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

10 juin 2024, 09:00 → 14 juin 2024, 18:00 Europe/Paris

Amphithéâtre Hermite / Darboux (Institut Henri Poincaré)

Group Actions and Rigidity: Around the Zimmer Program

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

June 10 to 14, 2024 - IHP, Paris

Analyzing actions of semisimple Lie groups and their lattices has been a main stay of rigidity in dynamics and geometry, and has been studied since Zimmer’s call to action in the 1980s. It ties together ergodic theory and smooth dynamics, geometry and Lie theory, via geometric structures, coarse geometry, order structures, measure and cocycle rigidity amongst others.  In turn it has motivated major problems in those areas.

The workshop will emphasize advances and connections between smooth dynamics and the Zimmer program over the last decade, such as the work on hyperbolic actions of higher rank abelian groups. It  brought about major advances, especially breakthroughs in the Zimmer conjecture about actions in low dimensions, the classification of lattice actions on tori and nilmanifolds in the presence of Anosov elements, and recently on arbitrary manifolds, on conformal Lorentz structures as well as local rigidity results for actions of lattices on boundaries. Crucially it will feature new tools developed.  

Many exciting problems remain, especially assuming invariant dynamical or geometric structures for the action, or equivalently studying such structures with lots of symmetry. 

Speakers:

• Danijela Damjanovic (KTH)
• Tim De Laat (University of Münster)
• Bassam Fayad (University of Maryland)
• Charles Frances (University of Strasbourg)
• Cyril Houdayer (ENS Paris)
• Mitul Islam (MPI Leipzig)
• Homin Lee (Northwestern University)
• Karin Melnick (University of Luxembourg)
• Vincent Pecastaing (Université Côte d'Azur)
• Federico Rodriguez-Hertz (PennState University)
• Victoria Sadovskaya (PennState University)
• Sven Sandfeldt (KTH)
• Alp Uzman (University of Utah)
• Wouter Van Limbeek (University of Illinois Chicago)
• Kurt Vinhage (University of Utah)
• Zhenqi Wang (Michigan State University)
• Amanda Wilkens (University of Texas)
• Abdelghani Zeghib (ENS Lyon)

Organsing committe:

• Aaron Brown (Northwestern University)
• David Fisher (Rice University)
• Karin Melnick (University of Luxembourg)
• Vincent Pecastaing (Université Côte d'Azur)

 

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lundi 10 juin

09:00 → 09:30 Registration/Welcome coffee

09:30 → 10:30 On global rigidity of Anosov G-actions

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.

Orateur: Danijela Damjanovic (KTH)

10:30 → 11:00 Coffee Break

11:00 → 12:00 Constructing super-expanders from actions of higher rank lattices

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 review some
recent constructions of (usual) expanders, for which we do not know
whether they are super-expanders.

Orateur: Tim De Laat (University of Münster)

12:00 → 14:00 Lunch Break

14:00 → 15:00 Arithmeticity for Smooth Maximal Rank Positive Entropy Actions of R^k

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 a constant time change of the suspension of some action of Z^k on the (k+1)-torus or the (k+1)-torus modulo {id,-id} by affine automorphisms with linear parts hyperbolic. Further, the constructed conjugacy has certain smoothness properties. This in particular answers a problem and a conjecture from a prequel paper of Katok and Rodriguez Hertz, joint with B. Kalinin [Ann. of Math. (2) 174 (2011), no. 1, 361–400; MR2811602].

Orateur: Alp Uzman (University of Utah)

15:00 → 15:30 Coffee Break

15:30 → 16:30 Zimmer's embedding theorem for tractor solutions and conformal actions of simple groups on closed pseudo-Riemannian manifolds

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 associated vector bundles for connections which can encode a very wide array of geometric PDEs. An application for the conformal-to-Einstein tractor connection is a rigidity theorem for conformal actions of SU(p',q') on closed (p,q)-pseudo-Riemannian manifolds in the real-analytic setting: 2p' <= p+1 and if 2p'=p+1, then the metric is conformally flat. This is work in progress with K. Neusser.

Orateur: Karin Melnick (University of Luxembourg)

16:40 → 17:40 Higher rank lattice actions with positive entropy

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.

