Workshop: Groups of dynamical origin
2019, January 21-25, Instituto de Matemáticas, UNAM,
Ciudad de México
Damien Gaboriau (lecture): Introduction to measured group theory
Measured Group Theory studies groups through their measurable actions. It
investigates classification, rigidity and invariants of (actions
of) discrete countable or locally compact groups and the ways these are
encoded in the orbit structure of the actions.
The roots of the subject can be found in ergodic theory and operator
Some milestones are Ornstein-Weiss' theorem on amenability and Zimmer's work
on super-rigidity for lattices in higher rank Lie groups.
The subject gained a new trend when Gromov suggested measured group theory
as a relative to geometric group theory and as a common framework
to better understand the family of (cocompact and non-cocompact) lattices in
a fixed Lie group.
In the last 20 years, measured group theory has known dramatic developments
with the discovery of several new invariants such as cost and L2 Betti
the discovery of several new rigidity phenomena such as Popa's
super-rigidity for Bernoulli shift actions.
It has also established new and sometimes quite unexpected connections
between various areas, including arithmetic groups, geometric group theory,
Neumann algebras, asymptotic group theory, Borel dynamics and combinatorics,
percolation and graph convergence.
I will survey this theory with some digressions depending on the knowledge
of the audience.
Thomas Koberda (lecture):
Regularity of groups acting on the circle
There is a rich interplay between the degree of regularity of a
group action on the circle and the allowable algebraic structure of the
group. In this series of lectures, I will outline some highlights of this
theory, culminating in a construction due to Kim and myself of groups of
every possible critical regularity α ∈ [1,∞).
Tianyi Zheng (lecture): Random walks and isoperimetry on groups
The study of random walks and isoperimetry on discrete groups goes
back to the works of Varopoulos in the early 80s. We will explain some basic
isoperimetric inequalities, connections to behavior of random walks and
tools such as comparison techniques for proving such inequalities. We will
describe several constructions of groups with prescribed isoperimetric
profiles. In the last lecture we will explain how isoperimetric
considerations on induced Schreier graphs play a role in finding efficient
probability measures with non-trivial Poisson boundary on Grigorchuk groups.
François Dahmani: Lamplighters among interval exchange
(joint with V. Guirardel and Koji Fujiwara)
In studying the possible subgroups of the groups IET of all Interval
Exchange Transformations, one can focus on the solvable ones.
It turns out that, although the torsion-free solvable subgroups of IET are
not so interesting, constuctions of the type of lamplighters produces many
(uncountably many) non-isomorphic solvable subgroups, containing a lot of
Bertrand Deroin: Locally discrete groups of diffeomorphisms
of the circle
I will present the classification of locally discrete finitely generated
groups of analytic diffeomorphisms of the circle having the expansion
property. If time permits I will survey the current knowledge on this
problem when expansion does not hold.
Rostislav Grigorchuk: On the question: "Can one hear the
shape of a group?" and Hulancki type theorem for graphs
In my talk I will address the famous question of M.Kac (traced back to
L.Bers and A.Weyl) "Can one hear the shape of a drum?" in the context of
groups viewed as geometric objects. I will explain why the answer in NO
in a strong sense: there is a continuum family of 4-generated pairwise
not quasi-isometric groups with the same spectrum of discrete Laplacian.
Moreover, each of these groups has an uncountable family of amenable
covering groups with the same spectrum.
The arguments will be based
on the construction by the speaker of groups of intermediate growth
(between polynomial and exponential), and the result in the spectral
theory of graphs which somehow is related to the famous Hulanicki
criterion of amenability of groups in terms of weak containment of
unitary representations. The talk is based on joint results with A.Dudko
and with T.Nagnibeda and A.Peres.
David Kerr: Dynamical alternating groups, property Γ,
and inner amenability
I will show that the alternating group of a topologically free action of a
infinite amenable group on the Cantor set has property Γ (and in
is inner amenable) and that there are large classes of such groups which are
simple, finitely generated, and nonamenable. This is joint work with Robin
Nicolás Matte Bon: Homomorphisms between groups of
Given a class of groups, on can ask to describe all possible homomorphisms
between groups in the class.
I will explain the answer to this question for various classes of groups of
dynamical origin: topological full groups of Cantor minimal systems, a class
groups of interval exchange transformations, the Higman-Thompson groups and
their higher-dimensional cousins.
After doing so, I will explain how these results are deduced from a common
theorem, stated in the language of pseudogroups and étale groupoids.
Volodymyr Nekrashevych: Tail groupoids of Bratteli diagrams
and group theory
Topological full groups of tail groupoids of Bratteli diagrams (a.k.a.
AF groupoids) form a well-known class of locally finite groups. We
will discuss how small modifications of these groupoids produce groups
with interesting properties: infinite finitely generated torsion
groups, groups of intermediate growth, non-elementary amenable groups,
etc.. The precise conditions on the modifications producing these
properties are not well understood, and several open questions will be
Jean Raimbault: Some applications of invariant random subgroups
Invariant random subgroups of a topological (e.g. discrete) group are
conjugation-invariant Borel probability measures on its Chabauty space of
closed subgroups. They are thus in some sense "dynamical systems of
group-theoretical origin". Their study in various contexts proves to be
quite inspiring, and they have some spectacular applications to group theory
and geometry. After a general introduction I will shortly discuss their
known properties in the setting of Lie groups and their discrete subgroups,
and outline some applications (most of this is joint work with many people,
and some work where I was not involved).
Cristobal Rivas: Positive cones in finitely generated groups
Abstract: A positive cone in a group is the set of group elements which are
positive in a total, left-multiplication invariant order of the group.
Naturally, the set of all positive cones in a group is a compact set, and a
basic question is to try to understand isolated points in that space and
groups supporting such isolated orders.
In this talk, I will review some of the results regarding the above
questions, and try to relate/compare them to regularity properties of the
languages describing such positive cones. We will put special attention when
the underlying group is word-hyperbolic.
Olga Salazar Díaz:
On subgroups of Thompson's group V
Thompson's group V has a large amount of subgroups. We consider
V as a universal algebra, with its group operations, plus the two
unary shift operations s0 and s1
considered by Geoghegan and Guzman . The subgroups of V which
are closed under these additional operations, are called subalgebras by
Lawson , and shift invariant subgroups by . In , it is shown
that there are exactly three subgroups of V that contain
F. We will discuss subalgebras of V that do not contain
F. This is joint work with Fernando Guzmán.
 R. Geoghegan and
F. Guzmán, Associativity and Thompson's group, Contemporary
Mathematics 394 (2006), 113-135
M.V. Lawson, A class of subgroups of Thompson's group V, Semigroup Forum
75 (2007), 241-252.
Strong amenability and the infinite conjugacy class property
A group is said to be strongly amenable if each of
its proximal topological actions has a fixed point. We show that
a finitely generated group is strongly amenable if and only if it
is virtually nilpotent. More generally, a countable discrete group
is strongly amenable if and only if none of its nontrivial quotients have the
infinite conjugacy class property.
Joshua Frisch and Pooya Vahidi Ferdowsi.