Relative property (T) and linear groups
Annales de l'Institut Fourier, Volume 56 (2006) no. 6, pp. 1767-1804.

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 Γ 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 (ΓA,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.

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 Γ de type fini admet une représentation dans SL n (R) dont la fermeture de Zariski n’est pas moyennable si et seulement si Γ agit par automorphismes sur un groupe A abélien de rang rationnel fini et non nul, de telle façon que la paire (ΓA,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.

DOI: 10.5802/aif.2227
Classification: 20F99,  20E22,  20G25,  46G99
Keywords: Relative property (T), group extensions, linear algebraic groups
Fernós, Talia 1

1 University of Illinois at Chicago Dept. of MSCS (m/c 249) 851 South Morgan Street Chicago, IL 60607-7045 (USA)
@article{AIF_2006__56_6_1767_0,
     author = {Fern\'os, Talia},
     title = {Relative property {(T)} and linear groups},
     journal = {Annales de l'Institut Fourier},
     pages = {1767--1804},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {6},
     year = {2006},
     doi = {10.5802/aif.2227},
     mrnumber = {2282675},
     zbl = {1175.22004},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2227/}
}
TY  - JOUR
AU  - Fernós, Talia
TI  - Relative property (T) and linear groups
JO  - Annales de l'Institut Fourier
PY  - 2006
DA  - 2006///
SP  - 1767
EP  - 1804
VL  - 56
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2227/
UR  - https://www.ams.org/mathscinet-getitem?mr=2282675
UR  - https://zbmath.org/?q=an%3A1175.22004
UR  - https://doi.org/10.5802/aif.2227
DO  - 10.5802/aif.2227
LA  - en
ID  - AIF_2006__56_6_1767_0
ER  - 
%0 Journal Article
%A Fernós, Talia
%T Relative property (T) and linear groups
%J Annales de l'Institut Fourier
%D 2006
%P 1767-1804
%V 56
%N 6
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2227
%R 10.5802/aif.2227
%G en
%F AIF_2006__56_6_1767_0
Fernós, Talia. Relative property (T) and linear groups. Annales de l'Institut Fourier, Volume 56 (2006) no. 6, pp. 1767-1804. doi : 10.5802/aif.2227. https://aif.centre-mersenne.org/articles/10.5802/aif.2227/

[1] Bass, H.; Milnor, J.; Serre, J.-P. Solution of the congruence subgroup problem for SL n (n3) and Sp 2n (n2), Inst. Hautes Études Sci. Publ. Math. (1967) no. 33, pp. 59-137 | DOI | Numdam | MR | Zbl

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

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

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

[5] Burger, Marc Kazhdan constants for SL 3 (), J. reine angew. Math., Volume 413 (1991), pp. 36-67 | DOI | MR | Zbl

[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 | Zbl

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

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

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

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

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

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

[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) | DOI | Numdam | MR | Zbl

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

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

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

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

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

[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

[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 | Zbl

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

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

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

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

[28] Törnquist, Asger Orbit equivalence and F n actions (2004) (Preprint)

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

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

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

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

Cited by Sources: