Workshop on Kazhdan's Property (T)
August 25-27, 2004

to be held at the University of Ottawa

Abstracts

---

Wednesday, August 25

Bachir Bekka
Introduction to Property (T)
We will introduce Property (T) according to Kazhdan's original definition, in terms of the unitary dual of a topological group. We will draw the first consequences and compare property (T) to amenability. We will present the first non-trivial examples of groups with Property (T).

Alain Valette
Property (T) and affine actions. I
1-cohomology with values in unitary representations. Affine isometric actions. Characterization of 1-coboundaries. The fixed point property for affine actions. Property (T) is equivalent to the fixed point property (i.e. the Delorme-Guichardet theorem)

Claire Anantharaman-Delaroche
Type II₁ factors in relation with group and ergodic theory
The first talk will give the basic definitions concerning type II₁-factors as well as some historical backgrounds. Several invariants will be introduced. The accent will be put on examples constructed from discrete groups and discrete measured equivalence relations.

Talia Fernos
New examples of group pairs with Kazhdan's Relative Property (T).
Relative property (T) has recently been rediscovered as a useful property in its own right. S. Popa (2004) used relative property (T) to construct examples of $II_1$ factors with rigid cartan subalgebra inclusion. A. Navas (2004) showed that relative property (T) group pairs acting on the circle by $C^2$ diffeomorphisms are trivial, in a suitable sense. The scarcity of examples of group pairs with relative property (T) motivated the following theorem: Let $\Gamma$ be a finitely generated group. If there exists a homomorphism $f : \Gamma \to SL_n(R)$ such that the Zariski-closure (over $\mathbb R$) of $f(\Gamma)$ is nonamenable then there exists a discrete Abelian group A of finite $Q$-rank such that $\Gamma$ acts on A by automorphisms and the corresponding group pair $(\Gamma \ltimes A, A)$ has relative property (T). The proof is constructive. The main ingredients are Furstenberg's celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.

---

Thursday, August 26

Bachir Bekka
Applications of Property (T)
As a first striking application, we will present the by now classical construction of expander graphs by Margulis. We then discuss the solution by Margulis and Sullivan of the Banach-Ruziewicz theorem about uniqueness of rotation-invariant means on the unit sphere. Following Lubotzky-Pak, we show the relevance of Property (T) for the so-called Product Replacement Algorithm in computational group theory.

Alain Valette
Property (T) and affine actions. II
Actions on trees and on the circle. Reduced 1-cohomology. Shalom's characterization of property (T). First applications.

Talia Fernos/ Steffan Vaes
There is no universal II₁ factor
I present the new proof of Nicoara, Popa and Sasyk of Ozawa's result on the non-existence of a universal II₁ factor : a (separable) II₁ factor that contains any (separable) II₁factor as a subfactor. The proof consists in constructing a concrete continuous family M_t of II₁ factors and uses property (T) to show that a separable II₁ factor only contains countably many M_t as a subfactor.

Claire Anantharaman-Delaroche
Property (T) for type II₁ factors.
Property (T) for type II₁ factors will be introduced. It will be shown that the inner automorphism group of such a factor is open and that its fundamental group is countable.

---

Friday, August 27

Bachir Bekka
Some new developments
We will discuss the spectral criterion for groups acting on 2-dimensional complexes (Garland, Zuk), following the proof of Gromov as exposed by Ghys. Finally, we will present a few non locally compact groups which have property (T): the loop group of SL_n(C) for n>2 (Shalom) and the unitary group of a Hilbert space.

Alain Valette
Property (T) and affine actions. III
Finite presentability and property (T). Harmonic maps and property (T) for Sp(n,1)

Claire Anantharaman-Delaroche
(HT) type II₁ factors
Relative property (T) and (H) (Haagerup approximation property) will be defined, giving rise to the notion of (HT) Cartan subalgebra. We shall give an idea of the proof of the following result of Popa: two (HT) Cartan subalgebras of a type II₁ factor are conjugate by an inner automorphism. Some examples and consequences will conclude the talk.

back to top

Back to workshop main page