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**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.

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.

Reviewer: Bachir Bekka (Metz)

### MSC:

22D10 | Unitary representations of locally compact groups |

22E40 | Discrete subgroups of Lie groups |

51E24 | Buildings and the geometry of diagrams |