Searcharxiv⌕ Search

arXiv subjects

Uri Bader

Publications and source records attributed to Uri Bader.

At least 19 recordsLinked to original sources
arXiv2026

On the unitary cohomology of semisimple groups

We study the continuous homology and cohomology of semisimple Lie groups with coefficients in arbitrary unitary representations. The case of irreducible representations was determined by Vogan and Zuckerman; we focus on reducible representations. The (co)homology splits into its Hausdorff and torsion parts. The Hausdorff part is governed by containment of irreducible cohomological representations. We show that the torsion part is governed by weak containment of irreducible cohomological representations. Precisely, we show that a unitary representation admits non-zero torsion if and only if there is a cohomological point which is not isolated in its support. We discuss in length examples of rank-$1$ groups, completely determining unitary cohomology for the group $\mathrm{SO}^\circ(n,1)$. For a simple Lie group, the first and fourth named authors showed that the first degree in which it obtains non-trivial cohomology for some unitary representation with no invariant vectors is related to the rank of the group. We discuss the analogous question regarding torsion cohomology, and show that the corresponding first degree could be much higher: in the presence of property (T), it is bounded below by the square root of the dimension of the symmetric space. A technical device that we use is the restriction to well chosen dense subgroups which satisfy finiteness properties, which we call cohomological witnesses. We combine it with results on the unitary cohomology of groups which satisfy finiteness properties. These results are of an independent interest.

math.GR↗
arXiv2025

Higher property T and below-rank phenomena of lattices

The purpose of this paper is twofold. We explore higher property T as an abstract group-theoretic property. In particular, we provide new operator-algebraic characterizations of higher property T. Then we turn to lattices in semisimple Lie groups. We relate higher property T to other cohomological, rigidity and geometric phenomena below the real rank. The second part outlines a conjectural framework that unifies these aspects and reviews recent advances.

math.GR↗
arXiv2025

Lyapunov spectrum via boundary theory I

This paper is concerned with the Lyapunov spectrum for measurable cocycles over an ergodic pmp system taking values in semi-simple real Lie groups. We prove simplicity of the Lyapunov spectrum and its continuity under certain perturbations for a class systems that includes many familiar examples. Our framework uses some soft qualitative assumptions, and does not rely on symbolic dynamics. We use ideas from boundary theory that appear in the study of super-rigidity to deduce our results. This gives a new perspective even on the most studied case of random matrix products. The current paper introduces the general framework and contains the proofs of the main results and some basic examples. In a follow up paper we discuss further examples.

math.DS↗
arXiv2024

Spectral gap for products and a strong normal subgroup theorem

We establish a general spectral gap theorem for actions of products of groups which may replace Kazhdan's property (T) in various situations. As a main application, we prove that a confined subgroup of an irreducible lattice in a higher rank semisimple Lie group is of finite index. This significantly strengthens the classical normal subgroup theorem of Margulis and removes the property (T) assumption from the recent counterpart result of Fraczyk and Gelander. We further show that any confined discrete subgroup of a higher rank semisimple Lie group satisfying a certain irreducibility condition is an irreducible lattice. This implies a variant of the Stuck-Zimmer conjecture under a strong irreducibility assumption of the action.

math.GR↗
arXiv2023

Compact Median Algebras are $\mu$-Boundaries in a unique way

We consider group actions on compact median algebras. We show that, given a generating probability measure $\mu$ on the acting group and under suitable conditions on the median algebra, it could be realized in a unique way as a $\mu$-boundary in the sense of Furstenberg. Along the way, we prove some structural results.

math.GR↗
arXiv2023

Higher Kazhdan property and unitary cohomology of arithmetic groups

Notions of higher Kazhdan property can be defined in terms of vanishing of unitary group cohomology in higher degrees. Garland's theorem for simple groups over non-archimedean fields provides the first examples of a higher Kazhdan property. We prove a version of Garland's theorem for simple Lie groups and their lattices. We generalize theorems of Borel and Borel-Yang about the invariance of the cohomology of lattices in semisimple Lie groups and adelic groups by improving the stability range and allowing for arbitrary unitary representations as coefficients. A novelty of our approach is the use of methods from geometric group theory and -- in the case of rank 1 -- from Clozel's work on the spectral gap property.

math.RT↗
arXiv2023

Geometric representations of group actions

We study equivariant morphisms from zero dimensional schemes to varieties and show that, under suitable assumptions, all such morphisms factor via a canonical one. We relate the above to Algebraic Representations of Ergodic Actions.

math.AG↗
arXiv2022

Homomorphic images of algebraic groups

We study topological group theoretic properties of algebraic groups over local fields. In particular, we find conditions under which such groups have closed images under arbitrary continuous homomorphisms into arbitrary topological groups.

math.GR↗
arXiv2022

Charmenability and Stiffness of Arithmetic Groups

We characterize charmenability among arithmetic groups and deduce dichotomy statements pertaining normal subgroups, characters, dynamics, representations and associated operator algebras. We do this by studying the stationary dynamics on the space of characters of the amenable radical, and in particular we establish stiffness: any stationary probability measure is invariant. This generalizes a classical result of Furstenberg for dynamics on the torus. Under a higher rank assumption, we show that any action on the space of characters of a finitely generated virtually nilpotent group is stiff.

math.GR↗
arXiv2022

Characterising acylindrical hyperbolicity via permutation actions

We characterise acylindrical hyperbolicity of a group in terms of properties of an action of the group on a set (without any extra structure). In particular, this applies to the action of the group on itself by left multiplication, as well as the action on a (full measure subset of the) Furstenberg-Poisson boundary.

math.GR↗
arXiv2021

Charmenability of higher rank arithmetic groups

We complete the study of characters on higher rank semisimple lattices initiated in [BH19,BBHP20], the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary characteristics. More precisely, we investigate dynamical properties of the conjugation action of such lattices on their space of positive definite functions. Our main results deal with the existence and the classification of characters from which we derive applications to topological dynamics, ergodic theory, unitary representations and operator algebras. Our key theorem is an extension of the noncommutative Nevo-Zimmer structure theorem obtained in [BH19] to the case of simple algebraic groups defined over arbitrary local fields. We also deduce a noncommutative analogue of Margulis' factor theorem for von Neumann subalgebras of the noncommutative Poisson boundary of higher rank arithmetic groups.

math.OA↗
arXiv2021

Balanced Measures on Compact Median Algebras

We initiate a systematic investigation of group actions on compact medain algebras via the corresponding dynamics on their spaces of measures. We show that a probability measure which is invariant under a natural push forward operation must be a uniform measure on a cube and use this to show that every amenable group action on a locally convex compact median algebra fixed a sub-cube.

math.GN↗
arXiv2020

Charmenability of arithmetic groups of product type

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees.

math.GR↗