Uri Bader

A fixed point theorem for L1 spaces

We prove a fixed point theorem for a family of Banach spaces including notably L1 and its non-commutative analogues. Several applications are given, e.g. the optimal solution to the “derivation problem” studied since the 1960s.

Simple groups without lattices

We show that the group of almost automorphisms of a d-regular tree does not admit lattices. As far as we know, this is the first such example among (compactly generated) simple locally compact groups.

On the cohomology of weakly almost periodic group representations

We initiate a study of cohomological aspects of weakly almost periodic group representations on Banach spaces, in particular, isometric representations on reflexive Banach spaces. Using the Ryll-Nardzewski fixed point Theorem, we prove a vanishing result for the restriction map (with respect to a subgroup) in the reduced cohomology of weakly periodic representations. Combined with the Alaoglu-Birkhoff decomposition theorem, this generalizes and complements theorems on continuous group cohomology by several authors.

Efficient subdivision in hyperbolic groups and applications

We identify the images of the comparision maps from ordinary homology and Sobolev homology, respectively, to the

On Some Geometric Representations of GL N (𝔬)

l^1

Cohomology of deformations

-homology of a word-hyperbolic group with coefficients in complete normed modules. The underlying idea is that there is a subdivision procedure for singular chains in negatively curved spaces that is much more efficient (in terms of the

Furstenberg maps for CAT(0) targets of finite telescopic dimension

l^1

Rigidity of group actions on homogeneous spaces, III

-norm) than barycentric subdivision. The results of this paper are an important ingredient in a forthcoming proof of the authors that hyperbolic lattices in dimension at least 3 are rigid with respect to integrable measure equivalence. Moreover, we prove a proportionality principle for the simplicial volume of negatively curved manifolds with regard to integrable measure equivalence.

Property (T) and rigidity for actions on Banach spaces

We study a family of complex representations of the group GL n (𝔬), where 𝔬 is the ring of integers of a non-archimedean local field F. These representations occur in the restriction of the Grassmann representation of GL n (F) to its maximal compact subgroup GL n (𝔬). We compute explicitly the transition matrix between a geometric basis of the Hecke algebra associated with the representation and an algebraic basis that consists of its minimal idempotents. The transition matrix involves combinatorial invariants of lattices of submodules of finite 𝔬-modules. The idempotents are p-adic analogs of the multivariable Jacobi polynomials.

Factor and normal subgroup theorems for lattices in products of groups

In this article we study cohomology of a group with coefficients in representations on Banach spaces and its stability under deformations. We show that small, metric deformations of the representation preserve vanishing of cohomology. As applications we obtain deformation theorems for fixed point properties on Banach spaces. In particular, our results yield fixed point theorems for affine actions in which the linear part is not uniformly bounded. Our proofs are effective and allow for quantitative estimates.