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with multiplicities 1, q2 + q, q2 + q, 1. In particular, the smallest non-zero eigenvalue of is 1 in .NET Connect Code 39 Extended in .NET with multiplicities 1, q2 + q, q2 + q, 1. In particular, the smallest non-zero eigenvalue of is 1




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with multiplicities 1, q2 + q, q2 + q, 1. In particular, the smallest non-zero eigenvalue of is 1 use .net vs 2010 uss code 39 printing toconnect ansi/aim code 39 for .net Microsoft Office Excel Website q > 1/2. q+1 Proof Recal l that P as well as L have n = q2 + q + 1 elements, so that G(P,L) has 2n vertices. Recall also that every point p P (respectively, every line L) is incident with q + 1 lines (respectively, q + 1 points). With respect to the basis { p : p P} { : has the matrix I 1 q+1 0 At A 0 , L},.

A spectral 3 of 9 for .NET criterion for Property (T). where A is the (n n) matrix (ap )(p, ap = 1 0 ) P L with if p is incident with otherwise. It suf ces to determine the eigenvalues of the matrix B= 0 At A 0 ,. which is the so-called adjacency matrix of the graph G(P,L) . We have B2 = AAt 0 0 At A . For two poi nts p, p P, the entry (p, p ) of AAt is equal to the number of lines incident with both p and p . Similarly, for two lines , L, the entry ( , ) of At A is equal to the number of points incident with both and . Hence AA = A A = .

q+1 1 1 q+1 1 1 1 1 1 1 q+1 1 1 q+1 . . The eigenva lues of AAt = At A are (q + 1)2 , with multiplicity 1, and q, with multiplicity n 1 = q2 + q. Indeed, the vector t (1, 1, , 1) is an eigenvector with eigenvalue (q + 1)2 , and the n 1 linearly independent vectors. (1, 1, 0, barcode code39 for .NET 0, . .

. , 0), t (1, 0, 1, 0, . .

. , 0), . .

. , t (1, 0, 0, . .

. , 0, 1). are eigenve ctors with eigenvalue q. Hence, the eigenvalues of B2 are (q + 1)2 q with multiplicity 2 . with multiplicity 2(q2 + q).. It follow s that the eigenvalues of B are contained in { (q + 1), q}. On the other hand, due to the special structure of the matrix B (the graph G(P,L) being bipartite), the spectrum of B is symmetric about 0. Indeed,.

5.7 Groups acting on A2 -buildings if t (x1 , . . .

, xn , y1 , . . .

, yn ) is an eigenvector of B with eigenvalue , it is straightforward to see that t (x1 , . . .

, xn , y1 , . . .

, yn ) is an eigenvector of B with eigenvalue . Therefore, { (q + 1), q} is the spectrum of B and q q 0, 1 ,1 + ,2 q+1 q+1 is the spectrum of , with multiplicities 1, q2 + q, q2 + q, 1..

Theorem 5.7 .net vs 2010 Code 39 .

7 Let X be an A2 -building and let G be a unimodular locally compact group acting on X . Assume that the stabilisers of the vertices of X are compact and open subgroups of G and that the quotient G\X is nite. Then G has Property (T).

Moreover, ( q 1)2 2 ( q 1)2 + q is a Kazhdan constant for the compact generating set S as in Property (Pvi) of the last section. Proof The rst statement is an immediate consequence of Proposition 5.7.

6 and Theorem 5.5.4.

The second statement follows from Proposition 5.7.6 and Remark 5.

5.3. Example 5.

7.8 Using the previous theorem, we obtain a new proof of Property (T) for SL3 (K), when K is a non-archidemean local eld. Indeed, the natural action of GL3 (K) on the set of all lattices in K 3 is transitive.

This gives rise to a transitive action of GL3 (K) on the A2 -building XK from Example 5.7.5.

There are three SL3 (K)-orbits in XK : the orbit of the equivalence class of the standard lattice L0 = O3 of K 3 , the orbit of the equivalence class of the lattice L1 = g1 L0 and the orbit of the equivalence class of the lattice L2 = g2 L0 , where 1 0 g1 = 0 0 0 1 0 0 1 0 and g2 = 0 0 0 0 0 . . The stabili sers of [L0 ], [L1 ] and [L2 ] in SL3 (K) are the compact subgroups 1 1 SL3 (O), g1 SL3 (O)g1 and g2 SL3 (O)g2 . This shows that the assumptions of Theorem 5.7.

7 are ful lled and SL3 (K) has Property (T). Remark 5.7.

9 The subgroup B of all triangular matrices in SL3 (K) has the same orbits on XK as SL3 (K) and the stabilisers in B of vertices are open and. A spectral ANSI/AIM Code 39 for .NET criterion for Property (T). compact. Ho wever, B does not have Property (T), since it is solvable and noncompact (see Theorem 1.1.

6). This shows that the unimodularity assumption of G in Theorem 5.7.

7 is necessary. Example 5.7.

10 A (discrete) group is called an A2 -group if it acts freely and transitively on the vertices of an A2 -building, and if it induces a cyclic permutation of the type of the vertices. These groups were introduced and studied in [CaMSZ 93]. Some A2 -groups, but not all, can be embedded as cocompact lattices in PGL3 (K) for a non-archimedean local eld K.

It was shown in [CaMlS 93] through a direct computation that an A2 -group has Property (T) in the case where the underlying projective plane is associated to a nite eld. Theorem 5.7.

7 is a generalisation of this result. Moreover, the Kazhdan constant from Theorem 5.7.

7 for the given generating set S of coincides with the one found in [CaMlS 93]; as shown in [CaMlS 93], this is the optimal Kazhdan constant for S in this case. Example 5.7.

11 The following examples of groups described by their presentations and satisfying the spectral criterion in Theorem 5.5.4 are given in [BalSw 97].

Let G be a nite group, and let S be a set of generators of G with e S. Assume that the Cayley graph G = G(G, S) of G has girth at least 6. / (Recall that the girth of a nite graph G is the length of a shortest closed circuit in G.

) Let S . R be a pres 3 of 9 for .NET entation of G. Then the group given by the presentation S { } .

R { 2 } {(s )3 : s S} acts transitively on the vertices of a CAT(0) two-dimensional simplicial complex, with nite stabilisers of the vertices, such that the link at every vertex of X is isomorphic to G. Therefore, if 1 (G) > 1/2, then has Property (T). Examples of nite groups G satisfying the conditions above are the groups PSL2 (F) over a nite eld F with a certain set of generators (see [Lubot 94] or [Sarna 90]).

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