Property (T) and A ¯ 2 groups

Donald I. Cartwright; Wojciech Młotkowski; Tim Steger

Annales de l'institut Fourier (1994)

  • Volume: 44, Issue: 1, page 213-248
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Cartwright, Donald I., Młotkowski, Wojciech, and Steger, Tim. "Property (T) and $\overline{A}_2$ groups." Annales de l'institut Fourier 44.1 (1994): 213-248. <http://eudml.org/doc/75056>.

@article{Cartwright1994,
abstract = {We show that each group $\Gamma $ in a class of finitely generated groups introduced in [2] and [3] 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 (in [9]), p. 133, Questions 1 and 2.},
author = {Cartwright, Donald I., Młotkowski, Wojciech, Steger, Tim},
journal = {Annales de l'institut Fourier},
keywords = {finitely generated groups; Kazhdan’s property (); Kazhdan constant},
language = {eng},
number = {1},
pages = {213-248},
publisher = {Association des Annales de l'Institut Fourier},
title = {Property (T) and $\overline\{A\}_2$ groups},
url = {http://eudml.org/doc/75056},
volume = {44},
year = {1994},
}

TY - JOUR
AU - Cartwright, Donald I.
AU - Młotkowski, Wojciech
AU - Steger, Tim
TI - Property (T) and $\overline{A}_2$ groups
JO - Annales de l'institut Fourier
PY - 1994
PB - Association des Annales de l'Institut Fourier
VL - 44
IS - 1
SP - 213
EP - 248
AB - We show that each group $\Gamma $ in a class of finitely generated groups introduced in [2] and [3] 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 (in [9]), p. 133, Questions 1 and 2.
LA - eng
KW - finitely generated groups; Kazhdan’s property (); Kazhdan constant
UR - http://eudml.org/doc/75056
ER -

References

top
  1. [1]M. BURGER, Kazhdan constants for SL(3, ℤ), J. reine angew. Math., 413 (1991), 36-67. Zbl0704.22009MR92c:22013
  2. [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. Zbl0784.51010MR95b:20053
  3. [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. Zbl0784.51011MR95b:20054
  4. [4]D.I. CARTWRIGHT, W. MLOTKOWSKI, Harmonic analysis for groups acting on triangle buildings, to appear, J. Aust. Math. Soc. Zbl0808.51014
  5. [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. Zbl0488.43007MR84j:22006
  6. [6]M. COWLING and T. STEGER, The irreducibility of restrictions of unitary representations to lattices, J. reine angew. Math., 420 (1991), 85-98. Zbl0760.22014MR93e:22019
  7. [7]A. FIGƒ-TALAMANCA and M.A. PICARDELLO, Harmonic Analysis on Free Groups, Lect. Notes Pure Appl. Math., 87 (1983). Zbl0536.43001MR85j:43001
  8. [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. Zbl0791.43008MR94j:22007
  9. [9]P. DE LA HARPE and A. VALETTE, La propriété (T) de Kazhdan pour les groupes localment compacts, Astérisque, Soc. Math. France, 175 (1989). Zbl0759.22001
  10. [10]R. HOWE, ENG CHYE TAN, Non-Abelian Harmonic Analysis, Applications of SL(2, ℝ), Universitext, Springer-Verlag, New York (1992). Zbl0768.43001
  11. [11]D.R. HUGHES, F.C. PIPER, Projective Planes, Graduate Texts in Mathematics, 6 (1973). Zbl0267.50018MR48 #12278
  12. [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). Zbl0535.43005MR85c:43009
  13. [13]S. LANG, SL2 (ℝ), Graduate Texts in Mathematics 105, Springer Verlag, New York Berlin Tokyo (1985). Zbl0583.22001
  14. [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. Zbl0815.51010
  15. [15]W. MLOTKOWSKI, Positive Definite Radial Functions on Free Product of Groups, Bollettino Un. Mat. Ital. (7), 2-B (1988), 53-66. Zbl0658.43004MR89g:43005
  16. [16]I. PAYS and A. VALETTE, Sous-groupes libres dans les groupes d'automorphismes d'arbres, L'Enseignement Mathématique, 37 (1991), 151-174. Zbl0744.20024MR92f:20028
  17. [17]M. RONAN, Lectures on Buildings, Perspectives in Math., vol. 7., Academic Press, (1989). Zbl0694.51001MR90j:20001
  18. [18]H.H. SCHAEFER, Banach lattices and positive operators, Grundlehren der Math. Wiss., Springer-Verlag, Berlin, (1974). Zbl0296.47023MR54 #11023
  19. [19]J. TITS, Buildings of spherical type and finite BN-pairs, Lecture Notes in Math., 386 (1974). Zbl0295.20047MR57 #9866
  20. [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. Zbl0611.20026

Citations in EuDML Documents

top
  1. Sylvain Barré, Immeubles de Tits triangulaires exotiques
  2. Sylvain Barré, Sur les polyèdres de rang 2
  3. Alain Valette, On the Haagerup inequality and groups acting on A ˜ n -buildings
  4. Alain Valette, Graphes de Ramanujan et applications
  5. Pierre Pansu, Formules de Matsushima, de Garland et propriété (T) pour des groupes agissant sur des espaces symétriques ou des immeubles
  6. Yehuda Shalom, Explicit Kazhdan constants for representations of semisimple and arithmetic groups
  7. Alain Valette, Nouvelles approches de la propriété (T) de Kazhdan

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.