Uri Bader (Weizmann Institute, Rehovot) and Roman Sauer (KIT, Karlsruhe)
Title:
Higher property (T) of arithmetic lattices, applications and conjectures
Abstract:
We prove that arithmetic lattices in a semisimple Lie group G satisfy a higher-degree version of property T below the rank of G. The proof relies on functional analysis and the polynomiality of higher Dehn functions of arithmetic lattices below the rank and avoids any automorphic machinery. We describe applications to the cohomology and stability of arithmetic groups (the latter being joint work with Alex Lubotzky and Shmuel Weinberger). Further, we describe some interesting connections and conjectures that naturally emerge from our work and are related to the spectral gap property and character rigidity.
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Marc Bourdon (U Lille)
Title:
L^p-cohomology of Lie groups
Abstract:
L^p-cohomology is a potentially interesting quasi-isometric invariant. For rank-one simple Lie groups, it has been studied in detail by Pansu. In higher rank, its study is less developed. Motivated by some questions by Gromov, we show that for p large enough, the L^p-cohomology is null in degree larger than the rank and non-null in degree equal to the rank. These results extend also to certain Lie groups of type AN. (Joint work with Bertrand Rémy).
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Martin Bridson (U of Oxford)
Title:
Profinite rigidity, central extensions, and finiteness properties
Abstract:
A finitely generated, residually finite group is profinitely rigid if its set of finite quotients distinguishes it from all other finitely generated, residually finite groups. A few years ago, McReynolds, Reid, Spitler and I proved that certain arithmetic lattices in PSL(2,R) and PSL(2,C) are profinitely rigid. In this talk, I will explain how to extend the ideas from the Fuchsian case to obtain infinite families of Seifert fibre spaces with profinitely rigid fundamental groups.
For certain of these Seifert fibre spaces, the fundamental group G has the property that no other finitely presented, residually finite group has the same finite quotients as G x G, but there are infinitely many non-isomorphic finitely generated subgroups H < G x G that have the same finite quotients as G x G. I will explain the origins of this contrast.
This is joint work with Alan Reid and Ryan Spitler.
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Marc Burger (ETH Zürich)
Title:
Real spectrum and character varieties
Abstract:
We discuss a compactification of the G-character variety of a finitely generated group L that is mindful of the variety's homological properties. Here G is a semisimple real algebraic group and the methods come from real algebraic geometry. We then give applications to structural properties of associated Weyl chamber valued length functions defined on L. When L is a compact surface group and G=PSL(2,R), we discuss properties of the mapping class group action on the corresponding compactification of Teichmueller space.
Joint work with A. Iozzi, A. Parreau, and B. Pozzetti.
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Pierre-Emmanuel Caprace (UC Louvain)
Title:
Hyperbolic locally compact groups of Type I
Abstract:
This talk, based on joint work with Mehrdad Kalantar and Nicolas Monod, concerns the study of the relations between the structure of a locally compact group and the properties of its unitary representations. I will discuss a conjecture on the structure of locally compact groups of Type I, and present an overview of its proof in the special case of hyperbolic locally compact groups containing a cocompact lattice.
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Ilaria Castellano (University of Milan - Bicocca)
Title:
Unimodular t.d.l.c. groups of type FP, Euler characteristic and double coset zeta-functions
Abstract:
Let G be an infinite group. Whenever P is a finitely generated projective kG-module the so-called Hom-Tensor identity is satisfied and it leads to a natural trace function, whence a definition of rank for P, the so-called Hattori-Stallings rank. In 1976, using such a rank, H. Bass defined the Euler characteristic of infinite groups of type FP over the commutative ring k.
A similar approach can be used to define an Euler characteristic for unimodular t.d.l.c. groups of type FP over the rationals. In this context the Euler characteristic is no longer just a number, but a rational multiple of a Haar measure normalized at some compact open subgroup.
In the second part of the talk I will focus on t.d.l.c. groups satisfying a different type of finiteness condition that allow to introduce a formal Dirichlet series for any compact open subgroup. In many cases it happens that such a formal series defines a meromorphic function of the complex plane whose value in -1 is related to the Euler characteristic of the t.d.l.c. group.
Based on joint work with Thomas Weigel and Gianmarco Chinello.
