Introduction in .NET Attach Code-39 in .NET Introduction

Introduction using visual studio .net tomake barcode 39 for web,windows application Data Capacity of QR Code the group was ess VS .NET barcode code39 entially an algebraic group of the appropriate kind or a lattice in such a group. In particular, there were only countably many known examples of countable groups with Property (T).

Gromov has announced that a non-elementary hyperbolic group has uncountably many pairwise non-isomorphic quotient groups with all elements of nite order; see Corollary 5.5.E of [Gromo 87], as well as [IvaOl 98] and [Ozawa 04].

If moreover has Property (T), for example if is a cocompact lattice in Sp(n, 1) for some n 2, all its quotients have Property (T). Let us digress to mention that this gave new examples (even if not the rst ones) of non-amenable groups without free subgroups.21 In particular, there exist groups with Property (T) which are not nitely presented; this answers a question of [Kazhd 67].

A rst concrete example is SL3 (Fp [X ]), a group which appears in [Behr 79]; it is isomorphic to the lattice SL3 (Fp [X 1 ]) in the locally compact group SL3 (Fp ((X ))). A second example, shown to us by Cornulier, is Sp4 (Z[1/p]) Z[1/p]4 ; see Theorem 3.4.

2. There are also examples of nitely presented groups with Property (T) which are non-Hop an [Cornu 07]. For a topological space, there is a natural topology on the space of its closed subspaces, which has been considered by Vietoris (1922), Michael (1951), Fell (1962), and others; see for example [Engel 89], as well as Section 4.

F and 12.C in [Kechr 95]. For a locally compact group (possibly discrete), the Vietoris Fell topology induces a topology on the space of closed subgroups that we like to call the Chabauty topology; the original reference is [Chaba 50], and there is an account from another point of view in [Bou Int2].

As a nice example, it is known that the space of closed subgroups of R 2 is a 4-sphere [HubPo 79]. In particular, for an integer m 1, the space Gm of normal subgroups of the non-abelian free group Fm on a set Sm of m generators, with the Chabauty topology, is a totally disconnected compact metric space. There is a natural bijection between the set of normal subgroups of Fm and the set of marked groups with m generators, namely the set of groups given together with an ordered set of m generators (up to isomorphisms of marked groups).

Thus Gm is also known as the space of marked groups with m generators. Here is a sample genericity result, from [Champ 91] (see also [Ol sh 92], [Champ 00], and [Ghys 04]). For m 2, let H be the subset of Gm de ned by torsion-free non-elementary hyperbolic groups.

Then the closure H contains a. 21 This answers a question which can reasonably be attributed to von Neumann. Twisting the question, several .net vs 2010 3 of 9 people but not von Neumann! have imagined a conjecture that groups without non-abelian free subgroups should be amenable. But the conjecture was wrong, as these and other examples show.

. Introduction dense G consisti ANSI/AIM Code 39 for .NET ng of pairs ( , S) with an in nite group having Property (T) and S a generating set. Here is another result: in Gm , the subset of Kazhdan groups is open (Theorem 6.

7 in [Shal 00a]). A completely different family of examples of groups with Property (T) is provided by many of the Kac Moody groups as shown in [DymJa 00] and [DymJa 02]. More information on Kac Moody lattices can be found in [CarGa 99], [R my 99], [R my 02] and [BadSh 06].

Random groups A model for random groups is a family (Xt )t , where each Xt is a nite set of presentations of groups and where the parameter t is either a positive integer or a positive real number. Given a Property (P) of countable groups and a value t of the parameter, let A(P, t) denote the quotient of the number of groups with Property (P) appearing in Xt by the total number of elements in Xt . Say Property (P) is generic for the model if, for any > 0, there exists t0 such that A(P, t) 1 for all t t0 .

(This is a rather unsatisfactory de nition since it does not capture what makes a model good or interesting .) There are several models for random groups currently in use, for which Property (T) has been shown to be generic. Historically, the rst models appeared in [Gromo 87, Item 0.

2.A]; another model is proposed in the last chapter of [Gromo 93], where Gromov asks whether it makes Property (T) generic. Positive results are established in [Champ 91], [Ol sh 92], [Champ 95], [Gromo 00] (see in particular page 158), [Zuk 03], [Olliv 04], and [Ghys 04].

There is a review in [Olliv 05]. Finite presentations with Property (T), examples beyond locally compact groups, and other new examples Other types of arguments independent of Lie theory, including analysis of particular presentations, can be used to study groups with Property (T). Here is a typical criterion.

Let be the fundamental group of a nite simplicial 2-complex X with the two following properties: (i) each vertex and each edge is contained in a triangle, and (ii) the link of each vertex is connected. For each vertex v of X , let v denote the smallest positive eigenvalue of the combinatorial Laplacian on the link of v. Here is the result: if v + w > 1 for each pair (v, w) of adjacent vertices of X , then has Property (T).

See [Zuk 96], [BalSw 97], [Pansu 98], [Wang 98], and [Zuk 03]. Proofs are strongly inspired by [Garla 73] and [Borel 73]; see also [Matsu 62] and [Picho 03]. Garland s paper has been an inspiration for much more work, including [DymJa 00] and [DymJa 02].

There are related spectral criteria for groups acting on A2 -buildings [CaMlS 93]..
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