Actions of Large Groups, Geometric Structures, and the Zimmer Program

Liste des Contributions

1. On global rigidity of Anosov G-actions

Danijela Damjanovic (KTH)

10/06/2024 09:30

I will present some recent progress on the global rigidity for Anosov actions of semisimple Lie groups of higher rank. This is joint work with Spatzier, Vinhage and Xu.

2. Constructing super-expanders from actions of higher rank lattices

Tim De Laat (University of Münster)

10/06/2024 11:00

Super-expanders are sequences of finite, d-regular graphs that
satisfy some nonlinear form of spectral gap with respect to all
uniformly convex Banach spaces. This notion vastly strengthens the
classical notion of expander. In this talk I will explain some recent
constructions of super-expanders, coming from actions of higher rank
lattices on Banach spaces and on manifolds. I will also...

3. Arithmeticity for Smooth Maximal Rank Positive Entropy Actions of R^k

Alp Uzman (University of Utah)

10/06/2024 14:00

We prove an arithmeticity theorem in the context of nonuniform measure rigidity. Adapting machinery developed by A. Katok and F. Rodriguez Hertz [J. Mod. Dyn. 10 (2016), 135–172; MR3503686] for Z^k systems to R^k systems, we show that any maximal rank positive entropy system on a manifold generated by k>=2 commuting vector fields of regularity C^r for r>1 is measure theoretically isomorphic to...

4. Zimmer's embedding theorem for tractor solutions and conformal actions of simple groups on closed pseudo-Riemannian manifolds

Karin Melnick (University of Luxembourg)

10/06/2024 15:30

Zimmer's embedding theorem concerns actions of connected Lie groups by automorphisms of differential-geometric structures and has yielded important restrictions on which groups can act on a manifold with a given structure. It has a useful version for Cartan geometries which generalizes rather easily to tractor solutions on parabolic-type geometries. Tractor solutions are parallel sections of...

9. Higher rank lattice actions with positive entropy

Homin Lee (Northwestern University)

10/06/2024 16:40

In this talk, I will discuss about smooth higher rank lattice actions on manifolds with positive entropy. From dynamical information, we can detect information on groups and manifolds. For instance, when lattices in SL(n,R) act on an n-dimensional manifold with positive entropy, we can see that the lattice is abstractly commensurable with SL(n,Z).
This is joint work with Aaron Brown.

5. Continuity of a measurable conjugacy between linear cocycles

Victoria Sadovskaya (PennState University)

11/06/2024 09:30

We consider Holder continuous GL(d,R)-valued cocycles over hyperbolic and partially hyperbolic diffeomorphisms. We discuss results on continuity of a measurable conjugacy between two cocycles. We focus on perturbations of constant cocycles and on cocycles with one Lyapunov exponent. We also mention related results on continuity of measurable invariant geometric structures. As an application,...

6. Higher-rank lattices actions on conformal structures

Vincent Pecastaing (Université Côte d'Azur)

11/06/2024 11:00

In this talk I will discuss both Lie groups and lattices actions by conformal transformation of a pseudo-Riemannian manifold, related to the Lorentzian Lichnerowicz' conjecture.
I will first discuss dynamics of SL(2,R)-actions on closed Lorentzian manifolds, and then I will detail recent advances for higher-rank lattices conformal actions, and ongoing works with Thierry Barbot in the...

10. Systems with rank one factors

Kurt Vinhage (University of Utah)

12/06/2024 09:30

Lie group actions with rank one factors have natural families of perturbations arising from modifying each factor independently. I will explain ways to obtain semi-rigid settings, in particular, various characterizations of product systems. Partially based on work with R. Spatzier, and work in-progress with A. Uzman.

11. Non tame cocycle rigidity above affine unipotent abelian actions on the torus

Bassam Fayad (University of Maryland)

12/06/2024 11:00

Cocycle rigidity with tame solutions is a crucial ingredient in KAM theory. We are interested in cocycle rigidity above affine unipotent abelian actions on the torus with Diophantine translation data. We consider unlocked actions whose rank one factors are non vanishing translations (the locked actions do not have any kind of stability).

