By Bachir Bekka, Pierre de la de la Harpe, Alain Valette

Estate (T) is a tension estate for topological teams, first formulated by means of D. Kazhdan within the mid 1960's with the purpose of demonstrating that an enormous category of lattices are finitely generated. Later advancements have proven that estate (T) performs an immense function in an amazingly huge number of matters, together with discrete subgroups of Lie teams, ergodic idea, random walks, operator algebras, combinatorics, and theoretical desktop technology. This monograph bargains a complete creation to the speculation. It describes the 2 most vital issues of view on estate (T): the 1st makes use of a unitary team illustration technique, and the second one a set element estate for affine isometric activities. through those the authors speak about a number of vital examples and functions to numerous domain names of arithmetic. a close appendix offers a scientific exposition of components of the speculation of team representations which are used to formulate and increase estate (T).

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**Example text**

2 Property (T) and Fell’s topology 35 and a + b = 1. We then have ϕ = aψ + bψ . We claim that either b = 0 or ψ = 0. Indeed, assume by contradiction that b = 0 and ψ (e) = 0. Then ψ (e) = 1, since ϕ(e) = 1 and ψ(e) ≤ 1, ψ (e) ≤ 1. 6 that limi ψi = ψ uniformly on compact subsets of G. 8 shows that ψ is a sum of functions of positive type associated to π0 . 1) and this is a contradiction. Therefore, we have limi ai ψi = ϕ and limi ai = 1. Thus, limi ψi = ϕ in the weak* topology; by Raikov’s Theorem, this holds also uniformly on compact subsets of G.

12). 12 again. Since π0 ⊗ π 0 = π1 ⊕ · · · ⊕ πn , it follows that σi is unitarily equivalent to one of the πk ’s. This is a contradiction to the choice of (σi )i∈I . P. 1]. 3 Compact generation and other consequences The ﬁrst spectacular application of Property (T) is the following result, due to Kazhdan. 1 Let G be a locally compact group with Property (T). Then G is compactly generated. In particular, a discrete group with Property (T) is ﬁnitely generated. 3 Compact generation and other consequences 37 Proof Let C be the set of all open and compactly generated subgroups of G.

Indeed, SLn (Q) is not ﬁnitely generated, since every ﬁnite subset {x1 , . . , xm } of SLn (Q) is contained in SLn (Z[1/N ]), where N is a common multiple of the denominators of the matrix coefﬁcients of x1 , . . , xm . 1. 4 Property (T) for SLn (K), n ≥ 3 Let K be a local ﬁeld . 4, for more details). 6). 4. Some general facts We collect the common ingredients used in the proofs of Property (T) for SLn (K) and Sp2n (K). 1 Let G be a topological group, and let (π , H) be a unitary representation of G with 1G ≺ π.