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Yves Cornulier (CNRS, U 1)
Title:
On piecewise homographic groups on the circle
Abstract:
I'll introduce Exel's notion of partial actions and explain how they can be used to show that the group of piecewise homographic transformations of the circle has no infinite subgroup with Kazhdan's property T, and that the group of piecewise affine transformations of the circle has no infinite subgroup with Property FW. The idea is to use the FW property to obtain a model on which the homographic (resp. affine) is preserved, and not only piecewise preserved.
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Cornelia Drutu (U of Oxford)
Title:
Effective equidistribution of expanding horospheres
Abstract: In this talk I will explain how an equivalence between lattice point counting inside dilated d-dimensional ellipsoids (Euclidean counting) and counting of lifts of a closed horosphere that intersect a ball of increasing radius can be used to obtain two types of results: effective equidistribution results for a family of expanding horospheres in a locally symmetric space, and error estimates for counting orbit points of a lattice in a symmetric space, and for counting the number of lifts of a closed horosphere that intersect a ball of large radius.
This is on joint work with N. Peyerimhoff.
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Alejandra Garrido (UA Madrid)
Title:
Detecting free factors in profinite completions
Abstract:
If one is interested in finitely generated residually finite groups, it is natural to ask to what extent it is determined by its profinite completion. Perhaps the boldest question in this area is that attributed to Remeslennikov: "If G has the same profinite completion as a free group of finite rank, must G be isomorphic to that free group?" The answer to this still seems remote, given current knowledge, but some success has been found on variants of it, restricting the type of group G is allowed to be. Another natural variant of Remeslennikov's question is whether a free factor of a group G can be detected from its profinite completion: if G has a subgroup H, whose closure in the profinite completion of G is a profinite free factor, must H be a free factor of G? This question is still hard, with positive known answer only when G itself is a free group.
I will report on joint work with A. Jaikin in which we provide a new proof of the above positive answer and extend it to the case when G is virtually free. The methods may be extended to other classes of groups if some interesting questions are answered on their completed group algebras.
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Tsachik Gelander (Northwestern U)
Title:
Discrete Stationary Random Subgroups and application to discrete subgroups of Lie groups
Abstract:
The notion of invariant random subgroups (IRS) has proven extremely useful during the last decade, particularly to the study of asymptotic invariants of lattices. However, the scope of problems that one can investigate when restricting to invariant measures (on the space of subgroups) is limited. It was recently realised that the notion of stationary random subgroups (SRS), which is much more general, is still extremely powerful and opens up new paths to attacking problems that previously seemed to be out of our reach.
In this talk, I will describe a fundamental inequality concerning the injectivity radius of discrete subgroups of semisimple Lie groups, which was obtained recently (jointly with A. Levit and G.A. Margulis). I will explain some applications to SRS, and in particular the proof (obtained jointly with M. Fraczyk) of a conjecture of Margulis about manifolds of infinite volume which is a far-reaching generalisation of the Normal Subgroup Theorem.
This is based on a joint work with A. Levit and G.A. Margulis and a joint work with M. Fraczyk.
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Mikael de la Salle (CNRS, U Lyon 1)
Title:
Actions of higher-rank lattices on uniformly convex Banach spaces
Abstract:
Elaborating on a recent breakthrough by Izhar Oppenheim who dealt with groups containing locally SL_4, I will explain that every action by isometries of a lattice in a higher-rank connected simple Lie group on a uniformly convex Banach space has a global fixed point. This confirms a conjecture by Bader, Furman, Gelander and Monod. Joint work with Tim de Laat.
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Jean Lécureux (U Paris Saclay)
Title:
Normal Subgroup Theorem for Ã_2 lattices
Abstract:
Buildings of type Ã_2 are classically associated to groups such as PGL(3,K) where K is a local field. However, there are many more buildings, some of them with automorphism groups which are not so well understood. In this talk I will explain the following result : if G is a cocompact lattice of an Ã_2 building, then any non-trivial normal subgroup of G is of finite index. This generalizes a theorem of Margulis in the classical case, and the main ideas of the proof are similar. This is a joint work with U. Bader and A. Furman.
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Waltraud Lederle (UC Louvain)
Title:
Boomerang subgroups
Abstract:
The set of subgroups of a locally compact group, with an appropriate topology, is called its Chabauty space. The group acts on it by conjugation. We call a subgroup a boomerang subgroup if, considered as element in the Chabauty space, it is strongly recurrent in a specific sense. Poincaré recurrence then implies that every IRS gives full measure to the set of boomerang subgroups. Therefore, if a statement is almost surely true for every IRS, we can hope that it is deterministically true for every boomerang subgroup. We present a few results in this direction.
