Property (T) and Kazhdan constants for discrete groups
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 \textit{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 \textit{W. Ballmann} and \textit{J. Swiatkowski} [Geom. Funct. Anal. 7, 615--645 (1997; Zbl 0897.22007)] and by \textit{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 \textit{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.
scientific article; zbMATH DE number 1973269