unitary group .

One can get an upper bound on from this corollary using Jordan’s theorem, but the resulting bound is a bit weaker than that in Corollary 2 (and the best bounds on Jordan’s theorem require the CFSG!).

*Proof:* Let be the set of all involutions in , then as discussed above . We may assume that has no non-trivial unitary representation of dimension less than (since such representations are automatically faithful by the simplicity of ); thus, in the language of quasirandomness, is -quasirandom, and is also non-abelian. We have the basic convolution estimate

(see Exercise 10 from this previous blog post). In particular,

and so there are at least pairs such that , i.e. involutions whose product is also an involution. But any such involutions necessarily commute, since

Thus there are at least pairs of non-identity elements that commute, so by the pigeonhole principle there is a non-identity whose centraliser has order at least . This centraliser cannot be all of since this would make central which contradicts the non-abelian simple nature of . But then the quasiregular representation of on has dimension at most , contradicting the quasirandomness.

The classification of finite simple groups (CFSG), first announced in 1983 but only fully completed in 2004, is one of the monumental achievements of twentieth century mathematics. Spanning hundreds of papers and tens of thousands of pages, it has been called the “enormous theorem”. A “second generation” proof of the theorem is nearly completed which is a little shorter (estimated at about five thousand pages in length), but currently there is no reasonably sized proof of the classification.

An important precursor of the CFSG is the Feit-Thompson theorem from 1962-1963, which asserts that every finite group of odd order is solvable, or equivalently that every non-abelian finite simple group has even order. This is an immediate consequence of CFSG, and conversely the Feit-Thompson theorem is an essential starting point in the proof of the classification, since it allows one to reduce matters to groups of even order for which key additional tools (such as the Brauer-Fowler theorem) become available. The original proof of the Feit-Thompson theorem is 255 pages long, which is significantly shorter than the proof of the CFSG, but still far from short. While parts of the proof of the Feit-Thompson theorem have been simplified (and it has recently been converted, after six years of effort, into an argument that has been verified by the proof assistant Coq), the available proofs of this theorem are still extremely lengthy by any reasonable standard.

However, there is a significantly simpler special case of the Feit-Thompson theorem that was established previously by Suzuki in 1957, which was influential in the proof of the more general Feit-Thompson theorem (and thus indirectly to the proof of CFSG). Define a CA-group to be a group with the property that the centraliser of any non-identity element is abelian; equivalently, the *commuting relation* (defined as the relation that holds when commutes with , thus ) is an equivalence relation on the non-identity elements of . Trivially, every abelian group is CA. A non-abelian example of a CA-group is the group of invertible affine transformations on a field . A little less obviously, the special linear group over a finite field is a CA-group when is a power of two. The finite simple groups of Lie type are not, in general, CA-groups, but when the rank is bounded they tend to behave as if they were “almost CA”; the centraliser of a generic element in , for instance, when is bounded and is large), is typically a maximal torus (because most elements in are regular semisimple) which is certainly abelian. In view of the CFSG, we thus see that CA or nearly CA groups form an important subclass of the simple groups, and it is thus of interest to study them separately. To this end, we have

Theorem 1 (Suzuki’s theorem on CA-groups)Every finite CA-group of odd order is solvable.

Of course, this theorem is superceded by the more general Feit-Thompson theorem, but Suzuki’s proof is substantially shorter (the original proof is nine pages) and will be given in this post. (See this survey of Solomon for some discussion of the link between Suzuki’s argument and the Feit-Thompson argument.) Suzuki’s analysis can be pushed further to give an essentially complete classification of all the finite CA-groups (of either odd or even order), but we will not pursue these matters here.

