January 27, 2022

August 8-11
Workshop on Profinite Groups and Applications

Carleton University, Ottawa


Jorge Almeida(University of Porto)

Maximal subgroups of finitely generated free profinite semigroups
The most developed part of finite semigroup theory has drawn its motivation and finds most of its applications in computer science. Its programme was put forward by S. Eilenberg in the mid-1970's in the form of the theory of pseudovarieties, that is of classes of finite semigroups which are closed under taking homomorphic images, subsemigroups and finite direct products. Over the next decade, it was realized that many problems in the theory could be successfully handled by translating them into structural problems on finitely generated free profinite semigroups relative to suitable pseudovarieties. Yet, very little is known about the structure of finitely generated (absolutely) free profinite semigroups.

The maximal subgroups of profinite semigroups are easily seen to be profinite groups. Thus, a natural question about the structure of (finitely generated) free profinite semigroups is which profinite groups can appear as its maximal subgroups. Since the division (quasi-)ordering in semigroups plays an important role in their structure, it is also of interest to determine where in this ordering those groups are found.

In this talk, we address this question and present some partial answers. The technique which we have developed consists in producing maximal subgroups by studying the dynamics of the monoid of continuous endomorphisms of finitely generated free profinite semigroups, which is a profinite monoid under the pointwise convergence topology. For instance, under simple and easily checked hypotheses, if we take an endomorphism f of the free monoid on a finite set A, then the images of the elements of A under all infinite iterates of the unique extension of f to a continuous endomorphism of the free profinite semigroup on A divide each other, that is they determine a unique associate class, which is known as a J-class in semigroup theory. This J-class contains maximal subgroups, which are all isomorphic. A natural question is then to compute these maximal subgroups, say in terms of a profinite presentation. A first general result in this direction is that, in case f induces an automorphism of the free group on A, these groups are free profinite groups for which free generators can be computed.

Dan Haran (Tel Aviv University)

Algebraic patching over rings
Let E be the field of rational functions in one variable over a complete field K. Patching (formal, rigid analytic) of covers has been used to obtain a Galois extension F of E with group G from Galois extensions F_1, F_2 of E with subgroups G_1, G_2 of G that generate G. The algebraic version of it is a quite simple procedure. We will discuss some recent extensions of this method, among them replacement of K by the (not necessarily complete) quotient field of a complete domain.

Wolfgang Herfort (University of Vienna)

Profinite groups with a CC-subgroup
A CC-subgroup M of a finite group G is a proper subgroup containing the centralizers of its nontrivial elements. Such groups have been classified through work of many authors. Frobenius groups and Suzuki groups are examples. As an application, one can classify totally disconnected topologically locally finite groups, containing a topological analogue of a CC-subgroup. Profinite groups with a CC-subgroup play the central role during the proofs -- in particular profinite Frobenius groups. This is joint work of Zvi Arad.

Alex Lubotzky (Hebrew University)

Counting primes, groups and manifolds
Let D be a finitely generated group. Let G be a semisimple Lie group, K a maximal compact subgroup and Y=G/K the associated symmetric space. Let x be a positive real number (going to infinity).

We will discuss questions of the following type: How many primes are there which are smaller than x ? How many subgroups D has of index at most x ? How many quotient mainfolds Y has, of volume at most x? We will show that these seemingly unrelated questions are actually connected in several different ways.

Florian Pop (University of Pennsylvania)

Galois theory and the Riemann-Zariski space
Recall that the Riemann--Zariski space of a function field K|k is the space of all the k-valuations on K. This is a birational invariant of the function field K|k, and it has a "nice" geometric interpretation. We will indicate a group theoretic procedure by which one can recover a good portion of this space from the pro-q Galois theory of K, where q is any prime number different from char(k). This result is one of the key facts used in recovering function fields from their Galois theory (which is the subject of the so called birational anabelian geometry).

Luis Ribes (Carleton University)

Minicourse: Introduction to Profinite Groups
This short course will cover basic concepts and methods in profinite groups. It will contain some important motivating examples for the development of the theory, connections with other parts of mathematics and some applications. This will provide some background for the rest of the courses as well as some hint to topics not covered in the workshop.

Benjamin Steinberg (Carleton University)

Profinite groups in automata and semigroup theory
Recently surprising connections between the theory of profinite groups and the theory of automata and finite semigroups have emerged. Reutenauer and Pin conjectured in 1990 that if H_1,...,H_n are finitely generated subgroups of a finite rank free group F, then the product H_1...H_n is a closed subset of F in the profinite topology. Their motivation for considering this conjecture is that they showed that it implied a then long-standing conjecture of Rhodes about finite semigroups. They also showed that a positive solution to this conjecture leads to an algorithm for computing the closure of rational subset of the free group in this topology.

Ribes and Zalesskii established this conjecture using the theory of profinite groups acting on profinite trees. Ash independently established
the conjecture via semigroup theoretic methods at approximately the same time. It was observed by the speaker that one of Ash's results can be directly translated into the Ribes and Zalesskii product theorem and so the two results are essentially one and the same.

Ribes and Zalesskii also proved, motivated by a semigroup theoretic question of Pin, that if V is a variety of finite groups closed under extension and if H_1,...,H_n are pro V-closed finitely generated subgroups of the free group F, then H_1...H_n is pro-V closed.

In the last several years, Margolis, Sapir, Weil, the speaker and Auinger (amongst others) have found some very strong connections
between Stallings Foldings, the pro-V topologies on a free group, the geometry of profinite Cayley graphs, freely indexed profinite groups and problems in finite semigroup and automata theory.

John Wilson (Oxford University)

Separability properties of groups
A group G is called conjugacy separable if any two non-conjugate elements map to non-conjugate elements in some finite quotient of G; and G is called subgroup separable if failure of an element to lie in a given finitely generated subgroup can again be detected by passage to some finite quotient. These are two of the best known separability properties of groups; like all separability properties they are statements about the richness of the collection of finite quotients of a group and can be reformulated for residually finite groups G as assertions that certain subsets of G coincide with the intersections with G of their closures in the profinite completion of G. Separability properties also have relevance to algorithmic problems for groups (for the two properties above, the conjugacy problem and the generalized word problem). Very diverse techniques have been used to establish separability properties of groups. These three lectures will illustrate some of the ideas that have played a role in the proofs of separability properties of soluble groups, free products, and branch groups.

Daniel Wise (McGill University)

Residual Finiteness in Geometric Group Theory
In my talk I will survey known results and problems about residual finiteness in groups that naturally arise in Geometric Group Theory.