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Zbl 0792.43002
Cartwright, D.I.; Młotkowski, W.; Steger, T.
Property $(T)$ and $\widetilde{A}\sb 2$ groups. (English)
[J] Ann. Inst. Fourier 44, No.1, 213-248 (1994). ISSN 0373-0956; ISSN 1777-5310/e

We show that each group $\Gamma$ in a class of finitely generated groups introduced in two recent papers by {\it D. I. Cartwright, A. M. Mantero, T. Steger} and {\it 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 {\it P. de la Harpe} and {\it A. Valette} ``La propriété ($T$) de Kazhdan pour les groupes localement compacts'' (Astérisque 175, 1989; Zbl 0759.22001).
[D.I.Cartwright (Sydney)]

MSC 2000:: *43A35 Positive definite functions on groups, etc.
43A90 Spherical functions (abstract harmonic analysis)
51E24 Buildings and the geometry of diagrams
43A65 Representations of groups, etc. (abstract harmonic analysis)
22E50 Repres. of Lie and linear algebraic groups over local fields

Keywords: finitely generated groups; Kazhdan's property ($T$); Kazhdan constant

Citations: Zbl 0784.51010; Zbl 0784.51011; Zbl 0759.22001

Cited in: Zbl 0840.22008

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