## Property $$(T)$$ and $$\widetilde{A}_ 2$$ groups.(English)Zbl 0792.43002

We show that each group $$\Gamma$$ in a class of finitely generated groups introduced in two recent papers by D. I. Cartwright, A. M. Mantero, T. Steger and A. Zappa [Geom. Dedicata 47, No. 2, 143-166, 167-223 (1993; Zbl 0784.51010 and Zbl 0784.51011)], 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 Questions 1 and 2 on p. 133 of P. de la Harpe and A. Valette “La propriété ($$T$$) de Kazhdan pour les groupes localement compacts” (Astérisque 175, 1989; Zbl 0759.22001).

### MSC:

 43A35 Positive definite functions on groups, semigroups, etc. 43A90 Harmonic analysis and spherical functions 51E24 Buildings and the geometry of diagrams 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 22E50 Representations of Lie and linear algebraic groups over local fields

### Citations:

Zbl 0784.51010; Zbl 0784.51011; Zbl 0759.22001
Full Text:

### References:

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