# Kazhdan's Property by Bachir Bekka, Pierre de la de la Harpe, Alain Valette

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

Estate (T) is a tension estate for topological teams, first formulated through 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 a tremendous position in an amazingly huge number of topics, together with discrete subgroups of Lie teams, ergodic idea, random walks, operator algebras, combinatorics, and theoretical machine technology. This monograph deals a complete creation to the idea. It describes the 2 most crucial issues of view on estate (T): the 1st makes use of a unitary workforce illustration procedure, and the second one a hard and fast element estate for affine isometric activities. through those the authors speak about a number of vital examples and purposes to a number of domain names of arithmetic. a close appendix presents a scientific exposition of components of the speculation of staff representations which are used to formulate and increase estate (T).

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In particular, this establishes a conjecture from [VerKa–82]: for G compactly generated, Property (T) is equivalent to the vanishing of H 1 (G, π) for all irreducible representations π of G. One more consequence is that any countable group with Property (T) is a quotient of a ﬁnitely presented group with Property (T). This can be viewed as the deﬁnitive answer to the question written as “Hypothesis 1” in [Kazhd–67] (see the paper in its original Russian version, since the English translation does not make sense at this point).

The next lemma will be used to establish invariance of vectors under copies of SL2 (K). 9 Let K be a non-discrete topological ﬁeld, and let (π , H) be a unitary representation of SL2 (K). Let ξ be a vector in H which is invariant under the subgroup N . Then ξ is invariant under SL2 (K). Proof It sufﬁces to show that ξ is invariant under A. 7 and from Mautner’s Lemma that ξ is invariant under N − and, hence, under SL2 (K) since N ∪ N − generates SL2 (K). To show the A-invariance of ξ , consider the function of positive type ϕ on SL2 (K) deﬁned by ϕ(x) = π(x)ξ , ξ , x ∈ SL2 (K).

N Hence, B → E(gB) and B → π(g)E(B)π(g −1 ) are projection valued measures associated to the unitary representation x → π(gxg −1 ) of N . By uniqueness, it follows that (∗) π(g)E(B)π(g)−1 = E(gB), for all B ∈ B(N ), g ∈ G. 1, there exists a positive linear functional ϕ on L(H) such that ϕ(I ) = 1, and such that ϕ(π(g)T ) = ϕ(T π(g)) = ϕ(T ) for all g ∈ G and T ∈ L(H). In particular, ϕ(T ) = ϕ(π(g)T π(g −1 )), for all g ∈ G, T ∈ L(H). Deﬁne m : B(N ) → R by m(B) = ϕ(E(B)), B ∈ B(N ). Then m is a mean on B(N ).