Moving even further down the ladder of simple precursors of CSFG is the following theorem of Frobenius from 1901. Define a Frobenius group to be a finite group which has a subgroup (called the *Frobenius complement*) with the property that all the non-trivial conjugates of for , intersect only at the origin. For instance the group is also a Frobenius group (take to be the affine transformations that fix a specified point , e.g. the origin). This example suggests that there is some overlap between the notions of a Frobenius group and a CA group. Indeed, note that if is a CA-group and is a maximal abelian subgroup of , then any conjugate of that is not identical to will intersect only at the origin (because and each of its conjugates consist of equivalence classes under the commuting relation , together with the identity). So if a maximal abelian subgroup of a CA-group is its own normaliser (thus is equal to ), then the group is a Frobenius group.

Frobenius’ theorem places an unexpectedly strong amount of structure on a Frobenius group:

Theorem 2 (Frobenius’ theorem)Let be a Frobenius group with Frobenius complement . Then there exists a normal subgroup of (called theFrobenius kernelof ) such that is the semi-direct product of and .

Roughly speaking, this theorem indicates that all Frobenius groups “behave” like the example (which is a quintessential example of a semi-direct product).

Note that if every CA-group of odd order was either Frobenius or abelian, then Theorem 2 would imply Theorem 1 by an induction on the order of , since any subgroup of a CA-group is clearly again a CA-group. Indeed, the proof of Suzuki’s theorem does basically proceed by this route (Suzuki’s arguments do indeed imply that CA-groups of odd order are Frobenius or abelian, although we will not quite establish that fact here).

Frobenius’ theorem can be reformulated in the following concrete combinatorial form:

Theorem 3 (Frobenius’ theorem, equivalent version)Let be a group of permutations acting transitively on a finite set , with the property that any non-identity permutation in fixes at most one point in . Then the set of permutations in that fix no points in , together with the identity, is closed under composition.

Again, a good example to keep in mind for this theorem is when is the group of affine permutations on a field (i.e. the group for that field), and is the set of points on that field. In that case, the set of permutations in that do not fix any points are the non-trivial translations.

To deduce Theorem 3 from Theorem 2, one applies Theorem 2 to the stabiliser of a single point in . Conversely, to deduce Theorem 2 from Theorem 3, set to be the space of left-cosets of , with the obvious left -action; one easily verifies that this action is faithful, transitive, and each non-identity element of fixes at most one left-coset of (basically because it lies in at most one conjugate of ). If we let be the elements of that do not fix any point in , plus the identity, then by Theorem 3 is closed under composition; it is also clearly closed under inverse and conjugation, and is hence a normal subgroup of . From construction is the identity plus the complement of all the conjugates of , which are all disjoint except at the identity, so by counting elements we see that

As normalises and is disjoint from , we thus see that is all of , giving Theorem 2.

Despite the appealingly concrete and elementary form of Theorem 3, the only known proofs of that theorem (or equivalently, Theorem 2) in its full generality proceed via the machinery of group characters (which one can think of as a version of Fourier analysis for nonabelian groups). On the other hand, once one establishes the basic theory of these characters (reviewed below the fold), the proof of Frobenius’ theorem is very short, which gives quite a striking example of the power of character theory. The proof of Suzuki’s theorem also proceeds via character theory, and is basically a more involved version of the Frobenius argument; again, no character-free proof of Suzuki’s theorem is currently known. (The proofs of Feit-Thompson and CFSG also involve characters, but those proofs also contain many other arguments of much greater complexity than the character-based portions of the proof.)

It seems to me that the above four theorems (Frobenius, Suzuki, Feit-Thompson, and CFSG) provide a ladder of sorts (with exponentially increasing complexity at each step) to the full classification, and that any new approach to the classification might first begin by revisiting the earlier theorems on this ladder and finding new proofs of these results first (in particular, if one had a “robust” proof of Suzuki’s theorem that also gave non-trivial control on “almost CA-groups” – whatever that means – then this might lead to a new route to classifying the finite simple groups of Lie type and bounded rank). But even for the simplest two results on this ladder – Frobenius and Suzuki – it seems remarkably difficult to find any proof that is not essentially the character-based proof. (Even trying to replace character theory by its close cousin, representation theory, doesn’t seem to work unless one gives in to the temptation to take traces everywhere and put the characters back in; it seems that rather than abandon characters altogether, one needs to find some sort of “robust” generalisation of existing character-based methods.) In any case, I am recording here the standard character-based proofs of the theorems of Frobenius and Suzuki below the fold. There is nothing particularly novel here, but I wanted to collect all the relevant material in one place, largely for my own benefit.

