Property $\left(T\right)$ and ${\tilde{A}}_2$ groups

Cartwright, Donald I.; Młotkowski, Wojciech; Steger, Tim

Annales de l'Institut Fourier, Tome 44 (1994) no. 1, pp. 213-248.

Nous montrons que chaque groupe $\Gamma$ dans une classe des groupes introduits dans [2] et [3] possède la propriété (T) de Kazhdan, et nous calculons la constante exacte de Kazhdan par rapport à l’ensemble naturel de ses générateurs. Ceux-ci sont les premiers groupes infinis pour lesquels on montre la propriété (T) sans faire aucun usage de la théorie des groupes semi-simples et de leurs représentations. Aussi, ces groupes sont les premiers pour lesquels la constante exacte de Kazhdan a été calculée. Ceci donne une réponse aux questions 1 et 2, de [9], p. 133.

We show that each group $\Gamma$ in a class of finitely generated groups introduced in [2] and [3] has Kazhdan’s property (T), and calculate the exact Kazhdan constant of $\Gamma$ with respect to its natural set of generators. These are the first infinite groups shown to have property (T) without making essential use of the theory of representations of linear groups, and the first infinite groups with property (T) for which the exact Kazhdan constant has been calculated. These groups therefore provide answers to (in [9]), p. 133, Questions 1 and 2.

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