5th HLF – Lecture: Asymptotic Group Theory
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Transcript: English(auto-generated)
00:01
So, the second speaker this morning, it's Efim Selmanov, so you can please come already up to the stage. He received the Fields Medal in 1994 for his work in abstract algebra and group theory and in particular for his solution to the restricted burn site problem.
00:21
And it's a real pleasure to introduce his talk on asymptotic group theory, which is a topic which is also very close to my own heart. So thank you very much, Efim.
00:41
Thank you and thank you for a nice introduction. It is a non-trivial task to give a talk at this forum. On the one hand, there is Sir Michael Atiyeh in the audience, Stephen Smale and the most knowledgeable people in the world. On the other hand, there are many young researchers with very different mathematical backgrounds
01:07
and there is a risk to look an idiot in the eyes of the first or the second or both. I will assume and know the concept of a group, the great unifying concept in mathematics.
01:27
If you go for a trip and you are allowed to take only two or three mathematical concepts with you, give a serious consideration to this one.
01:40
I won't have any picture in my talk. I don't know how to make a picture of a group. So let G be a group, possibly an infinite group, and suppose that there exists a family of homomorphisms of this group onto finite groups, GIs, such that the intersection of
02:06
kernels is trivial. It means that for any two distinct elements, there is a projection that distinguishes between them. Projection is like a photo or photo is like a projection.
02:21
When you make a photo, you lose some information. But here, we make photos from all sides of the group. So all together, they give us a pretty good idea how the group looks like. Such groups are called residually finite.
02:40
Examples, if you have a finitely generated linear group that is a group of matrices, it is residually finite. The free group on m generators is residually finite. If you have an infinite Galois extension, it is also residually finite.
03:05
In general, the whole class of infinite groups can be subdivided into two classes, hopelessly infinite groups. They arise mostly in geometric group theory and residually finite groups.
03:22
They arise in number theory, combinatorics. It's like a difference between hopeless infinity and infinity, which is still a limit of finite. And these classes behave differently. As an example, I will mention the Burnside problem.
03:43
Maybe the first problem on infinite groups, formulated in 1902 by William Burnside, a colorful professor at Cambridge. The problem sounds as follows. Suppose that you have a finitely generated group.
04:02
So an arbitrary element of the group can be represented as a product of these generators and their inverses. And suppose that there exists n such that an arbitrary element to the power n is the identity. Does it imply that the group is finite?
04:22
If a group is finite, then clearly it has finitely many generators and those who had a course of abstract algebra and still remember Lagrange theorem know that this condition also holds. The question is, is it enough to make a group finite?
04:42
Or more generally, what makes a group finite? Positive solutions are known only for n equal to 2, 3, 4, and 6. That's it. For n equal to 5, it's a widely open problem.
05:03
In 1968, Novikov and Adyan constructed counterexamples. That was an amazing work on more than 300 pages with a simultaneous induction on more than 100 parameters. It was the first hopelessly infinite group that I referred to on the previous page.
05:30
On the other hand, for residual finite groups, the answer is yes. And maybe I owe, it is to this work, I owe the honor of being at this stage.
05:44
So I will discuss several subjects related to residual finite groups that have been pretty active recently. The first subject is about expanders.
06:05
At this conference, even pure mathematicians start talking about computer science. So let gamma be a finite connected graph and let W be a non-empty set of vertices.
06:24
The boundary of that, of W, consists of those points, of those vertices that do not lie in W, but they are connected by a simple H to some element from W. So if you think of vertices as electric nodes, the current spreads from W to its boundary to the next
06:47
boundary and so on. So you have this expansion process. Now, what is a good graph? A good graph is a graph where current spreads fast.
07:03
So the best graph is a graph where all vertices are connected, but it is very expensive because you pay for edges. Now let epsilon be a positive number. We know that if it is epsilon, then it should be a small positive number. Well, a graph is called an epsilon expander if for every non-empty subset, W, which
07:28
say contains less than one half of all vertices. If you add the boundary, it expands, it increases at least one plus epsilon times. So that is a uniform measure of how good your graph is.