I’ve just uploaded to the arXiv my joint paper with Vitaly Bergelson, “Multiple recurrence in quasirandom groups“, which is submitted to Geom. Func. Anal.. This paper builds upon a paper of Gowers in which he introduced the concept of a quasirandom group, and established some mixing (or recurrence) properties of such groups. A -quasirandom group is a finite group with no non-trivial unitary representations of dimension at most . We will informally refer to a “quasirandom group” as a -quasirandom group with the quasirandomness parameter large (more formally, one can work with a *sequence* of -quasirandom groups with going to infinity). A typical example of a quasirandom group is where is a large prime. Quasirandom groups are discussed in depth in this blog post. One of the key properties of quasirandom groups established in Gowers’ paper is the following “weak mixing” property: if are subsets of , then for “almost all” , one has

where denotes the density of in . Here, we use to informally represent an estimate of the form (where is a quantity that goes to zero when the quasirandomness parameter goes to infinity), and “almost all ” denotes “for all in a subset of of density “. As a corollary, if have positive density in (by which we mean that is bounded away from zero, uniformly in the quasirandomness parameter , and similarly for ), then (if the quasirandomness parameter is sufficiently large) we can find elements such that , , . In fact we can find approximately such pairs . To put it another way: if we choose uniformly and independently at random from , then the events , , are approximately independent (thus the random variable resembles a uniformly distributed random variable on in some weak sense). One can also express this mixing property in integral form as

for any bounded functions . (Of course, with being finite, one could replace the integrals here by finite averages if desired.) Or in probabilistic language, we have

where are drawn uniformly and independently at random from .

As observed in Gowers’ paper, one can iterate this observation to find “parallelopipeds” of any given dimension in dense subsets of . For instance, applying (1) with replaced by , , and one can assert (after some relabeling) that for chosen uniformly and independently at random from , the events , , , , , , are approximately independent whenever are dense subsets of ; thus the tuple resebles a uniformly distributed random variable in in some weak sense.

However, there are other tuples for which the above iteration argument does not seem to apply. One of the simplest tuples in this vein is the tuple in , when are drawn uniformly at random from a quasirandom group . Here, one does *not* expect the tuple to behave as if it were uniformly distributed in , because there is an obvious constraint connecting the last two components of this tuple: they must lie in the same conjugacy class! In particular, if is a subset of that is the union of conjugacy classes, then the events , are perfectly correlated, so that is equal to rather than . Our main result, though, is that in a quasirandom group, this is (approximately) the *only* constraint on the tuple. More precisely, we have

Theorem 1Let be a -quasirandom group, and let be drawn uniformly at random from . Then for any , we havewhere goes to zero as , are drawn uniformly and independently at random from , and is drawn uniformly at random from the conjugates of for each fixed choice of .

This is the probabilistic formulation of the above theorem; one can also phrase the theorem in other formulations (such as an integral formulation), and this is detailed in the paper. This theorem leads to a number of recurrence results; for instance, as a corollary of this result, we have

for almost all , and any dense subsets of ; the lower and upper bounds are sharp, with the lower bound being attained when is randomly distributed, and the upper bound when is conjugation-invariant.

To me, the more interesting thing here is not the result itself, but how it is proven. Vitaly and I were not able to find a purely finitary way to establish this mixing theorem. Instead, we had to first use the machinery of ultraproducts (as discussed in this previous post) to convert the finitary statement about a quasirandom group to an infinitary statement about a type of infinite group which we call an *ultra quasirandom group* (basically, an ultraproduct of increasingly quasirandom finite groups). This is analogous to how the Furstenberg correspondence principle is used to convert a finitary combinatorial problem into an infinitary ergodic theory problem.

