Lyon

A group acting on a regular rooted tree has the congruence subgroup property if every subgroup of finite index contains a level stabilizer. The congruence subgroup problem then asks to quantitatively describe the kernel of the surjection from the profinite completion to the topological closure as a subgroup of the automorphism group of the tree. We will study the congruence subgroup property for a family of branch groups whose construction generalizes that of the Hanoi Towers group, which models the game “The Towers of Hanoi".

 

Globally Valued Fields, studied jointly with E. Hrushovski, are a formalism for fields in which the sum formula for valuations holds, such as number fields or function fields of curves. They form an elementary class (in continuous first order logic), and model-theoretic questions regarding this class give rise to difficult yet fascinating geometric questions.
I intend to present « Lyon school » approach to studying GVFs. This consists of reducing as much as possible to local considerations, among other things via the "fullness" axiom.
 

In joint work with T. Tsankov we study a (yet other) point at which model theory and dynamics intersect. On the one hand, a (metric) aleph_0-categorical structure is determined, up to bi-interpretability, by its automorphism group, while on the other hand, such automorphism groups are exactly the Roelcke precompact ones. One can further identify formulae on the one hand with Roelcke-continuous functions on the other hand, and similarly stable formulae with WAP functions, providing an easy tool for proving that a group is Roelcke precompact and for calculating its Roelcke/WAP compactification. Model-theoretic techniques, transposed in this manner into the topological realm, allow one to prove further that if R(G) = W(G); then G is totally minimal.

I shall present a geometric property valid in many Hrushovski
amalgamation constructions, relative CM-triviality, and derive
consequences on definable groups: modulo their centre they are already
products of groups interpretable in the initial theories used for the
construction. For the bad field constructed in this way, I shall
moreover classify all interpretable groups up to isogeny.

( IN: LADY MARGARET HALL)

As part of the Conference on Geometric Model Theory in honour of Professor Boris Zilber

A group is called boundedly generated if it is the product of a finite sequence of its cyclic subgroups. Bounded generation is a property possessed by finitely generated abelian groups and by some other linear groups.

Apparently it was not known before whether all boundedly generated groups are linear. Another question about such groups has also been open for a while: If a torsion-free group $G$ has a finite sequence of generators $a_1,\dotsc,a_n$ such that every element of $G$ can be written in a unique way as $a_1^{k_1}\dotsm a_n^{k_n}$, where $k_i\in\mathbb Z$, is it true then that $G$ is virtually polycyclic? (Vasiliy Bludov, Kourovka Notebook, 1995.)

Counterexamples to resolve these two questions have been constructed using small-cancellation method of combinatorial group theory. In particular boundedly generated simple groups have been constructed.

If $G$ is a group and $g$ an element of the derived subgroup $[G,G]$, the commutator length of $g$ is the least positive integer $n$ such that $g$ can be written as a product of $n$ commutators. The commutator width of $G$ is the maximum of the commutator lengths of elements of $[G,G]$. Until 1991, to my knowledge, it has not been known whether there exist simple groups of commutator width greater than $1$. The same question for finite simple groups still remains unsolved. In 1992, Jean Barge and Étienne Ghys showed that the commutator width of certain simple groups of diffeomorphisms is infinite. However, those groups are not finitely generated. Finitely generated infinite simple groups of infinite commutator width can be constructed using "small cancellations." Additionally, finitely generated infinite boundedly simple groups of arbitrarily large (but necessarily finite) commutator width can be constructed in a similar way.

H. Jerome Keisler suggested to associate to each classical structure M a family of "random" structures consisting of random variables with values in M . Viewing the random structures as structures in continuous logic one is able to prove preservation results of various "good" model theoretic properties e.g., stability and dependence, from the original structure to its randomisation. On the other hand, simplicity is not preserved by this construction. The work discussed is mostly due to H.

Jerome Keisler and myself (given enough time I might discuss some applications obtains in joint work with Alex Usvyatsov).