Kazhdan sets in groups and equidistribution properties

Kazhdan and Haagerup Properties in Algebraic Groups over Local Fields

We prove some results about solvable Lie algebras endowed with a reductive action of a fixed Lie algebra. The first application consists in proving that if g is a Lie algebra over a local field of characteristic zero whose “amenable radical” is not a direct factor, then g contains a subalgebra which is isomorphic to the semidirect product of sl2 by either a nontrivial irreducible representation...

New Kazhdan groups

Introduction A locally compact, second countable group G is called Kazhdan if for any unitary representation of G, the first continuous cohomology group is trivial H 1 ct (G, ρ) = 0. There are several other equivalent definitions, the reader should consult [6], esp. 1.14 and 4.7. For some time now Kazhdan groups have attracted attention. One of the main challenges is to understand them geometri...

Mixing, counting and equidistribution in Lie groups

Kazhdan–lusztig Cells in Infinite Coxeter Groups

Groups defined by presentations of the form 〈s1, . . . , sn | si = 1, (sisj)i,j = 1 (i, j = 1, . . . , n)〉 are called Coxeter groups. The exponents mi,j ∈ N ∪ {∞} form the Coxeter matrix, which characterizes the group up to isomorphism. The Coxeter groups that are most important for applications are the Weyl groups and affine Weyl groups. For example, the symmetric group Sn is isomorphic to the...

Kazhdan Constants of Hyperbolic Groups ∗

Let H be an infinite hyperbolic group with Kazhdan property (T ) and let κ(H,X) denote the Kazhdan constant of H with respect to a generating set X . We prove that infX κ(H,X)= 0, where the infimum is taken over all finite generating sets of H . In particular, this gives an answer to a Lubotzky question.