Ultra quasirandom groups come equipped with a finite, countably additive measure known as *Loeb measure* , which is very analogous to the Haar measure of a compact group, except that in the case of ultra quasirandom groups one does not quite have a topological structure that would give compactness. Instead, one has a slightly weaker structure known as a *-topology*, which is like a topology except that open sets are only closed under countable unions rather than arbitrary ones. There are some interesting measure-theoretic and topological issues regarding the distinction between topologies and -topologies (and between Haar measure and Loeb measure), but for this post it is perhaps best to gloss over these issues and pretend that ultra quasirandom groups come with a Haar measure. One can then recast Theorem 1 as a mixing theorem for the left and right actions of the ultra approximate group on itself, which roughly speaking is the assertion that

for “almost all” , if are bounded measurable functions on , with having zero mean on all conjugacy classes of , where are the left and right translation operators

To establish this mixing theorem, we use the machinery of *idempotent ultrafilters*, which is a particularly useful tool for understanding the ergodic theory of actions of countable groups that need not be amenable; in the non-amenable setting the classical ergodic averages do not make much sense, but ultrafilter-based averages are still available. To oversimplify substantially, the idempotent ultrafilter arguments let one establish mixing estimates of the form (2) for “many” elements of an infinite-dimensional parallelopiped known as an *IP system* (provided that the actions of this IP system obey some technical mixing hypotheses, but let’s ignore that for sake of this discussion). The claim then follows by using the quasirandomness hypothesis to show that if the estimate (2) failed for a large set of , then this large set would then contain an IP system, contradicting the previous claim.

Idempotent ultrafilters are an extremely infinitary type of mathematical object (one has to use Zorn’s lemma no fewer than *three* times just to construct one of these objects!). So it is quite remarkable that they can be used to establish a finitary theorem such as Theorem 1, though as is often the case with such infinitary arguments, one gets absolutely no quantitative control whatsoever on the error terms appearing in that theorem. (It is also mildly amusing to note that our arguments involve the use of ultrafilters in two completely different ways: firstly in order to set up the ultraproduct that converts the finitary mixing problem to an infinitary one, and secondly to solve the infinitary mixing problem. Despite some superficial similarities, there appear to be no substantial commonalities between these two usages of ultrafilters.) There is already a fair amount of literature on using idempotent ultrafilter methods in infinitary ergodic theory, and perhaps by further development of ultraproduct correspondence principles, one can use such methods to obtain further finitary consequences (although the state of the art for idempotent ultrafilter ergodic theory has not advanced much beyond the analysis of two commuting shifts currently, which is the main reason why our arguments only handle the pattern and not more sophisticated patterns).

We also have some miscellaneous other results in the paper. It turns out that by using the triangle removal lemma from graph theory, one can obtain a recurrence result that asserts that whenever is a dense subset of a finite group (not necessarily quasirandom), then there are pairs such that all lie in . Using a hypergraph generalisation of the triangle removal lemma known as the *hypergraph removal lemma*, one can obtain more complicated versions of this statement; for instance, if is a dense subset of , then one can find triples such that all lie in . But the method is tailored to the specific types of patterns given here, and we do not have a general method for obtaining recurrence or mixing properties for arbitrary patterns of words in some finite alphabet such as .

We also give some properties of a model example of an ultra quasirandom group, namely the ultraproduct of where is a sequence of primes going off to infinity. Thanks to the substantial recent progress (by Helfgott, Bourgain, Gamburd, Breuillard, and others) on understanding the expansion properties of the finite groups , we have a fair amount of knowledge on the ultraproduct as well; for instance any two elements of will almost surely generate a spectral gap. We don’t have any direct application of this particular ultra quasirandom group, but it might be interesting to study it further.

Given a function between two sets , we can form the graph

which is a subset of the Cartesian product .

