Group Actions and Rigidity: Around the Zimmer Program, Paris

Apr 15, 2024, 8:30 AM → Jul 5, 2024, 8:30 PM Europe/Paris

Amphitheater Hermite / Darboux (IHP)

Thematic trimester programme with short courses, seminars and workshops

April 15 to July 5, 2024 at the Institut Henri Poincaré, Paris

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Presentation of the trimester

This term focuses on rigidity of group actions with some focus on issues that arise out of Zimmer's groundbreaking work in the 1980's. Areas of interest range from local and global rigidity of group actions to special rigidity phenomena in small dimensions to symmetry groups of geometric structures to measure rigidity results for quite general group actions with hyperbolicity.

 

This broad area has seen a very large number of dramatic breakthroughs in recent years. The goal of the term is to bring together experts and people in adjacent areas to share ideas and techniques and to foster additional breakthroughs. There will also be a significant component of the term dedicated to introducing younger participants to the broad area.

Introductory School:

April 15 to 19, 2024: Rigidity, Dynamics and Geometric Structures, held at CIRM, Marseille

Workshops: 

April 29 to Saturday, May 4, 2024: Low Dimensional Actions (IHP, Paris)

May 27 to 31, 2024: Group Actions with Hyperbolicity and Measure Rigidity (IHP, Paris)

June 10 to 14, 2024: Actions of Large Groups, Geometric Structures, and the Zimmer Program (IHP, Paris)

Mini-courses:

Mini-courses will be held in amphitheater Darboux in IHP. The standard format will be three 2h long lectures during one week for each mini-course. 

Titles and abstracts of mini-courses: click here.

• Bertrand Deroin (Université de Cergy & CNRS): Week of May 6
• Thang Nguyen (Florida State University): Week of May 13
• Romain Dujardin (Sorbonne Université): Week of May 20
• Cyril Houdayer (ENS Paris): Week of June 3
• Tim de Laat (University of Münster): Week of June 3
• Karin Melnick (University of Luxembourg): Week of June 17
• Kurt Vinhage (University of Utah): Week of June 17

Weekly seminar

In weeks not containing workshops there will be a research seminar on Mondays and Wednesdays at 3pm. Webpage of the seminar: link

Organising committe:

• Aaron Brown (Northwestern University)
• David Fisher (Rice University)
• Kathryn Mann (Cornell University)
• Vincent Pecastaing (Université Côte d’Azur)
• Ralf Spatzier (University of Michigan)

Scientific committe:

• Serge Cantat (Université de Rennes I)
• Alex Eskin (University of Chicago)
• Étienne Ghys (ENS Lyon)
• Federico Rodriguez Hertz (Pennsylvania State University)
• Amie Wilkinson (University of Chicago)

 

Back to the thematic trimester main page 

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Programme coordinated by the Centre Emile Borel at IHP

 

Sorbonne Université, CNRS

Poster T2 2024 GAR.pdf

Wed, April 24

3:00 PM → 6:00 PM Finite Fourier transform by Yves Benoist

Abstract: On a cyclic group of prime order, the non-principal Dirichlet characters together with their Fourier transforms are functions that vanish at 0 and whose modulus is constant outside 0. Do there exist other functions with these properties? I will explain the history of this question and give a positive answer for non-safe primes. The proof relies on Floer homology.

Mon, April 29

9:00 AM → 9:20 AM Registration/ Welcome coffee

9:20 AM → 9:30 AM Opening Speech by the Deputy Director of IHP, Prof Domonique Mouhanna

9:30 AM → 5:30 PM Workshop 'Low dimensional actions' - April 29 to May 4 - 1st day

Tue, April 30

9:30 AM → 5:30 PM Workshop 'Low dimensional actions' - April 29 to May 4 - 2nd day

6:00 PM → 9:00 PM Cocktail dinner party

Wed, May 1

9:00 AM → 5:30 PM Public holiday

Thu, May 2

9:30 AM → 5:30 PM Workshop 'Low dimensional actions' - April 29 to May 4 - 3rd day

Fri, May 3

9:30 AM → 5:30 PM Workshop 'Low dimensional actions' - April 29 to May 4 - 4th day

