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).


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
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[1] M. BURGER, Kazhdan constants for SL(3, ℤ), J. reine angew. Math., 413 (1991), 36-67. · Zbl 0704.22009
[2] D.I. CARTWRIGHT, A.M. MANTERO, T. STEGER, A. ZAPPA, Groups acting simply transitively on the vertices of a building of type ã2 I, Geom. Ded., 47 (1993), 143-166. · Zbl 0784.51010
[3] D.I. CARTWRIGHT, A.M. MANTERO, T. STEGER, A. ZAPPA, Groups acting simply transitively on the vertices of a building of type ã2 II : the cases q = 2 and q = 3, Geom. Ded., 47 (1993), 167-223. · Zbl 0784.51011
[4] D.I. CARTWRIGHT, W. MLOTKOWSKI, Harmonic analysis for groups acting on triangle buildings, to appear, J. Aust. Math. Soc. · Zbl 0808.51014
[5] J.M. COHEN, L. DE MICHELE, The radial Fourier-Stieltjes algebra of free groups, Operator Algebras and Theory Contemporary Mathematics, 10, Am. Math. Soc., Providence (1982), 33-40. · Zbl 0488.43007
[6] M. COWLING and T. STEGER, The irreducibility of restrictions of unitary representations to lattices, J. reine angew. Math., 420 (1991), 85-98. · Zbl 0760.22014
[7] A. FIGƒ-TALAMANCA and M.A. PICARDELLO, Harmonic analysis on free groups, Lect. Notes Pure Appl. Math., 87 (1983). · Zbl 0536.43001
[8] P. DE LA HARPE, A.G. ROBERTSON and A. VALETTE, On the spectrum of the sum of generators for a finitely generated group, Israel J. Math., 81 (1993), 65-96. · Zbl 0791.43008
[9] P. DE LA HARPE and A. VALETTE, La propriété (T) de Kazhdan pour LES groupes localment compacts, Astérisque, Soc. Math. France, 175 (1989). · Zbl 0759.22001
[10] R. HOWE, ENG CHYE TAN, Non-abelian harmonic analysis, Applications of SL(2, ℝ), Universitext, Springer-Verlag, New York (1992). · Zbl 0768.43001
[11] D.R. HUGHES, F.C. PIPER, Projective planes, Graduate Texts in Mathematics, 6 (1973). · Zbl 0267.50018
[12] A. IOZZI, M.A. PICARDELLO, Spherical functions on symmetric graphs, p. 344-386 in Harmonic Analysis, Lecture Notes in Math. 992, Springer Verlag, Berlin Heidelberg New York (1983). · Zbl 0535.43005
[13] S. LANG, SL2 (ℝ), Graduate texts in mathematics 105, Springer Verlag, New York Berlin Tokyo (1985). · Zbl 0583.22001
[14] A.M. MANTERO and A. ZAPPA, Spherical functions and spectrum of the Laplacian operators on buildings of rank 2, to appear, Boll. Un. Mat. Ital. · Zbl 0815.51010
[15] W. MLOTKOWSKI, Positive definite radial functions on free product of groups, Bollettino Un. Mat. Ital. (7), 2-B (1988), 53-66. · Zbl 0658.43004
[16] I. PAYS and A. VALETTE, Sous-groupes libres dans LES groupes d’automorphismes d’arbres, L’Enseignement Mathématique, 37 (1991), 151-174. · Zbl 0744.20024
[17] M. RONAN, Lectures on buildings, Perspectives in Math., vol. 7., Academic Press, (1989). · Zbl 0694.51001
[18] H.H. SCHAEFER, Banach lattices and positive operators, Grundlehren der Math. Wiss., Springer-Verlag, Berlin, (1974). · Zbl 0296.47023
[19] J. TITS, Buildings of spherical type and finite BN-pairs, Lecture Notes in Math., 386 (1974). · Zbl 0295.20047
[20] J. TITS, Immeubles de type affine in buildings and the geometry of diagrams, Proc. CIME Como 1984 (L.A. Rosati, ed), Lecture Notes in Math., 1181, Springer-Verlag, Berlin (1986), 159-190. · Zbl 0611.20026
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