There are a number of “closed graph theorems” in mathematics which relate the regularity properties of the function with the closure properties of the graph , assuming some “completeness” properties of the domain and range . The most famous of these is the closed graph theorem from functional analysis, which I phrase as follows:

Theorem 1 (Closed graph theorem (functional analysis))Let be complete normed vector spaces over the reals (i.e. Banach spaces). Then a function is a continuous linear transformation if and only if the graph is both linearly closed (i.e. it is a linear subspace of ) and topologically closed (i.e. closed in the product topology of ).

I like to think of this theorem as linking together qualitative and quantitative notions of regularity preservation properties of an operator ; see this blog post for further discussion.

The theorem is equivalent to the assertion that any continuous linear bijection from one Banach space to another is necessarily an isomorphism in the sense that the inverse map is also continuous and linear. Indeed, to see that this claim implies the closed graph theorem, one applies it to the projection from to , which is a continuous linear bijection; conversely, to deduce this claim from the closed graph theorem, observe that the graph of the inverse is the reflection of the graph of . As such, the closed graph theorem is a corollary of the open mapping theorem, which asserts that any continuous linear *surjection* from one Banach space to another is open. (Conversely, one can deduce the open mapping theorem from the closed graph theorem by quotienting out the kernel of the continuous surjection to get a bijection.)

It turns out that there is a closed graph theorem (or equivalent reformulations of that theorem, such as an assertion that bijective morphisms between sufficiently “complete” objects are necessarily isomorphisms, or as an open mapping theorem) in many other categories in mathematics as well. Here are some easy ones:

Theorem 2 (Closed graph theorem (linear algebra))Let be vector spaces over a field . Then a function is a linear transformation if and only if the graph is linearly closed.

Theorem 3 (Closed graph theorem (group theory))Let be groups. Then a function is a group homomorphism if and only if the graph is closed under the group operations (i.e. it is a subgroup of ).

Theorem 4 (Closed graph theorem (order theory))Let be totally ordered sets. Then a function is monotone increasing if and only if the graph is totally ordered (using the product order on ).

Remark 1Similar results to the above three theorems (with similarly easy proofs) hold for other algebraic structures, such as rings (using the usual product of rings), modules, algebras, or Lie algebras, groupoids, or even categories (a map between categories is a functor iff its graph is again a category). (ADDED IN VIEW OF COMMENTS: further examples include affine spaces and -sets (sets with an action of a given group ).) There are also various approximate versions of this theorem that are useful in arithmetic combinatorics, that relate the property of a map being an “approximate homomorphism” in some sense with its graph being an “approximate group” in some sense. This is particularly useful for this subfield of mathematics because there are currently more theorems about approximate groups than about approximate homomorphisms, so that one can profitably use closed graph theorems to transfer results about the former to results about the latter.

A slightly more sophisticated result in the same vein:

Theorem 5 (Closed graph theorem (point set topology))Let be compact Hausdorff spaces. Then a function is continuous if and only if the graph is topologically closed.

Indeed, the “only if” direction is easy, while for the “if” direction, note that if is a closed subset of , then it is compact Hausdorff, and the projection map from to is then a bijective continuous map between compact Hausdorff spaces, which is then closed, thus open, and hence a homeomorphism, giving the claim.

Note that the compactness hypothesis is necessary: for instance, the function defined by for and for is a function which has a closed graph, but is discontinuous.

A similar result (but relying on a much deeper theorem) is available in algebraic geometry, as I learned after asking this MathOverflow question:

Theorem 6 (Closed graph theorem (algebraic geometry))Let be normal projective varieties over an algebraically closed field of characteristic zero. Then a function is a regular map if and only if the graph is Zariski-closed.

*Proof:* (Sketch) For the only if direction, note that the map is a regular map from the projective variety to the projective variety and is thus a projective morphism, hence is proper. In particular, the image of under this map is Zariski-closed.

