The algebraization of Kazhdan’s property (T)

Abstract

Abstract. We present the surge of activity since 2005, around what we call the algebraic (as contrasted with the geometric) approach to Kazhdan's property (T). The discussion includes also an announcement of a recent result (March 2006) regarding property (T) for linear groups over arbitrary finitely generated rings.

Subspace arrangements and property T

“…Usually this is proved using the representation theory of the ambient Lie group [20], but this approach does not produce any bounds for the Kazhdan constants of this groups. In the last ten years several algebraic methods for proving property T have been developed [13], [18], [19], [23], [24]. One main advantage of these methods is that they provide explicit bounds for the Kazhdan constants of these groups, another is that these methods are applicable in a more general setting.…”

The bi-embeddability relation for finitely generated groups II

“…In this section, making use of results of Gao [8] and Shalom [21], we will prove that the isomorphism relation on the space G kaz of Kazhdan groups is weakly universal. It should be pointed out that the basic strategy of our proof is based on induces an associated Borel action on the powerset P(Z 2 ), defined by…”

“…in [25], Williams constructed a subspace X of the space G f g of finitely generated groups such that ∼ = X is smooth and ≈ em X is countable universal.) However, in Section 4, making use of results of Gao [8] and Shalom [21], we will prove the following result. Theorem 1.5.…”

Examples of hyperlinear groups without factorization property

“…First of all, A. Suslin proved in [Sus77] that the group of elementary 3 3-matrices EL 3 .F p OEt; t 1 / coincides with SL 3 .F p OEt; t 1 / (see for example Proposition 5.4 in [Lam06]). Secondly, Y. Shalom showed in [Sha06], Theorem 1.1, that for any finitely generated and commutative ring R, the group EL n .R/ is a Kazhdan group for n 2 C dim R, where dim denotes the Krull dimension of the ring R. has the relative Kazhdan property, meaning that every 1-cocycle on the crossed product will be bounded on F p OEt; t 1 2 . Let now W G !…”