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On property (T) for discrete groups. (English) Zbl 1007.22011

Burger, Marc (ed.) et al., Rigidity in dynamics and geometry. Contributions from the programme Ergodic theory, geometric rigidity and number theory, Isaac Newton Institute for the Mathematical Sciences, Cambridge, UK, January 5-July 7, 2000. Berlin: Springer. 473-482 (2002).
Let \(\Gamma\) be a finite symmetric (i.e. \(S = S^{-1}\)) set and suppose that the identity element \(e\) does not belong to \(S\). The associated graph \(L(S)\) (Def. 3.1) is defined with the vertices \(\{s; s\in S\}\) and the edges \(\{(s,s'); s,s',s^{-1}s' \in S\}\). The degree of a vertex \(s\in S\) is \(\operatorname {deg}(s) =\) # edges adjacent to \(s\), the Laplace operator \(\Delta\) is defined as a nonnegative, selfadjoint operator on \(\ell^2(L(S),\operatorname {deg})\) acting on functions \(f\in \ell^2(L(S),\operatorname {deg})\) satisfying \[ \Delta f(s) = f(s) - \frac{1}{\operatorname {deg}(s)}\sum_{s'\sim s} f(s'), \] where \(s' \sim s\) iff the vertex \(s'\) is adjacent to \(s\). The main result (Theorem 3.2): Let \(\lambda_1(L(S))\) be the smallest non-zero eigenvalue of \(\Delta\), \(\lambda_1(L(S)) > \frac{1}{2}\), then \(\Gamma\) has Kazhdan’s property (T). Moreover, \(\frac{2}{\sqrt{3}}(2 - \frac{1}{\lambda_1(L(S))})\) is a Kazhdan constant with respect to the set \(S\). This condition applies to some lattices for which property (T) was known and gives a new elementary proof (in Section 3). In §4 the author proves that using this condition one can prove that random groups in the sense of Gromov are infinite, hyperbolic and have property (T) (Theorems 4.1-4.3, Corollary 4.4).
For the entire collection see [Zbl 0987.00036].

MSC:

22D10 Unitary representations of locally compact groups
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