Conversely, if is Zariski-closed, then it is also a projective variety, and the projection is a projective morphism from to , which is clearly quasi-finite; by the characteristic zero hypothesis, it is also separated. Applying (Grothendieck’s form of) Zariski’s main theorem, this projection is the composition of an open immersion and a finite map. As projective varieties are complete, the open immersion is an isomorphism, and so the projection from to is finite. Being injective and separable, the degree of this finite map must be one, and hence and are isomorphic, hence (by normality of ) is contained in (the image of) , which makes the map from to regular, which makes regular.

The counterexample of the map given by for and demonstrates why the projective hypothesis is necessary. The necessity of the normality condition (or more precisely, a weak normality condition) is demonstrated by (the projective version of) the map from the cusipdal curve to . (If one restricts attention to smooth varieties, though, normality becomes automatic.) The necessity of characteristic zero is demonstrated by (the projective version of) the inverse of the Frobenius map on a field of characteristic .

There are also a number of closed graph theorems for topological groups, of which the following is typical (see Exercise 3 of these previous blog notes):

Theorem 7 (Closed graph theorem (topological group theory))Let be -compact, locally compact Hausdorff groups. Then a function is a continuous homomorphism if and only if the graph is both group-theoretically closed and topologically closed.

The hypotheses of being -compact, locally compact, and Hausdorff can be relaxed somewhat, but I doubt that they can be eliminated entirely (though I do not have a ready counterexample for this).

In several complex variables, it is a classical theorem (see e.g. Lemma 4 of this blog post) that a holomorphic function from a domain in to is locally injective if and only if it is a local diffeomorphism (i.e. its derivative is everywhere non-singular). This leads to a closed graph theorem for complex manifolds:

Theorem 8 (Closed graph theorem (complex manifolds))Let be complex manifolds. Then a function is holomorphic if and only if the graph is a complex manifold (using the complex structure inherited from ) of the same dimension as .

Indeed, one applies the previous observation to the projection from to . The dimension requirement is needed, as can be seen from the example of the map defined by for and .

(ADDED LATER:) There is a real analogue to the above theorem:

Theorem 9 (Closed graph theorem (real manifolds))Let be real manifolds. Then a function is continuous if and only if the graph is a real manifold of the same dimension as .

This theorem can be proven by applying invariance of domain (discussed in this previous post) to the projection of to , to show that it is open if has the same dimension as .

Note though that the analogous claim for *smooth* real manifolds fails: the function defined by has a smooth graph, but is not itself smooth.

(ADDED YET LATER:) Here is an easy closed graph theorem in the symplectic category:

Theorem 10 (Closed graph theorem (symplectic geometry))Let and be smooth symplectic manifolds of the same dimension. Then a smooth map is a symplectic morphism (i.e. ) if and only if the graph is a Lagrangian submanifold of with the symplectic form .

In view of the symplectic rigidity phenomenon, it is likely that the smoothness hypotheses on can be relaxed substantially, but I will not try to formulate such a result here.

There are presumably many further examples of closed graph theorems (or closely related theorems, such as criteria for inverting a morphism, or open mapping type theorems) throughout mathematics; I would be interested to know of further examples.

This is a sequel to my previous blog post “Cayley graphs and the geometry of groups“. In that post, the concept of a Cayley graph of a group was used to place some geometry on that group . In this post, we explore a variant of that theme, in which (fragments of) a Cayley graph on is used to describe the basic algebraic structure of , and in particular on elementary word identities in . Readers who are familiar with either category theory or group homology/cohomology will recognise these concepts lurking not far beneath the surface; we wil remark briefly on these connections later in this post. However, no knowledge of categories or cohomology is needed for the main discussion, which is primarily focused on elementary group theory.

Throughout this post, we fix a single group , which is allowed to be non-abelian and/or infinite. All our graphs will be directed, with loops and multiple edges permitted.

In the previous post, we drew the entire Cayley graph of a group . Here, we will be working much more locally, and will only draw the portions of the Cayley graph that are relevant to the discussion. In this graph, the vertices are elements of the group , and one draws a directed edge from to labeled (or “coloured”) by the group element for any ; the graph consisting of all such vertices and edges will be denoted . Thus, a typical edge in looks like this:

Figure 1.

