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Introduction generate, create uss code 39 none in .net projects Microsoft Official Website only if it ANSI/AIM Code 39 for .NET has no simple factor of K-rank one.9 In the general case, S.

P. Wang has determined when G has Property (T) in terms of a Levi decomposition of G (assuming that such a decomposition exists, this being always the case in characteristic zero); see [Wang 82], as well as [Shal 99b] and [Cornu 06d, Corollary 3.2.

6].. Constructio n of expanding graphs and Property (T) for pairs The rst application of the Kazhdan Property outside group theory was the explicit construction by Margulis of remarkable families of nite graphs. In particular, for any degree k 3, there are constructions of families of nite k-regular graphs which are expanders; this means that there exists a so-called isoperimetric constant > 0 such that, in each graph of the family, any nonempty subset A of the set V of vertices is connected to the complementary set by at least min{#A, #(V \ A)} edges. While the existence of such graphs is easily established on probabilistic grounds, explicit constructions require other methods.

A basic idea of [Margu 73] is that, if an in nite group generated by a nite set S has Property (T) and is residually nite, then the nite quotients of have Cayley graphs with respect to S which provide a family of the desired kind. Margulis construction is explicit for the graphs, but does not provide explicit estimates for the isoperimetric constants. Constructions given together with lower bounds for these constants were given later, for example in [GabGa 81]; see also the discussion below on Kazhdan constants.

10 Rather than Property (T) for one group, Margulis used there a formulation for a pair consisting of a group and a subgroup. This Property (T) for pairs, also called relative Property (T), was already important in Kazhdan s paper, even though a name for it appears only in [Margu 82]. It has since become a basic notion, among other reasons for its role in operator algebras, as recognized by Popa.

Recent progress involves de ning Property (T) for a pair consisting of a group and a subset [Cornu 06d]. There is more on Property (T) for pairs and for semidirect products in [Ferno 06], [Shal 99b], [Valet 94], and [Valet 05]..

9 If G is c Code 39 Extended for .NET onnected, simple, and of K-rank one, then G acts properly on its Bruhat Tits building 10 More recently, there has been important work on nding more expanding families of graphs,. [BruTi 72], which is a tree, and it follows that G does not have Property (T).. sometimes w Code 39 Extended for .NET ith optimal or almost optimal constants. We wish to mention the so-called Ramanujan graphs, rst constructed by Lubotzky Phillips Sarnak and Margulis (see the expositions of [Valet 97] and [DaSaV 03]), results of J.

Friedman [Fried 91] based on random techniques, and the zig-zag construction of [ReVaW 02], [AlLuW 01]. Most of these constructions are related to some weak form of Property (T)..

Introduction Group cohom .net vs 2010 Code 39 Full ASCII ology, af ne isometric actions, and Property (FH) Kazhdan s approach to Property (T) was expressed in terms of weak containment of unitary representations. There is an alternative approach involving group cohomology and af ne isometric actions.

In the 1970s, cohomology of groups was a very active subject with (among many others) an in uential paper by Serre. In particular (Item 2.3 in [Serre 71]), he conjectured that H i ( , R) = 0, i {1, .

. . , l 1}, for a cocompact discrete subgroup in an appropriate linear algebraic group G of rank l .

This conjecture was partially solved by Garland in an important paper [Garla 73] which will again play a role in the later history of Property (T). Shortly after, S.P.

Wang [Wang 74] showed that H 1 (G, ) = 0 for a separable locally compact group G with Property (T), where indicates here that the coef cient module of the cohomology is a nite-dimensional Hilbert space on which G acts by a unitary representation. In 1977, Delorme showed that, for a topological group G, Property (T) implies that H 1 (G, ) = 0 for all unitary representations of G [Delor 77]. Previously, for a group G which is locally compact and -compact, Guichardet had shown that the converse holds (see [Guic 72a], even if the expression Property (T) does not appear there, and [Guic 77a]).

Delorme s motivation to study 1-cohomology was the construction of unitary representations by continuous tensor products. A topological group G is said to have Property (FH) if every continuous action of G by af ne isometries on a Hilbert space has a xed point. It is straightforward to check that this property is equivalent to the vanishing of H 1 (G, ) for all unitary representations of G, but this formulation was not standard before Serre used it in talks (unpublished).

Today, we formulate the result of Delorme and Guichardet like this: a -compact11 locally compact group has Property (T) if and only if it has Property (FH). Recall that there is a Property (FA) for groups, the property of having xed points for all actions by automorphisms on trees. It was rst studied in [Serre 74] (see also I.

6 in [Serre 77]); it is implied by Property (FH) [Watat 82]. We make one more remark, in order to resist the temptation of oversimplifying history. Delorme and Guichardet also showed that a -compact locally compact group G has Property (T) if and only if all real-valued continuous.

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