Sat, May 4

9:30 AM → 12:00 PM Workshop 'Low dimensional actions' - April 29 to May 4 - 5th day

Mon, May 6

10:00 AM → 12:00 PM Orderability of lattices in semi-simple Lie groups. Mini course by Bertrand Deroin, Université de Cergy & CNRS

Abstract : There are several interesting ways of ordering groups, but we will focus here on the notion of left-orderability, namely the existence of a total ordering on the group invariant by left translations. In the lectures, we will provide the complete classification of left-orders on irreducible lattices in a semi-simple Lie group of real rank at least 2, based on work of Deroin and Hurtado: such orders exist only if the Lie group has a
-factor and are obtained by pull-back. This answers positively conjectures of Ghys and Witte in the beginning of the 90's. Techniques for the proof include 1) random walks on the group of homeomorphism of the real line, 2) group of almost-periodic homeomorphism of the real line 3) Furstenberg's Poisson formula, 4) dynamics of Cartan subgroups. If time permit, we will discuss the situation in the rank one case.

3:00 PM → 4:30 PM Seminar by Colin Guillarmou, Université Paris-Saclay

Tue, May 7

10:00 AM → 12:00 PM Orderability of lattices in semi-simple Lie groups. Mini course-2 by Bertrand Deroin

2:00 PM → 4:00 PM Orderability of lattices in semi-simple Lie groups. Mini course-3 by Bertrand Deroin

Mon, May 13

10:00 AM → 12:00 PM Local rigidity of higher rank lattice actions on flag manifolds. Mini course by Thang Nguyen, Florida State University

Abstract: The main focus of the minicourse is to study local rigidity of standard actions of uniform lattices in higher rank Lie groups on flag manifolds within the space of actions by homeomorphisms. The rigidity in more differentiable actions often relies on the stability of uniform hyperbolic systems. In the setting of actions by homeomorphisms, We explore a much weaker form of stability derived from coarse geometry. We will go over how dynamics and coarse geometry intertwine to achieve a desired local rigidity. At the end of the minicourse, we will extend our discussion to other natural boundary actions.
The outlined of topics in the minicourse include:
Local rigidity: problems and results.
Stability of uniform hyperbolic systems and uniformly transverse foliations.
Geometry and coarse geometry of symmetric spaces.
Weyl chamber flows.
Suspension and developing.
Barycenter and its regularity.
Proof of local rigidity for actions on flag manifolds.
Actions on geodesic boundaries and some related actions.

2:00 PM → 4:00 PM Seminar by Barbara Schapira, Université de Rennes

Tue, May 14

10:00 AM → 12:00 PM Local rigidity of higher rank lattice actions on flag manifolds. Mini course-2 by Thang Nguyen

Wed, May 15

10:00 AM → 12:00 PM Local rigidity of higher rank lattice actions on flag manifolds. Mini course-3 by Thang Nguyen

3:00 PM → 4:30 PM Seminar by Bruno Duchesne, Université Paris Saclay

Thu, May 16

3:00 PM → 4:30 PM Weekly ‘informal seminar. Talk about entropy and Pesin’s formula by Homin Lee

Tue, May 21

10:00 AM → 12:00 PM Dynamics of automorphism groups of real and complex projective surfaces. Mini course by Romain Dujardin, Sorbonne Université

Abstract: In a series of papers in collaboration with Serge Cantat, we have studied group actions by automorphisms on compact complex surfaces. In these lectures, I will review our results and discuss a number of meaningful examples. Depending on time, the following results will be discussed: action on cohomology and the notion of non-elementary group, classification of stationary and invariant measures, (non-)Zariski density of finite orbits, asymptotic distribution of infinite orbits.