07:45
But then it is easy to see that if you choose epsilon sufficiently small, then every finite connected graph is an epsilon expander. And the condition that less than one half is necessary, or some condition of this kind,
08:02
because if W is very close to the whole V, it has no room to expand. So really what we want is an infinite family of K regular graphs. A graph is said to be K regular if for every vertex there are exactly K edges adjacent
08:22
to it. We want a family of K regular graphs that are all epsilon expanders. And K and epsilon are fixed, and the size goes to infinity. In this way, we get highly connected because epsilon is fixed, and inexpensive graphs because
08:43
K is also fixed. There are such animals as random K regular graphs, and Pinsker and Erdos proved that this family of random graphs is an expander family. But there was still a problem of an explicit construction, not random.
09:03
If you asked me three years ago, I would say that the concept of an expander was introduced by Pinsker in the 70s, but in these three years, some people discovered proceedings of some conference in Latvia in mid-60s with a paper by Bardzin and Kolmogorov where
09:26
they introduced the notion of expander proof, Pinsker Erdos theorem, Kolmogorov was one of the predominant mathematicians of the 20th century, and I would assume that all his
09:42
papers are very well known. Apparently, this is not the case. Okay, now I am going to, I will talk about the explicit construction of expanders. So let G be a finite group. Generated by a set X, which again means that an arbitrary element can be represented as
10:06
a product of generators and their inverses. What is a Cayley graph? A Cayley graph, the set of vertices is the set of all elements of the group. Two vertices are connected if one of them can be obtained from another one by multiplication,
10:25
by a generator or its inverse. So it is a regular graph because at each point you can multiply by a fixed number of generators or their inverses. In 67, Kashdan made the following observation, that there exists groups that are finitely
10:46
generated and have the following property, there exists epsilon such that for any unitary representation without fixed, non-zero fixed points, unitary representation means a holomorphism
11:01
to the group of unitary linear transformations of some Hilbert space. So for any such unitary representation without non-zero fixed point, every element of H is moved significantly by one of the generators significantly compared to its length, precisely
11:21
this. You see, if H is zero then there is nothing to talk about, if H is non-zero then at least one generator has to move it, otherwise it's a fixed point, but we demand this condition and epsilon does not depend not only on the choice of the element H but it does
11:44
not even depend on the choice of the unitary representation as long as it doesn't have fixed points. So for example the group of n by n integer matrices of determinant one when n is greater or equal than three has this property.
12:03
He called it property T, I asked him why T, well at that point in functional analysis there was property S, property P, so that was property T, okay.
12:20
The first construction was due to Margulis. For various non-mathematical reasons Margulis could not get, after he made his PhD, couldn't get a job in university or mathematical institute so he got a job at the institute of information transmission, more applied, and he felt that sometimes he should do something
12:44
related to the subject of the institute. So he published this construction and in subsequent years he wrote many very deep papers and got Phil's medal in 78 but this paper on two pages is still his most cited paper.
13:08
Okay, he proved that if the group G is finitely generated and is residual finite and has property T, for example S, L, and Z that I talked about, and we consider this
13:23
family of homomorphisms into finite groups, generators, I'm apt to generate this, so we have got a family of calligraphs. He proved that this is an expander family. Well, I could give, with minimal cheating, I could give even a proof of this theorem
13:44
because it's not a theorem, it's a translation. The proof, let W be a subset of vertices. What does it mean that some generator moves it significantly? It means that it moves the characteristic function significantly.
14:04
And why does it move the characteristic function significantly? Because of the property T. Basically that's it, with minimal cheating. Later, Kasabov-Lubotsky-Briard-Green-Tao proved that given any infinite family of finite simple groups,
14:27
you can always find systems of generators of bounded number of elements that make this family an expander family. So every family is an expander family.
14:40
Okay, and recently, very recently, it's not on my transparency, Sarnak and Salekhi-Golsefidi proved a very important thing about these expanders. I will try to explain without having it on the transparencies. If you have a finitely generated subgroup of SLNZ, finitely generated group of integer matrices,
15:10
and it does not have a homorphism into an infinite abelian group, which is a necessary condition, then you consider that group modular,
15:21
you know, various positive integers. Since these are matrices over integers, you can consider them more modular five, modular seven, and so on. And you got a family of finite groups, and this is an expander family. This is a very important result with many consequences.
15:40
It's called super strong rigidity for linear groups. Okay, next subject. So far, I talked about growth of graphs. Now let's talk about growth of groups. So let G be a group, again, finitely generated.
