Property (T) and Kazhdan constants for discrete groups. (English) Zbl 1036.22004

A topological group \(\Gamma\) has Kazhdan’s property (T), if every unitary representation of \(G\) which almost has invariant vectors, actually has a non-zero invariant vector. Property (T) is a powerful tool with applications ranging from graph theory to operator algebras. As a rule, proving property (T) for an (infinite) group is usually difficult.
In this very interesting paper, a new criterion for property (T) for a discrete group \(\Gamma\) is given. This criterion is in terms of a presentation of \(\Gamma;\) it involves a spectral property of a finite graph associated with the given presentation. Previously, after fundamental work by H. Garland [Ann. Math. 97, 375–423 (1973; Zbl 0262.22010)], a criterion of this type, formulated in a cohomological context, was established by the author [C. R. Acad. Sci. Paris 323, 453–458 (1996; Zbl 0858.22007)], by W. Ballmann and J. Swiatkowski [Geom. Funct. Anal. 7, 615–645 (1997; Zbl 0897.22007)] and by P. Pansu [Bull. Soc. Math. Fr. 126, 107–139 (1998; Zbl 0933.22009)].
Using this criterion, the author proves that in some setting a “generic” group is an infinite hyperbolic group with property (T). This result is related to M. Gromov’s theory of random groups as developed in [Lond. Math. Soc. Lect. Note Ser. 182, 1–295 (1993; Zbl 0841.20039)] and [Geom. Funct. Anal. 13, 73–146 (2003; Zbl 1122.20021)]. Moreover, it is shown that property (T) is stable under small perturbations of the presentation of the group.


22D10 Unitary representations of locally compact groups
22E40 Discrete subgroups of Lie groups
51E24 Buildings and the geometry of diagrams
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