Wed, May 22

10:00 AM → 12:00 PM Dynamics of automorphism groups of real and complex projective surfaces. Mini course-2 by Romain Dujardin

3:00 PM → 4:30 PM Seminar by Thibault Lefeuvre, Sorbonne Université

Thu, May 23

3:00 PM → 4:30 PM Anosov diffeomorphism and an argument of Cawley by Alp Uzman. Weekly ‘informal seminar

Fri, May 24

10:00 AM → 12:00 PM Dynamics of automorphism groups of real and complex projective surfaces. Mini course-3 by Romain Dujardin

Mon, May 27

9:00 AM → 5:30 PM Workshop on Group Actions with Hyperbolicity and Measure Rigidity

Tue, May 28

9:30 AM → 5:00 PM Workshop on Group Actions with Hyperbolicity and Measure Rigidity

Wed, May 29

9:30 AM → 12:00 PM Workshop on Group Actions with Hyperbolicity and Measure Rigidity

Thu, May 30

9:30 AM → 5:30 PM Workshop on Group Actions with Hyperbolicity and Measure Rigidity

Fri, May 31

9:30 AM → 5:30 PM Workshop on Group Actions with Hyperbolicity and Measure Rigidity

Mon, June 3

10:00 AM → 12:00 PM Noncommutative rigidity of higher rank lattices by Cyril Houdayer, ENS Paris

Abstract: In this lecture series, I will survey recent results regarding the dynamics of positive definite functions and character rigidity of higher rank lattices. I will discuss the notion of noncommutative boundary structure and give the proof of the noncommutative Nevo-Zimmer structure theorem for ergodic actions of higher rank lattices on von Neumann algebras due to Boutonnet-Houdayer (2019). I will present several applications to ergodic theory, topological dynamics, unitary representation theory and operator algebras. I will also present a noncommutative analogue of Margulis' factor theorem for higher rank lattices and explain its relevance towards Connes’ celebrated rigidity conjecture.

2:00 PM → 4:00 PM Isometric actions of higher rank groups on uniformly convex Banach spaces by Tim de Laat, University of Münster

Abstract: Fixed point properties for isometric group actions on Banach
spaces are fundamental rigidity properties which can be viewed as Banach
space versions of Kazhdan's property (T). After an introduction to this
topic, I will explain that all isometric actions of higher rank
semisimple Lie/algebraic groups (over arbitrary local fields) and their
lattices on arbitrary uniformly convex Banach spaces have a fixed point.
This theorem confirmed a long-standing conjecture of Bader, Furman,
Gelander and Monod. The proof of this result is a combination of the
"Archimedean case", i.e. Lie groups, and the "non-Archimedean case",
i.e. semisimple groups over non-Archimedean local fields. These two
cases are proved with fundamentally different methods. The focus of this
mini-course will lie on the Lie group side of the above theorem, which I
recently proved in collaboration with Mikael de la Salle, vastly
generalizing work of Oppenheim. I will explain the strategy and key
steps of the proof and highlight certain new techniques.

Tue, June 4

10:00 AM → 12:00 PM Noncommutative rigidity of higher rank lattices by Cyril Houdayer

2:00 PM → 4:00 PM Isometric actions of higher rank groups on uniformly convex Banach spaces by Tim de Laat

Wed, June 5

9:30 AM → 11:30 AM Noncommutative rigidity of higher rank lattices by Cyril Houdayer

1:00 PM → 3:00 PM Isometric actions of higher rank groups on uniformly convex Banach spaces by Tim de Laat

3:30 PM → 5:30 PM Seminar by Jean Lecureux, Université Paris-Saclay

Thu, June 6

3:00 PM → 4:30 PM Arithmetic equidistribution and applications to dynamics by Ethan Cohen. Weekly ‘informal seminar

Mon, June 10

9:00 AM → 4:30 PM Workshop on Actions of large groups, geometric structures, and the Zimmer program from June 10 to 14

Tue, June 11

9:30 AM → 5:30 PM Workshop on Actions of large groups, geometric structures, and the Zimmer program from June 10 to 14

Wed, June 12

9:30 AM → 12:00 PM Workshop on Actions of large groups, geometric structures, and the Zimmer program from June 10 to 14

Thu, June 13

9:30 AM → 5:30 PM Workshop on Actions of large groups, geometric structures, and the Zimmer program from June 10 to 14

Fri, June 14

9:30 AM → 12:00 PM Workshop on Actions of large groups, geometric structures, and the Zimmer program from June 10 to 14