16:02
Let us consider the set of products of generators and their inverses of lengths up to N. If you think of a Cayley graph, this is a ball centered at the identity of radius N.
16:20
All elements at a distance no more than N from the identity. The union is the whole group. These balls are finite sets, and of course, their orders increase. And we can discuss how fast do they increase. In this way, we can compare two infinite groups.
16:44
If you have two finite groups, you can discuss which of them has a bigger order. But suppose that both groups are infinite. For example, you can discuss which one grows faster. And this concept comes from geometry. If you have a remaining manifold and fix a point
17:01
and consider the growth of walls centered at this point, it is the same growth as this growth in the fundamental group. The first is a geometric characteristic. The second is topological characteristic. So this thing was considered in mid-50s
17:20
by Schwarz, Yefremovich, Milner. Okay, so this is a growth function. Unfortunately, this growth function depends on the generating system. So we have to do something about it. Let f and g be two, say, increasing functions
17:41
on positive integers. We say that f is asymptotically less or equal to g if there exists a constant such that this inequality always holds. And if f is less or equal than g and g is less or equal than s, then we say that f and g are asymptotically equivalent.
18:03
Now, if you choose a different generating system for g but it is also finite, then the growth functions are asymptotically equivalent. So the growth of a group is a class of equivalence. In 68, Milner wrote a paper for Mathematics Monthly
18:26
for those who know what Mathematics Monthly is. It's a journal for undergraduates. Where he formulated two problems on the growth of the group. The first problem was, is it true
18:41
that a group has polynomially bounded growth? So there is an upper bound for the growth which is a polynomial. If and only if it has a subgroup of finite index which is nilpotent. It was known that nilpotent groups have polynomially bounded growth. So it was a natural question.
19:02
And the second question, do they exist groups of intermediate growth? That is groups that grow faster than any polynomial and slower than the exponential. Both questions were answered almost simultaneously. In, well, I don't know from 80 to 82.
19:24
The first question was answered by Gromov who proved that yes indeed, polynomially bounded groups are virtually nilpotent. And his proof was maybe even more interesting than the result. I will give a rough idea of the proof.
19:42
We consider the Cayley graph of the group G. A Cayley graph is a connected graph. Every connected graph is a metric space. The group G acts on that metric space by isometries. Now, suppose that we sit on this, this is a helicopter.
20:05
This is a Cayley graph. And this metric space is discrete, you know, points of the group. Now our helicopter goes up, points look closer and closer to us.
20:21
This process can be rigorously expressed as Hausdorff-Gromov limit of metric spaces. And as a limit, you get the Euclidean space Rn and the group acted on each member of the sequence.
20:41
Well, a little bit not so straightforward but it acts also on the limit. So now we have got an action of the group on the Euclidean space. So we have got a homorphism into linear group. And for linear groups, the problem has been solved before
21:01
by Jacques Tits and so everything was known. Well, Gromov told himself that when he did this proof he could not believe that it could be that simple. And another problem was solved by Grigarchuk.
21:21
He constructed groups of intermediate groups as groups of automorphisms of the graph. And these were very interesting groups, also finitely generated torsion groups which answered some question of Burnside. So initially these groups appeared as counterexamples to everything.
21:42
But later, it turned out that they are more than counterexamples. They naturally appear in number theory in dynamical systems. Now they are called fractal groups and there are reasons for that. The same idea works. You can talk about growth of any infinite system.
22:04
For example of algebras, if F is a field and A is an F algebra generated by a finite dimensional space, you could consider instead of balls, you could consider the span of all products of generators of lengths up to N, you have got ascending chain of finite dimensional spaces
22:24
and you can discuss how fast their dimensions grow. If it is polynomially bounded, then the minimal D with this property is called Galvan-Kirillov dimension of the algebra A and the study of non-commutative algebras
22:43
of finite Galvan-Kirillov dimension is what is known as one of versions of non-commutative algebraic geometry developed by Artin, Tate, Vanderberg and so on. Another subject related to this, approximate groups.
23:02
Let G be a group, okay. Let G be a group. Let A be a subset of G. Let us say symmetric subset, but it's not a restriction. We could add all inverses. And let K be a constant, greater or equal than one.
23:23
We know that A is a subgroup if for any two elements X and Y, X, Y inverse again lies in A. Let us say that it's not always the case, but it is the case with non-zero probability.