It follows from Katok and Robinson's observations...

7. Rigidity of lattice actions on boundaries

Mitul Islam (MPI Leipzig)

12/06/2024 14:00

Lattices in linear semi-simple Lie groups have a natural action on their (generalized) Furstenberg boundaries, e.g. a lattice in SL_n(R) acting on the different flag varieties. For such natural actions, I will discuss the local rigidity question---are these actions ‘stable’ under small deformations in the homeomorphism group of the boundary? We will answer this question for higher rank...

8. Poisson–Voronoi tessellations and fixed price in higher rank

Amanda Wilkens (University of Texas)

12/06/2024 15:30

We define and motivate the Poisson point process, which is, informally, a "maximally random" scattering of points in space. We introduce the ideal Poisson--Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT...

12. Rigidity of commensurators and its applications

Wouter Van Limbeek (University of Illinois Chicago)

13/06/2024 09:30

I will discuss a question raised independently by Greenberg and Shalom: Can an infinite discrete subgroup of a simple Lie group have dense commensurator and not be a lattice? I will explain the surprising connections between this question and other long-standing open problems, and discuss recent progress on special cases of the question. This is joint work with (subsets of) Brody, Fisher, and Mj.

13. Global smooth rigidity for toral automorphisms

Zhenqi Wang (Michigan State University)

13/06/2024 11:00

Suppose f is a diffeomorphism on torus whose linearization A is weakly irreducible. Let
H be a conjugacy between f and A. We prove the following: 1 if A is hyperbolic and H is weakly differentiable 2. if A is partially hyperbolic and H is C^1+holder. Then H is C^\infty. Our result shows that the conjugacy in all local and global rigidity results for irreducible A is $C^\infty$....

14. Rigidity of some higher rank partially hyperbolic actions

Sven Sandfeldt (KTH)

13/06/2024 14:00

Smooth rigidity of higher rank abelian and lattice actions with some hyperbolicity has been studied extensively. When the manifold is a nilmanifold, results by Rodriguez Hertz, Wang, and Brown, Rodriguez Hertz, Wang show that: If the action contains an Anosov diffeomorphism then the action is globally rigid. I will discuss rigidity of higher rank partially hyperbolic actions on nilmanifolds....

15. Strong primeness for equivalence relations arising from Zariski dense subgroups

Cyril Houdayer (ENS Paris)

13/06/2024 15:30

In this talk, I will describe an ongoing joint work with Daniel Drimbe in which we show that equivalence relations arising from essentially free ergodic probability measure preserving actions of Zariski dense discrete subgroups of simple algebraic groups are strongly prime. As a consequence, we obtain a unique prime factorization result for direct products of such equivalence relations. This...

16. Lichnerowicz conjecture(s)

Abdelghani Zeghib (ENS Lyon)

13/06/2024 16:40

A priori, the conformal group of a compact Riemannian manifold has no reason to be compact, since it only preserves angles and not distances. A posteriori, however, it turns out that this group is compact, with a single exception: the round sphere! The Lichnerowicz conjecture refers to similar rigidity statements in the cases of pseudo-Riemannian conformal and projective structures.

17. Rigidity for boundary actions and classification in low dimensions

Federico Rodriguez-Hertz (PennState University)

14/06/2024 09:30

The plan of the talk is to describe joint work with A. Brown and Z. Wang on the smooth classification of actions of lattices in SL(n,R) on n-1 dimensional manifolds. The method is also amenable to show rigidity for some boundary actions.

18. Coarse embeddings and group actions preserving rigid geometric structures

Charles Frances (University of Strasbourg)

14/06/2024 11:00

We will see how the notion of coarse embeddings allows to better understand discrete group actions preserving a rigid geometric structure. The focus will mainly be on isometric and conformal actions. In particular, we will discuss a Tits alternative for isometry groups of compact Lorentzian manifolds.

19. Miklós Abért

Miklós Abért (Rényi Institute)