Mon, June 17

10:00 AM → 12:00 PM Rigid Geometric Structures and their Automorphism Groups by Karin Melnick, University of Luxembourg

Abstract: This mini-course will be an introduction to rigid differential-geometric structures and their automorphisms. It will cover some of the basic theory of G-structures of finite type in the sense of Cartan and a variety of examples. A few major theorems on automorphisms of such structures, such as Zimmer’s Embedding Theorem and Gromov’s Open-Dense Theorem, will be presented, along with some significant applications. In the last section of the course, the plan is to present a 21st-century rigidity result for conformal actions of nilpotent groups on compact pseudo-Riemannian manifolds.

2:30 PM → 4:30 PM Parameterized accessibility classes and rigidity by Kurt Vinhage, University of Utah

Abstract: In this mini-course, I will describe the way in which accessibility classes of partially hyperbolic systems interact with rigidity properties. In particular, I will explain a setting where the action of a certain topological group parameterizes accessibility classes, and methods to control this action to produce rigidity theorems. Time permitting, I will explain how common rigidity assumptions can place us in this special setting, and how sub-accessibility classes play an important role in understanding the non-rigid setting. Based on joint works and ideas developed with D. Damjanovic, R. Spatzier, A. Uzman, Z.J. Wang, and D. Xu.

Tue, June 18

10:00 AM → 12:00 PM Rigid Geometric Structures and their Automorphism Groups by Karin Melnick

2:00 PM → 4:00 PM Parameterized accessibility classes and rigidity by Kurt Vinhage

Wed, June 19

9:30 AM → 11:30 AM Rigid Geometric Structures and their Automorphism Groups by Karin Melnick

1:00 PM → 3:00 PM Parameterized accessibility classes and rigidity by Kurt Vinhage

3:30 PM → 5:30 PM Seminar by Laurent Stolovitch. Local rigidity of actions of isometries on compact real analytic Riemannian manifolds

Abstract: In this work in collaboration with Z. Zhao (Nice), we consider analytic perturbations of isometries of an analytic Riemannian manifold $M$. We prove that, under some conditions, a finitely presented group of such small enough perturbations is analytically conjugate on $M$ to the same group of isometry it is a perturbation of. Our result relies on a ``Diophantine-like" condition, relating the actions of the isometry group and the eigenvalues of the Laplace-Beltrami operator. Our result generalizes Arnold-Herman's theorem about diffeomorphisms of the circle that are small perturbations of rotations.

Thu, June 20

3:00 PM → 4:30 PM Circles in the boundary of hyperbolic groups; application to Anosov representations by Blandine Gailiay. Weekly ‘informal seminar

Mon, June 24

2:00 PM → 3:00 PM EQUIDISTRIBUTION OF PERIODIC TORI. Seminar by Nguyen-Thi DANG, Université Paris Saclay

Abstract: Bowen and Margulis independently proved in the 70s that closed geodesics on compact hyper- bolic surfaces equidistribute towards the measure of maximal entropy. From a homogeneous dynamics point of view, this measure is the quotient of the Haar measure on PSL(2,R) modulo some discrete cocompact sugroup.
In a joint work with Jialun Li, we investigate the higher rank setting of this problem by taking a higher rank Lie group (like SL(d,R) for d ≥ 3) and by studying the dynamical properties of geodesic flows in higher rank : the so-called Weyl chamber flows and their induced diagonal action. We obtain an equidistribution formula of periodic tori (instead of closed orbits of the geodesic flow).

3:30 PM → 4:30 PM Rigidity of fibrewise Anosov diffeomorphisms on principal torus bundles. Seminar by Danyu Zhang, University of Luxembourg

Abstract: A fibrewise Anosov diffeomorphism is a fibre-preserving diffeomorphism on a fibre bundle that preserves an invariant stable and unstable splitting along the fibres. We discuss a result showing that every fibrewise Anosov diffeomorphism on a principal torus bundle is topologically conjugate to a map that is linear in the fibres, by a conjugacy that fibres over the identity on the base and is homotopic to the identity.