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

Cartwright, D. I.; Młotkowski, W.; Steger, T.

Ann. Inst. Fourier 44, No. 1, 213-248 (1994).

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).

Reviewer: D.I.Cartwright (Sydney)

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

Keywords:

finitely generated groups; Kazhdan’s property (\(T\)); Kazhdan constant

Citations:

Zbl 0784.51010; Zbl 0784.51011; Zbl 0759.22001

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