This is joint work with Yair Glasner.
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Nicolas Monod (EPFL)
Title:
Gelfand pairs, Iwasawa decompositions and geometric consequences
Abstract:
We will explain the classical notion of Gelfand pairs and give some examples such as semi-simple groups. We then prove that every Gelfand pair admits some sort of "Iwasawa decomposition". Although this result is not very hard to prove, I find it surprising. It motivates some fascinating questions and conjectures, for instance related to the talk of Pierre-Emmanuel in this conference. Moreover, earlier joint work with Pierre-Emmanuel implies that we can give a complete geometric description of Gelfand pairs in non-positive curvature.
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Shahar Mozes (Hebrew U, Jerusalem)
Title:
Locally testable codes
A locally testable code is an error correcting code that has a property-tester who when receiving a word reads q bits of it that are randomly chosen, and rejects the word with probability proportional to its distance from the code. The parameter q is called the locality of the tester.
In a joint work with Irit Dinur, Shai Evra, Ron Livne and Alex Lubotzky, we introduce and use a certain family of complexes associated with groups and which have good expansion properties to construct an infinite family of locally testable codes which have constant rate, constant distance and constant locality.
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Roman Sauer (KIT, Karlsruhe) and Uri Bader (Weizmann Institute, Rehovot)
Title:
Higher property (T) of arithmetic lattices, applications and conjectures
Abstract:
We prove that arithmetic lattices in a semisimple Lie group G satisfy a higher-degree version of property T below the rank of G. The proof relies on functional analysis and the polynomiality of higher Dehn functions of arithmetic lattices below the rank and avoids any automorphic machinery. We describe applications to the cohomology and stability of arithmetic groups (the latter being joint work with Alex Lubotzky and Shmuel Weinberger). Further, we describe some interesting connections and conjectures that naturally emerge from our work and are related to the spectral gap property and character rigidity.
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Lancelot Semal (UC Louvain)
Title:
Radu groups and the CCR conjecture on trees
Abstract:
CCR groups are the locally compact groups whose irreducible representations range into the compact operators. Those groups are of type I and hence have a representation theory that is well behaved with respect to irreducibles. In the 90's, Nebbia conjectured that any group of tree automorphisms acting transitively on the tree's boundary must be CCR. This conjecture was already verified for key examples, such as rank one semi-simple algebraic groups over local fields or Burger-Mozes groups. The purpose of this talk will be to talk about the much more recent contribution to this conjecture. In 2017, Radu provided a classification and an explicit description of all the groups of tree automorphisms acting transitively on the boundary and whose local action at each vertex contains the alternating group. Relying on his description, we were able to classify their irreducible and show that these groups are CCR solving Nebbia's conjecture on trees for a large amount of semi-regular trees. Our purpose is to explain these results.
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Rachel Skipper (ENS Paris)
Title:
Braiding the group of almost automorphisms of a tree
Abstract:
We will discuss how to lift the group of almost automorphisms of a tree (Neretin's group) to the mapping class group of a surface of infinite type, as well as what happens for certain subgroups, e.g. self-similar groups, Higman-Thompson groups, and Roever-Nekrashevych groups.
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Romain Tessera (CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche)
Titre:
Which groups admit a coarse embedding into a hyperbolic space?
Abstract:
I will characterize amenable groups that coarsely embed into a hyperbolic space, and more generally into a direct product of a Hyperbolic space and a Euclidean space.
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Stefan Witzel (Julius Liebig U, Gießen)
Title:
Strong property (T) for Ã_2-lattices
Abstract:
Lafforgue introduced strong property (T) as a strengthening of Kazhdan's property (T) to non-unitary representations and proved that it is satisfied by higher-rank Lie groups and their uniform lattices. Strong property (T) for non-uniform higher-rank arithmetic lattices was proven by de la Salle and subsequently used in the proof of Zimmer's conjecture by Brown, Fisher, Hurtado.
I will speak about the proof of strong property (T) for lattices on Ã_2 buildings that may not be arithmetic. This is joint work with Mikael de la Salle and Jean Lécureux.
