## August 8-11

Workshop on Profinite Groups and Applications

## Carleton University, Ottawa

### Abstracts

**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.