TBA
Organizers: Corentin Correia and Vincent Dumoncel.
Goal: Get a better understanding on topics interacting with geometric group theory
(dynamics, large-scale geometry, representations, operator algebras, logic, among others).
Format: "Mini-courses" intended to be elementary and accessible to a wide audience. They should last between one and two hours.
The working group takes place in
bâtiment Sophie Germain
(usual room to specify),
on Tuesday (sometimes Thursday), around 4pm.
You can also attend it
online.
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TBA
Property (T) is a central rigidity property in analytic group theory.
It can be formulated in many equivalent ways, for instance
as a spectral gap property or in terms of group cohomology.
Ozawa showed a remarkable characterization: Property (T) can be detected
by an equation in the group algebra involving sums of squares.
This was fundamental in proving that Aut(F_n) has
Property (T) for n at least 4 via computer assisted methods.
Recently, there has been increasing interest in higher versions of property (T).
Using group cohomology, there are at least
two possible non-equivalent definitions: ona via vanishing and
one via reducedness of cohomology. It is thus natural to explore
whether these higher analogues can be characterized
in terms of equations à la Ozawa. In this series of lectures we want
to discuss (not in 1-to-1 correspondence with lectures):
Scaling quasi-isometries are a refinment of quasi-isometries and
were introduced recently by Genevois and Tessera as a crucial ingredient
in their quasi-isometric classification of (amenable) lamplighters.
The goal of this series of talks is to introduce this class of maps and
to explain the main steps of the rigidity phenomenon proved
by Genevois and Tessera: under mild assumptions, any quasi-isometry
between lamplighters is scaling. If time permits,
we will also explain elementary applications to subgroups of lamplighters
and to the quasi-isometric rigidity of some iterated wreath products.
Let $K$ be a class of finitely generated structures. Fraïssé isolated conditions for which we can define the Fraïssé limit of $K$, a countable structure $M$ in which all structures in $K$ embeds and which is homogeneous in the sense that any isomorphism between finite substructures of $M$ can be extended to an automorphism of $M$. For instance, the order of the rationals is the Fraïssé limit of the class of finite linear orders, the Rado graph is the Fraïssé limit of the class of finite graphs and the atomless countable Boolean algebra is the Fraïssé limit of the class of finite Boolean algebras. The goal of this talk (and probably the next one) will be to give a formulation and a proof of the Fraïssé correspondence for metric structures, which extends the classical Fraïssé theory. We will be mainly following a proof due to Tsankov, which will be the occasion to introduce central notions in (metric) model theory, like metric structures, (quantifier-free) types, type spaces, ultrahomogeneity or existential closedness.
Median graphs, also known as CAT(0) cube complexes, are interesting mathematical objects : they have an interesting geometry and are very useful in geometric group theory. We will start by defining what median graphs are and take some fundamental examples of median graphs. Later we will study the most important notions relative to median graphs, namely hyperplanes, convex subgraphs and convex hulls, and projections on convex subgraphs.
I will go over the basics of abstract harmonic analysis with an emphasis on the context of permutation groups. Most of the examples of our groups will come from model theory and include S∞ (the group of all permutations of a countable set), Aut(Q) (the order preserving bijections of the rationals) and the isometry group of the Urysohn spaces (universal and homogeneous countable metric spaces for a prescribed distance set). All of the groups mentioned lie far away from the locally compact groups in the spectrum of Polish groups, hence the need for new techniques in place of tools such as the Haar measure in the classical theory. To that aim, I will develop the notion of dissociation: a definition, ways to obtain dissociated groups, rigidity results for dissociated groups. Based, in parts, on a joint work with Colin Jahel and Matthieu Joseph.
The goal of this series of lectures
is to give an overview of Bass-Serre theory and
to discuss some applications of it.
Roughly speaking, Bass-Serre theory gives a generalization
of the following well-known theorem in geometric group theory:
"A group is free iff it acts freely on a tree".
It is a powerful tool to study the structure of a group acting
(non necessarily freely) on a tree, and to get information on its subgroups.
Groups acting on trees arise in many contexts in mathematics.
If a topological space admits an open cover that satisfies suitable conditions,
the action of its fundamental group
on its universal cover leads to an action on a tree.
On the other hand, Van Kampen's theorem stands that this group can be
explicitly calculated using the fundamental groups of the given open sets,
and of their pairwise intersections. In that sense, Bass-Serre theory can be
thought of as a combinatorial generalisation of Van Kampen's theorem.
Following the first chapter of Arbres, amalgames, SL2 from
Jean-Pierre Serre, we will enter into the details of the main
theorems and we will give two applications :
Two groups are orbit equivalent if they admit free probability-measure preserving actions on a standard and atomless probability space which have the same orbits. This equivalence relation is a particular case of measure equivalence, introduced by Gromov as a measured analogue of quasi-isometry. Many rigidity results have been found in the non-amenable world, whereas Ornstein and Weiss proved that amenable groups are always orbit equivalent. During these sessions, I will talk about quantitative strenghtenings of orbit equivalence, introduced by Delabie, Koivisto, Le Maître and Tessera, which capture the geometry of amenable groups (in contrast to Ornstein-Weiss Theorem). I will finish by explaining some methods to build concrete orbit equivalences (using Følner tiling sequences).
Keywords : Odometers, Følner tiling sequences, cofinite equivalence relation
Keywords : Isoperimetric profile, regular embedding
Keywords : Standard probability space, Ornstein-Weiss, L^1 orbit equivalence, volume growth
In this last talk, we will see the links between upper bounds on the return probability and the L2-isoperimetric profile via Nash inequalities.
In this last talk, we will see the links between upper bounds on the return probability and the L2-isoperimetric profile via Nash inequalities.
I will introduce another coarse invariant, closely related to the L2-isoperimetric profile, namely the return probability to the origin. I will introduce such an invariant and prove its links to the L2-isoperimetric profile and Nash inequalities on graphs.
I will start with a brief motivation by discussing about Sobolev inequalities. I will proceed to discuss the notion of isoperimetric profile for graphs as well as proving its invariance under quasi-isometries. I will finish by proving the relationship between the isoperimetric profile and the growth for Cayley graphs (Coulhon Saloff-Coste inequality).
In this talk, we will introduce the uniform Roe algebras of uniform locally finite metric spaces. We will discuss some examples of these algebras and examine their basic properties. We conclude the series of talks with a brief introduction to the rigidity problem of uniform Roe algebras.
In this talk, we will examine two fundamental properties of coarse spaces: property A and coarse embeddability into a Hilbert space. These properties play a crucial role in the coarse Baum-Connes conjecture and the rigidity problem for Roe algebras. We will demonstrate that they are preserved under coarse equivalences and present illustrative examples. Their significance for the rigidity problem will be addressed in the third talk.
In this talk, we will explore the fundamental concepts of coarse geometry and compare them with the framework of geometric group theory introduced by Vincent in the preceding lecture series. We will introduce coarse structures as an abstraction of metric spaces viewed through the lens of large-scale geometry and discuss coarse equivalences, which are maps preserving large-scale properties.
A finitely generated group is naturally a metric space, that can be studied from a large-scale point of view. Quasi-isometries are the maps preserving this structure, and a natural problem is then to determine whether two finitely generated groups are quasi-isometric. We introduce two invariants that provide answers in some cases: the volume growth and the number of ends. The second part will be devoted to the quasi-isometric rigidity of ℤ.