23:43
Let us say if A is finite, we can talk about it. So let us say that this inclusion holds with probability greater or equal than one over K. So it's close to being a subgroup or another condition.
24:02
If A is finite, then it is a subgroup if and only if all products lie in A. So A squared is A. Let us say that this is not so, but the order is bounded by the order of A times some constant. Or a third condition that this square can be covered
24:23
by K translates of A. I am not saying that these three conditions are equivalent, but they lead to the same theory. And the third condition has some advantages because it doesn't require A to be finite. The first two do require.
24:42
Otherwise, which probability are we talking about? And then A is called K-approximate subgroup. Some examples. In the infinite cyclic group, let's consider a segment. It's a two-approximate subgroup,
25:01
or slight generalization, D-dimensional arithmetic progression. Again, we take this, well, shall I say, parallely pivot. This is two to the power D-approximate subgroups. And in general, approximate subgroups arise whenever we talk about growth,
25:20
because we take a ball. If the balls grow fast, so you have fast growth. If the balls do not grow fast, it means that many products lie in the same subset, but then you have an approximate subgroup. So either or.
25:41
In this way, you can get a much more conceptual proof of Gromov theorem, which with even some estimates. If you want some kind of classification, here is a typical example how a classification could look like.
26:03
This is Freiman-Ruzsa theorem, and it is a description of approximate subgroups in the infinite cyclic group. If A is a K-approximate subgroup, then it lies in some segment of the D-dimensional, some D-dimensional arithmetic progression.
26:22
And it is big there. The ratio of volumes is bounded. Now many, many names. This activity was started with the paper of Harold Helfgott, and then continued
26:41
by Gomborg-Borgen, Saronok-Krushevsky, Briard-Green-Tau, Piber-Sabo. It became a very fashionable subject, and by the names, you could see that it looks more like Fourier analysis, because these are the methods that have been used.
27:01
And side effect of this study was better understanding and even efficient version of Gromov theorem, and the new approach to Hilbert's fifth problem. Okay, now, profinite and propi groups.
27:22
Let's come back to the beginning. So let G be just a residual finite group, which means that the intersection of all normal subgroups of finite index is trivial. If the intersection is trivial, then we can view these subgroups as basis of neighborhoods of the identity element.
27:43
So G becomes a topological group. If this topology is complete, then the group can be represented as an inverse limit of finite groups. And then it is called a profinite group.
28:02
If the topology is not complete, we can embed it into a completion, which is denoted G hat, and called profinite completion of the group G. Finally, if the group is even not residual finite,
28:21
which means that the intersection of all subgroups of finite index is not trivial, we will factor it out. If the quotient group, it will become trivial, and then we take profinite completion. So the moral is that we can consider a profinite completion of any group,
28:41
but only residual finite groups are embeddable into their profinite completions. Example, again, an infinite Galois extension of fields, then the Galois group is profinite. And that's basically the source of problems related to profinite groups.
29:02
Now let P be a prime number. We can see the somewhat more restrictive situation. Now we can see the family of homomorphisms onto finite P groups, such that the intersection of kernels is trivial. If it exists, then we say that G is residual P.
29:25
Again, these kernels can be taken for a basis of neighborhoods of the identity element, G becomes a topological group. If the topology is complete, then it is called pro-P group. And then it is an inverse limit of finite P groups.
29:42
If it is not complete, we embed G into its pro-P completion. And again, for any group, we can consider the intersection of all normal subgroups of P power index. If it is not trivial, we factor it out, and then take a pro-P completion.
30:00
So we can talk about the pro-P completion of any group. Examples, M is the free group on M generators, then for an arbitrary prime number P, it is residual P. And we can take its pro-P completion. This group, this pro-P completion
30:21
is called the free pro-P group. It deserves this name, because an arbitrary mapping of the free generators X1, X2, XM to an arbitrary pro-P group uniquely extends to continuous homomorphism. Another example. Let Z sub P be the ring of periodic integers.
30:43
We can see the N by N matrices over periodic integers that are equal to the identity matrix, modulo P. It means that all diagonal entries are equal to one modulo P, and all off-diagonal elements are divisible by P.
