Relative property (T) and linear groups

[La propriété (T) relative et les groupes linéaires]

Fernós, Talia

Annales de l'Institut Fourier, Tome 56 (2006), p. 1767-1804 / Harvested from Numdam

Résumé

La propriété (T) relative a récemment été utilisée pour démontrer l’existence de divers nouveaux phénomènes de rigidité, par exemple dans la théorie des algèbres de von Neumann et dans l’étude des relations d’équivalence définies par les orbites d’un groupe. Cependant, jusqu’à récemment, il n’y avait pas beaucoup d’exemples dans la littérature de paires de groupes qui jouissent de la propriété (T) relative. Cette situation a motivé le théorème suivant : Un groupe $\Gamma $ de type fini admet une représentation dans $SL_n(R)$ dont la fermeture de Zariski n’est pas moyennable si et seulement si $\Gamma $ agit par automorphismes sur un groupe $A$ abélien de rang rationnel fini et non nul, de telle façon que la paire $(\Gamma \ltimesA, A)$ ait la propriété (T) relative.

La preuve de ce théorème est constructive. Les ingrédients principaux sont le lemme de Furstenberg sur les mesures invariantes sur l’espace projectif et le théorème spectral pour la décomposition des représentations unitaires de groupes abéliens. Des méthodes provenant de la théorie des groupes algébriques, telles que la restriction des scalaires, sont également employées.

Abstract

Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group $\Gamma $ admits a special linear representation with non-amenable $R$-Zariski closure if and only if it acts on an Abelian group $A$ (of finite nonzero $Q$-rank) so that the corresponding group pair $(\Gamma \ltimesA,A)$ has relative property (T).

The proof is constructive. The main ingredients are Furstenberg’s celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.

Détail

Publié le : 2006-01-01

MR 2282675 | Zbl 1175.22004 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2227
Classification:  20F99,  20E22,  20G25,  46G99
Mots clés: propriété (T) relative, extension de groupes, groupes algébriques linéaires

Bibliographie

[1] Bass, H.; Milnor, J.; Serre, J.-P. Solution of the congruence subgroup problem for ${\rm SL}_{n}\,(n\ge 3)$ and ${\rm Sp}_{2n}\,(n\ge 2)$, Inst. Hautes Études Sci. Publ. Math. (1967) no. 33, pp. 59-137 | Article | Numdam | MR 244257 | Zbl 0174.05203

[2] Bass, Hyman Groups of integral representation type, Pacific Journal of Math., Tome 86 (1980) no. 1, pp. 15-51 | MR 586867 | Zbl 0444.20006

[3] Borel, A.; Tits, J. Groupes réductifs, Publ. Math. IHÉS, Tome 27 (1965), pp. 55-150 | Numdam | MR 207712 | Zbl 0145.17402

[4] Borel, Armand Linear algebraic groups, Springer-Verlag (1991) | MR 1102012 | Zbl 0726.20030

[5] Burger, Marc Kazhdan constants for $\textnormal{SL}_{3}(\mathbb{Z})$, J. reine angew. Math., Tome 413 (1991), pp. 36-67 | Article | MR 1089795 | Zbl 0704.22009

[6] Gaboriau, Damien; Popa, Sorin An uncountable family of non-orbit equivalen actions of $F_n$ (2003) (arXiv:math.GR/0306011)

[7] Goldstein, Larry Joel Analytic number theory, Prentice-Hall, Englewood Cliffs, New Jersey (1971) | MR 498335 | Zbl 0226.12001

[8] De La Harpe, Pierre; Valette, Alain La propriété (T) de Kazhdan pour les groupes localement compacts, Asterisque, Tome 175 (1989), pp. 1-157 | Zbl 0759.22001

[9] Hochschild, G. The structure of Lie groups, Holden-Day Inc., San Francisco (1965) | MR 207883 | Zbl 0131.02702

[10] Humphreys, James E. Linear Algebraic Groups, Springer (1998) | MR 396773 | Zbl 0471.20029

[11] Jolissaint, P. Borel cocycles, approximation properties and relative property (T), Ergod. Th. & Dynam. Sys., Tome 20 (2000), pp. 483-499 | Article | MR 1756981 | Zbl 0955.22008