One usually does not work with the complete Cayley graph . It is customary to instead work with smaller Cayley graphs , in which the edge colours are restricted to a smaller subset of , such as a set of generators for . As we will be working locally, we will in fact work with even smaller fragments of at a time; in particular, we only use a handful of colours (no more than nine, in fact, for any given diagram), and we will not require these colours to generate the entire group (we do not care if the Cayley graph is connected or not, as this is a global property rather than a local one).

Cayley graphs are left-invariant: for any , the left translation map is a graph isomorphism. To emphasise this left invariance, we will usually omit the vertex labels, and leave only the coloured directed edge, like so:

Figure 2.

This is analogous to how, in undergraduate mathematics and physics, vectors in Euclidean space are often depicted as arrows of a given magnitude and direction, with the initial and final points of this arrow being of secondary importance only. (Indeed, this depiction of vectors in a vector space can be viewed as an abelian special case of the more general depiction of group elements used in this post.)

Let us define a *diagram* to be a finite directed graph , with edges coloured by elements of , which has at least one graph homomorphism into the complete Cayley graph of ; thus there exists a map (not necessarily injective) with the property that whenever is a directed edge in coloured by a group element . Informally, a diagram is a finite subgraph of a Cayley graph with the vertex labels omitted, and with distinct vertices permitted to represent the same group element. Thus, for instance, the single directed edge displayed in Figure 2 is a very simple example of a diagram. An even simpler example of a diagram would be a depiction of the identity element:

We will however omit the identity loops in our diagrams in order to reduce clutter.

We make the obvious remark that any directed edge in a diagram can be coloured by at most one group element , since implies . This simple observation provides a way to prove group theoretic identities using diagrams: to show that two group elements are equal, it suffices to show that they connect together (with the same orientation) the same pair of vertices in a diagram.

Remark 1One can also interpret these diagrams as commutative diagrams in a category in which all the objects are copies of , and the morphisms are right-translation maps. However, we will deviate somewhat from the category theoretic way of thinking here by focusing on the geometric arrangement and shape of these diagrams, rather than on their abstract combinatorial description. In particular, we view the arrows more as distorted analogues of vector arrows, than as the abstract arrows appearing in category theory.

Just as vector addition can be expressed via concatenation of arrows, group multiplication can be described by concatenation of directed edges. Indeed, for any , the vertices can be connected by the following triangular diagram:

In a similar spirit, inversion is described by the following diagram:

Figure 5.

We make the pedantic remark though that we do not consider a edge to be the reversal of the edge, but rather as a distinct edge that just happens to have the same initial and final endpoints as the reversal of the edge. (This will be of minor importance later, when we start integrating “-forms” on such edges.)

A fundamental operation for us will be that of *gluing* two diagrams together.

Lemma 1 ((Labeled) gluing)Let be two diagrams of a given group . Suppose that the intersection of the two diagrams connects all of (i.e. any two elements of are joined by a path in ). Then the union is also a diagram of .

*Proof:* By hypothesis, we have graph homomorphisms , . If they agree on then one simply glues together the two homomorphisms to create a new graph homomorphism . If they do not agree, one can apply a left translation to either or to make the two diagrams agree on at least one vertex of ; then by the connected nature of we see that they now must agree on all vertices of , and then we can form the glued graph homomorphism as before.

The above lemma required one to specify the label the vertices of (in order to form the intersection and union ). However, if one is presented with two diagrams with unlabeled vertices, one can identify some partial set of vertices of with a partial set of vertices of of matching cardinality. Provided that the subdiagram common to and after this identification connects all of the common vertices together, we may use the above lemma to create a glued diagram .

For instance, if a diagram contains two of the three edges in the triangular diagram in Figure 4, one can “fill in” the triangle by gluing in the third edge:

Figure 6.