31:01
This is a pro-P group. This is a very special pro-P group. Because, you know, we can talk about periodic lie groups. The ordinary lie groups are groups and manifolds, so locally near every point, each point there are coordinates, and in these coordinates the multiplication
31:21
is smooth or analytic, choose your favorite condition. And now, instead of real numbers or complex numbers, we can see the periodic integers, and we can talk about periodic lie groups. So every periodic lie group has an open subgroup
31:40
which is embeddable into this one. So this is a very special pro-P group. This is a result of Lazar. We can consider a more general situation instead of periodic integers. We can consider a more general commutative ring. Ah, this commutative ring, of course,
32:02
has to be Nasserian local complete ring, such that if you take a quotient module as a maximal ideal, you will get a finite field. Well, if you got tired of this list, consider, you can consider infinite series
32:20
over a finite field, or over periodic integers. Now, the last kind of result is that these groups
32:45
satisfy some periodic identity. What does it mean? It means that there exists a non-identical element of the free group, and then if you substitute arbitrary elements from the group for generators,
33:02
you all will get the identity. Why is it important? Because the most important groups are linear groups. Ah, subgroups of this group. But unfortunately, the homomorphic image of a linear group does not have to be a linear group.
33:24
But if the group satisfied an identity, then every homomorphic image still satisfies the identity. So if you want to study linear groups together, linear Pro-P groups, together with their homomorphic images, which is a must sometimes, this is a helpful notion.
33:46
And I want to mention possible application in number theory. Let S, now I'm afraid to become slightly technical. But that's number theory.
34:05
Let us consider the maximal Pro-P extension of the field of rational numbers. Pro-P extension means that the Galois group is Pro-P group which is infinitely many ramification points
34:22
which do not include P. Arguably, this is the most important group in number theory, and many classical problems boil down to it. There is a famous Fontaine-Meyzer conjecture that this group is strongly non-linear.
34:42
Moreover, that if you consider any linear representation, the image is finite. Now, this group is the so-called Golodshifarevich group. Golodshifarevich groups, you know,
35:03
they are the latest in relations. Max Dehn invented those presentations at the beginning of the 20th century. He represented fundamental groups in this way, and there is a story, I was not present there,
35:21
that after that, he repeated the phrase of Rene Descartes, I solved all problems, which presumably Rene Descartes said that after he invented coordinate system. This phrase was as correct as the phrase of Rene Descartes.
35:41
You know, if all problems can be reduced to presentation by generators and relations, it only shows that you can say very little by presentation, by generators and relations. For example, given a finite presentation
36:01
that doesn't exist in algorithm, that will tell you if this group is finite or infinite, if it is the identity group or not. In 64, Golodshifarevich found a sufficient condition for the group presented by generators and relations to be infinite, and this group satisfies it.
36:23
And so far, this is the only way to prove that it is infinite. Now, looking at this problem, it was not known even whether this group was linear itself or not. It's very difficult group.
36:42
Now, from the theorem above, it follows that it is not linear. Why? In 2000, I proved that every Golodshifarevich group contains a free-propi group. Free-propi group, by definition, does not satisfy any identity.
37:00
Since this group satisfies identity, free-propi group cannot be embedded into it, and this group cannot be embedded into it. I hope that further development of this theory of identities will be instrumental in solution of this problem.
37:23
Thank you. If you thank you very much for your nice talk, we again are well in time and have time for questions.
37:41
So if there are any questions from the audience, or did he lose you in the end?
38:03
What is the most important unsolved problem, where? In the area, okay. It's only my personal opinion, right? Speaking of the Burnside problem, it is widely open for finitely presented groups.
38:24
Nothing is known about finitely presented groups. Speaking about growth, groups of intermediate growth, there are no examples among finitely presented groups. You know, for more finitistically-oriented people,
38:41
only finitely presented groups exist, and nothing is known about them. Well, thank you for wonderful talk. I'm just curious, so what kind of other implications
39:02
does the periodic group have in mathematics, besides number theory and this group theory on its own? Oh, besides number theory and group theory? Well, more likely, in geometry. Do we study these, am I supposed to know about this, considering I'm learning about periodic heart theory? Oh yes, yes, periodic manifolds are widely studied.
39:24
But nowadays, when you look at the paper of number theory, it is very difficult to say if it's a number theory or in geometry. They converged. And many problems related to periodic analysis
39:41
or periodic groups. Thank you.