Property ( T ) and A ˜ 2 groups
Annales de l'Institut Fourier, Volume 44 (1994) no. 1, pp. 213-248.

We show that each group Γ in a class of finitely generated groups introduced in [2] and [3] has Kazhdan’s property (T), and calculate the exact Kazhdan constant of Γ 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 Γ 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]M. Burger, Kazhdan constants for S L ( 3 , ) , J. reine angew. Math., 413 (1991), 36-67. | MR | Zbl

[2]D.I. Cartwright, A.M. Mantero, T. Steger, A. Zappa, Groups acting simply transitively on the vertices of a building of type A ˜ 2 I, Geom. Ded., 47 (1993), 143-166. | MR | Zbl

[3]D.I. Cartwright, A.M. Mantero, T. Steger, A. Zappa, Groups acting simply transitively on the vertices of a building of type A ˜ 2 II : the cases q = 2 and q = 3 , Geom. Ded., 47 (1993), 167-223. | MR | Zbl

[4]D.I. Cartwright, W. Mlotkowski, Harmonic analysis for groups acting on triangle buildings, to appear, J. Aust. Math. Soc. | Zbl

[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. | MR | Zbl

[6]M. Cowling and T. Steger, The irreducibility of restrictions of unitary representations to lattices, J. reine angew. Math., 420 (1991), 85-98. | MR | Zbl

[7]A. Figƒ-Talamanca and M.A. Picardello, Harmonic Analysis on Free Groups, Lect. Notes Pure Appl. Math., 87 (1983). | MR | Zbl

[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. | MR | Zbl

[9]P. De La Harpe and A. Valette, La propriété ( T ) de Kazhdan pour les groupes localment compacts, Astérisque, Soc. Math. France, 175 (1989). | Numdam | Zbl

[10]R. Howe, Eng Chye Tan, Non-Abelian Harmonic Analysis, Applications of S L ( 2 , ) , Universitext, Springer-Verlag, New York (1992). | Zbl

[11]D.R. Hughes, F.C. Piper, Projective Planes, Graduate Texts in Mathematics, 6 (1973). | MR | Zbl

[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). | MR | Zbl

[13]S. Lang, S L 2 ( ) , Graduate Texts in Mathematics 105, Springer Verlag, New York Berlin Tokyo (1985). | Zbl

[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

[15]W. Mlotkowski, Positive Definite Radial Functions on Free Product of Groups, Bollettino Un. Mat. Ital. (7), 2-B (1988), 53-66. | MR | Zbl

[16]I. Pays and A. Valette, Sous-groupes libres dans les groupes d'automorphismes d'arbres, L'Enseignement Mathématique, 37 (1991), 151-174. | MR | Zbl

[17]M. Ronan, Lectures on Buildings, Perspectives in Math., vol. 7., Academic Press, (1989). | MR | Zbl

[18]H.H. Schaefer, Banach lattices and positive operators, Grundlehren der Math. Wiss., Springer-Verlag, Berlin, (1974). | MR | Zbl

[19]J. Tits, Buildings of spherical type and finite B N -pairs, Lecture Notes in Math., 386 (1974). | MR | Zbl

[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

Cited by Sources: