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.
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...
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...
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...
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.
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,...
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...
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.
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...
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...
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.
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 $C^\infty$....
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.
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.
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.
