We show that each group in a class of finitely generated groups introduced in [2] and [3] has Kazhdan’s property (T), and calculate the exact Kazhdan constant of 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.
Nous montrons que chaque groupe 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.
@article{AIF_1994__44_1_213_0, author = {Cartwright, Donald I. and M{\l}otkowski, Wojciech and Steger, Tim}, title = {Property $(T)$ and $\tilde{A}_2$ groups}, journal = {Annales de l'Institut Fourier}, pages = {213--248}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {44}, number = {1}, year = {1994}, doi = {10.5802/aif.1395}, mrnumber = {95j:20024}, zbl = {0792.43002}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.1395/} }
TY - JOUR AU - Cartwright, Donald I. AU - Młotkowski, Wojciech AU - Steger, Tim TI - Property $(T)$ and $\tilde{A}_2$ groups JO - Annales de l'Institut Fourier PY - 1994 SP - 213 EP - 248 VL - 44 IS - 1 PB - Institut Fourier PP - Grenoble UR - http://archive.numdam.org/articles/10.5802/aif.1395/ DO - 10.5802/aif.1395 LA - en ID - AIF_1994__44_1_213_0 ER -
%0 Journal Article %A Cartwright, Donald I. %A Młotkowski, Wojciech %A Steger, Tim %T Property $(T)$ and $\tilde{A}_2$ groups %J Annales de l'Institut Fourier %D 1994 %P 213-248 %V 44 %N 1 %I Institut Fourier %C Grenoble %U http://archive.numdam.org/articles/10.5802/aif.1395/ %R 10.5802/aif.1395 %G en %F AIF_1994__44_1_213_0
Cartwright, Donald I.; Młotkowski, Wojciech; Steger, Tim. Property $(T)$ and $\tilde{A}_2$ groups. Annales de l'Institut Fourier, Volume 44 (1994) no. 1, pp. 213-248. doi : 10.5802/aif.1395. http://archive.numdam.org/articles/10.5802/aif.1395/
[1]Kazhdan constants for , J. reine angew. Math., 413 (1991), 36-67. | MR | Zbl
,[2]Groups acting simply transitively on the vertices of a building of type I, Geom. Ded., 47 (1993), 143-166. | MR | Zbl
, , , ,[3]Groups acting simply transitively on the vertices of a building of type II : the cases and , Geom. Ded., 47 (1993), 167-223. | MR | Zbl
, , , ,[4]Harmonic analysis for groups acting on triangle buildings, to appear, J. Aust. Math. Soc. | Zbl
, ,[5]The radial Fourier-Stieltjes algebra of free groups, Operator Algebras and Theory Contemporary Mathematics, 10, Am. Math. Soc., Providence (1982), 33-40. | MR | Zbl
, ,[6]The irreducibility of restrictions of unitary representations to lattices, J. reine angew. Math., 420 (1991), 85-98. | MR | Zbl
and ,[7]Harmonic Analysis on Free Groups, Lect. Notes Pure Appl. Math., 87 (1983). | MR | Zbl
and ,[8]On the spectrum of the sum of generators for a finitely generated group, Israel J. Math., 81 (1993), 65-96. | MR | Zbl
, and ,[9]La propriété de Kazhdan pour les groupes localment compacts, Astérisque, Soc. Math. France, 175 (1989). | Numdam | Zbl
and ,[10]Non-Abelian Harmonic Analysis, Applications of , Universitext, Springer-Verlag, New York (1992). | Zbl
, ,[11]Projective Planes, Graduate Texts in Mathematics, 6 (1973). | MR | Zbl
, ,[12]Spherical functions on symmetric graphs, p. 344-386 in Harmonic Analysis, Lecture Notes in Math. 992, Springer Verlag, Berlin Heidelberg New York (1983). | MR | Zbl
, ,[13]Graduate Texts in Mathematics 105, Springer Verlag, New York Berlin Tokyo (1985). | Zbl
, ,[14]Spherical functions and spectrum of the Laplacian operators on buildings of rank , to appear, Boll. Un. Mat. Ital. | Zbl
and ,[15]Positive Definite Radial Functions on Free Product of Groups, Bollettino Un. Mat. Ital. (7), 2-B (1988), 53-66. | MR | Zbl
,[16]Sous-groupes libres dans les groupes d'automorphismes d'arbres, L'Enseignement Mathématique, 37 (1991), 151-174. | MR | Zbl
and ,[17]Lectures on Buildings, Perspectives in Math., vol. 7., Academic Press, (1989). | MR | Zbl
,[18]Banach lattices and positive operators, Grundlehren der Math. Wiss., Springer-Verlag, Berlin, (1974). | MR | Zbl
,[19]Buildings of spherical type and finite -pairs, Lecture Notes in Math., 386 (1974). | MR | Zbl
,[20]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
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