[12] Kassabov, Martin; Nikolov, Nikolay Universal lattices and property $\tau $ (2004) (http://arxiv.org/ pdf/math.GR/0502112) | Zbl 05052841

[13] Kazhdan, D. Connection of the dual space of a group with the structure of its closed subgroups, Functional Analysis and its Applications, Tome 1 (1967), pp. 63-65 | Article | MR 209390 | Zbl 0168.27602

[14] Lubotzky, Alexander; Mozes, Shahar; Raghunathan, M. S. The word and Riemannian metrics on lattices of semisimple groups, Inst. Hautes Études Sci. Publ. Math. (2000) no. 91, p. 5-53 (2001) | Article | Numdam | MR 1828742 | Zbl 0988.22007

[15] Mackey, George W. A theorem of Stone and von Neumann, Duke Math. J., Tome 16 (1949), pp. 313-326 | Article | MR 30532 | Zbl 0036.07703

[16] Mackey, George W. Induced representations of locally compact groups, I, Ann. of Math. (2), Tome 55 (1952), pp. 101-139 | Article | MR 44536 | Zbl 0046.11601

[17] Margulis, G. A. Discrete subgroups of semisimple Lie groups, Springer-Verlag, Berlin, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Tome 17 (1991) | MR 1090825 | Zbl 0732.22008

[18] Margulis, G.A. Explicit constructions of concentrators, Problems Information Transmission, Tome 9 (1973), pp. 325-332 | MR 484767 | Zbl 0312.22011

[19] Navas, Andrés Quelques nouveaux phénomènes de rang 1 pour les groupes de difféomorphismes du cercle, Comment. Math. Helv., Tome 80 (2005) no. 2, pp. 355-375 | Article | MR 2142246 | Zbl 1080.57002

[20] Popa, Sorin On a class of type II $_1$ factors with Betti numbers invariants (2003) (MSRI preprint no. 2001-0024 math.OA/0209310) | MR 1867564

[21] Popa, Sorin Strong rigidity of II $_1$ factors arising from malleable actions of w-rigid groups, Part I (2003) (arXiv:math.OA/0305306)

[22] Popa, Sorin Some computations of 1-cohomology groups and constructions of non-orbit equivalent actions (2004) (arXiv:math.OA/0407199)

[23] Shalom, Y. Measurable group theory (2005) (Preprint) | MR 2185757 | Zbl 02212149

[24] Shalom, Yehuda Bounded generation and Kazhdan’s property (T), Inst. Hautes Études Sci. Publ. Math. (1999) no. 90, p. 145-168 (2001) | Article | Numdam | MR 1813225 | Zbl 0980.22017

[25] Springer, T.A. Linear algebraic groups, 2nd edition, Birkhäuser (1998) | MR 1642713 | Zbl 0927.20024

[26] Suslin, A. A. The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat., Tome 41 (1977) no. 2, p. 235-252, 477 | MR 472792 | Zbl 0378.13002

[27] Tits, J. Classification of algebraic semisimple groups in algebraic groups and discrete subgroups, Proc. Symp. Pure Math. IX Amer. Math. Soc. (1966) | MR 224710 | Zbl 0238.20052

[28] Törnquist, Asger Orbit equivalence and $F_n$ actions (2004) (Preprint)

[29] Valette, Alain Group pairs with property (T), from arithmetic lattices, Geom. Dedicata, Tome 112 (2005), pp. 183-196 | Article | MR 2163898 | Zbl 1076.22012

[30] Wang, P. S. On isolated points in the dual spaces of locally compact groups, Mathematische Annalen, Tome 218 (1975), pp. 19-34 | Article | MR 384993 | Zbl 0332.22009

[31] Whitney, H. Elementary structure of real algebraic varieties, The Annals of Mathematics, Tome 66 (1957) no. 3, pp. 545-556 | Article | MR 95844 | Zbl 0078.13403

[32] Zimmer, R.J. Ergodic theory and semisimple groups, Birkhäuser (1984) | MR 776417 | Zbl 0571.58015