One can use glued diagrams to demonstrate various basic group-theoretic identities. For instance, by gluing together two copies of the triangular diagram in Figure 4 to create the glued diagram

Figure 7.

and then filling in two more triangles, we obtain a tetrahedral diagram that demonstrates the associative law :

Figure 8.

Similarly, by gluing together two copies of Figure 4 with three copies of Figure 5 in an appropriate order, we can demonstrate the Abel identity :

Figure 9.

In addition to gluing, we will also use the trivial operation of *erasing*: if is a diagram for a group , then any subgraph of (formed by removing vertices and/or edges) is also a diagram of . This operation is not strictly necessary for our applications, but serves to reduce clutter in the pictures.

If two group elements commute, then we obtain a parallelogram as a diagram, exactly as in the vector space case:

Figure 10.

In general, of course, two arbitrary group elements will fail to commute, and so this parallelogram is no longer available. However, various substitutes for this diagram exist. For instance, if we introduce the conjugate of one group element by another, then we have the following slightly distorted parallelogram:

Figure 11.

By appropriate gluing and filling, this can be used to demonstrate the homomorphism properties of a conjugation map :

Figure 12.

Figure 13.

Another way to replace the parallelogram in Figure 10 is to introduce the commutator of two elements, in which case we can perturb the parallelogram into a pentagon:

Figure 14.

We will tend to depict commutator edges as being somewhat shorter than the edges generating that commutator, reflecting a “perturbative” or “nilpotent” philosophy. (Of course, to fully reflect a nilpotent perspective, one should orient commutator edges in a different dimension from their generating edges, but of course the diagrams drawn here do not have enough dimensions to display this perspective easily.) We will also be adopting a “Lie” perspective of interpreting groups as behaving like perturbations of vector spaces, in particular by trying to draw all edges of the same colour as being approximately (though not perfectly) parallel to each other (and with approximately the same length).

Gluing the above pentagon with the conjugation parallelogram and erasing some edges, we discover a “commutator-conjugate” triangle, describing the basic identity :

Figure 15.

Other gluings can also give the basic relations between commutators and conjugates. For instance, by gluing the pentagon in Figure 14 with its reflection, we see that . The following diagram, obtained by gluing together copies of Figures 11 and 15, demonstrates that ,

Figure 16.

while this figure demonstrates that :

Figure 17.

Now we turn to a more sophisticated identity, the Hall-Witt identity

which is the fully noncommutative version of the more well-known Jacobi identity for Lie algebras.

The full diagram for the Hall-Witt identity resembles a slightly truncated parallelopiped. Drawing this truncated paralleopiped in full would result in a rather complicated looking diagram, so I will instead display three components of this diagram separately, and leave it to the reader to mentally glue these three components back to form the full parallelopiped. The first component of the diagram is formed by gluing together three pentagons from Figure 14, and looks like this:

This should be thought of as the “back” of the truncated parallelopiped needed to establish the Hall-Witt identity.

While it is not needed for proving the Hall-Witt identity, we also observe for future reference that we may also glue in some distorted parallelograms and obtain a slightly more complicated diagram:

Figure 19.

To form the second component, let us now erase all interior components of Figure 18 or Figure 19:

Figure 20.

Then we fill in three distorted parallelograms:

Figure 21.

This is the second component, and is the “front” of the truncated praallelopiped, minus the portions exposed by the truncation.

Finally, we turn to the third component. We begin by erasing the outer edges from the second component in Figure 21:

Figure 22.

We glue in three copies of the commutator-conjugate triangle from Figure 15:

Figure 23.

But now we observe that we can fill in three pentagons, and obtain a small triangle with edges :

Figure 24.

Erasing everything except this triangle gives the Hall-Witt identity. Alternatively, one can glue together Figures 18, 21, and 24 to obtain a truncated parallelopiped which one can view as a geometric representation of the *proof* of the Hall-Witt identity.

Among other things, I found these diagrams to be useful to visualise group cohomology; I give a simple example of this below, developing an analogue of the Hall-Witt identity for -cocycles.

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