Orateur: Homin Lee (Northwestern University)

mardi 11 juin

09:30 → 10:30 Continuity of a measurable conjugacy between linear cocycles

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, we bootstrap regularity of a conjugacy between an Anosov toral automorphism and its perturbation.

Orateur: Victoria Sadovskaya (PennState University)

10:30 → 11:00 Coffee Break

11:00 → 12:00 Higher-rank lattices actions on conformal structures

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 specific Lorentzian case.

Orateur: Vincent Pecastaing (Université Côte d'Azur)

12:00 → 12:10 Group Photo

12:10 → 17:45 Free afternoon

17:45 → 18:00 Doors open. Bâtiment Perrin - 2nd floor - Salon Emmy Noether

18:00 → 21:00 Cocktail Dinner Party. Bâtiment Perrin - 2nd floor - Salon Emmy Noether

mercredi 12 juin

09:30 → 10:30 Systems with rank one factors

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.

Orateur: Kurt Vinhage (University of Utah)

10:30 → 11:00 Coffee Break

11:00 → 12:00 Non tame cocycle rigidity above affine unipotent abelian actions on the torus

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 that when one generator of the action is of step less or equal to 2 then cocycle rigidity with tame solutions holds. Moreover, Damjanovic, Fayad and Saprykina
proved that in this case almost cocycles also have almost solutions (with a low regularity control on the error), and from there concluded KAM-rigidity of these actions.

In a joint work with S. Durham, we find examples of affine $\mathbb Z^2$-actions on the torus above which smooth cocycle rigidity holds but is not tame. The linear part of the action is generated by unipotent matrices of step 3. Our examples show that KAM-rigidity for higher rank actions by affine unipotent toral actions does not hold in general when no element of the actions is of step less or equal to 2.

Orateur: Bassam Fayad (University of Maryland)

12:00 → 14:00 Lunch Break

14:00 → 15:00 Rigidity of lattice actions on boundaries

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 uniform lattices by showing that any small deformation is semi-conjugate to the natural action. This is joint work with Chris Connell, Thang Nguyen, and Ralf Spatzier. Time permitting, I will discuss how local rigidity fails when we replace the Furstenberg boundary with the visual boundary of the symmetric space.

Orateur: Mitul Islam (MPI Leipzig)

15:00 → 15:30 Coffee Break

15:30 → 16:30 Poisson–Voronoi tessellations and fixed price in higher rank

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 to prove a result on the relationship between the volume of a manifold and the number of generators of its fundamental group. We give some intuition for the proof. No prior knowledge on Poisson point processes, fixed price, or higher rank will be assumed.

Orateur: Amanda Wilkens (University of Texas)

jeudi 13 juin

09:30 → 10:30 Rigidity of commensurators and its applications

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.

Orateur: Wouter Van Limbeek (University of Illinois Chicago)

10:30 → 11:00 Coffee Break

11:00 → 12:00 Global smooth rigidity for toral automorphisms

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 $C^\infty$.
This is a joint work with B. Kalinin and V Sadovskaya.

Orateur: Zhenqi Wang (Michigan State University)

12:00 → 14:00 Lunch Break

14:00 → 15:00 Rigidity of some higher rank partially hyperbolic actions

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. In particular, I will discuss global rigidity of abelian and higher rank lattice actions that contain one fibered partially hyperbolic element.

Orateur: Sven Sandfeldt (KTH)

15:00 → 15:30 Coffee Break

15:30 → 16:30 Strong primeness for equivalence relations arising from Zariski dense subgroups

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 extends and strengthens Zimmer's primeness result for equivalence relations arising from actions of lattices in simple Lie groups (1981). The key novelty in our approach relies on a combination of ergodic theory of algebraic group actions and Popa's intertwining theory for equivalence relations.

Orateur: Cyril Houdayer (ENS Paris)

16:40 → 17:40 Lichnerowicz conjecture(s)

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.

Orateur: Abdelghani Zeghib (ENS Lyon)

vendredi 14 juin

09:30 → 10:30 Rigidity for boundary actions and classification in low dimensions

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.

Orateur: Federico Rodriguez-Hertz (PennState University)

10:30 → 11:00 Coffee Break

11:00 → 12:00 Coarse embeddings and group actions preserving rigid geometric structures

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.

Orateur: Charles Frances (University of Strasbourg)