no. G8
Article de recherche - Analyse et géométrie complexes
SL 4 (Z) is not purely matricial field
[SL 4 (Z) n’est pas purement MF]
Magee, Michael1, 2 ; de la Salle, Mikael3, 2 
1Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, UK
2IAS Princeton, School of Mathematics, 1 Einstein Drive, Princeton 08540, USA
3Institut Camille Jordan, CNRS, Université Lyon 1, France
Comptes Rendus. Mathématique, Tome 362 (2024) no. G8, pp. 903-910

We prove that every non-zero finite dimensional unitary representation of SL 4 (Z) contains a non-zero SL 2 (Z)-invariant vector. As a consequence, there is no sequence of finite-dimensional representations of SL 4 (Z) that gives rise to an embedding of its reduced C * -algebra into an ultraproduct of matrix algebras.

Nous montrons que toute représentation unitaire de dimension finie non nulle de SL 4 (Z) a un vecteur SL 2 (Z)-invariant non nul. Il n’existe donc pas de suite de représentations de dimension finie de SL 4 (Z) qui permettent de réaliser sa C * -algèbre réduite dans un ultraproduit d’algèbres de matrices.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.617
Classification : 20C15, 20C33, 22D25
Keywords: Special linear groups, Finite dimensionnal unitary representations, Purely MF groups, MF $C^*$-algebra
Mots-clés : Groupes spéciaux linéaires, représentations unitaires de dimension finie, groupes purement MF, $C^*$-algèbres MF

Magee, Michael  1 , 2   ; de la Salle, Mikael  3 , 2

1 Department of Mathematical Sciences, Durham University, Lower Mountjoy, DH1 3LE Durham, UK
2 IAS Princeton, School of Mathematics, 1 Einstein Drive, Princeton 08540, USA
3 Institut Camille Jordan, CNRS, Université Lyon 1, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{CRMATH_2024__362_G8_903_0,
     author = {Magee, Michael and de la Salle, Mikael},
     title = {SL$_{4}(\textbf{Z})$ is not purely matricial field},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {903--910},
     year = {2024},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     number = {G8},
     doi = {10.5802/crmath.617},
     zbl = {07929052},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.617/}
}
TY  - JOUR
AU  - Magee, Michael
AU  - de la Salle, Mikael
TI  - SL$_{4}(\textbf{Z})$ is not purely matricial field
JO  - Comptes Rendus. Mathématique
PY  - 2024
SP  - 903
EP  - 910
VL  - 362
IS  - G8
PB  - Académie des sciences, Paris
UR  - https://www.numdam.org/articles/10.5802/crmath.617/
DO  - 10.5802/crmath.617
LA  - en
ID  - CRMATH_2024__362_G8_903_0
ER  - 
%0 Journal Article
%A Magee, Michael
%A de la Salle, Mikael
%T SL$_{4}(\textbf{Z})$ is not purely matricial field
%J Comptes Rendus. Mathématique
%D 2024
%P 903-910
%V 362
%N G8
%I Académie des sciences, Paris
%U https://www.numdam.org/articles/10.5802/crmath.617/
%R 10.5802/crmath.617
%G en
%F CRMATH_2024__362_G8_903_0
Magee, Michael; de la Salle, Mikael. SL$_{4}(\textbf{Z})$ is not purely matricial field. Comptes Rendus. Mathématique, Tome 362 (2024) no. G8, pp. 903-910. doi: 10.5802/crmath.617

[1] Bordenave, Charles; Collins, Benoît Eigenvalues of random lifts and polynomials of random permutation matrices, Ann. Math., Volume 190 (2019) no. 3, pp. 811-875 | DOI | MR | Zbl

[2] Bekka, Bachir Operator-algebraic superridigity for SL n (ℤ), n≥3, Invent. Math., Volume 169 (2007) no. 2, pp. 401-425 | DOI | MR | Zbl

[3] Blackadar, Bruce; Kirchberg, Eberhard Generalized inductive limits of finite-dimensional C * -algebras, Math. Ann., Volume 307 (1997) no. 3, pp. 343-380 | DOI | MR | Zbl

[4] Bass, H.; Milnor, J.; Serre, J.-P. Solution of the congruence subgroup problem for SL n (n≥3) and Sp 2n (n≥2), Publ. Math., Inst. Hautes Étud. Sci., Volume 33 (1967), pp. 59-137 | DOI | MR | Zbl

[5] Brown, Nathanial P.; Ozawa, Narutaka C * -algebras and finite-dimensional approximations, Graduate Studies in Mathematics, 88, American Mathematical Society, 2008, xvi+509 pages | DOI | MR | Zbl

[6] Bonnafé, Cédric Representations of SL 2 (𝔽 q ), Algebra and Applications, 13, Springer, 2011, xxii+186 pages | DOI | MR | Zbl

[7] Hide, Will; Magee, Michael Near optimal spectral gaps for hyperbolic surfaces, Ann. Math., Volume 198 (2023) no. 2, pp. 791-824 | DOI | MR | Zbl

[8] Haagerup, Uffe; Thorbjø rnsen, Steen A new application of random matrices: Ext (C red * (F 2 )) is not a group, Ann. Math., Volume 162 (2005) no. 2, pp. 711-775 | DOI | MR | Zbl

[9] Humphreys, J. E. Ordinary and modular characters of SL (3,p), J. Algebra, Volume 72 (1981) no. 1, pp. 8-16 | DOI | MR | Zbl

[10] Louder, Larsen; Magee, Michael Strongly convergent unitary representations of limit groups (2023) (with Appendix by Will Hide and Michael Magee, https://arxiv.org/abs/2210.08953)

[11] Magee, Michael; Thomas, Joe Strongly convergent unitary representations of right-angled Artin groups, 2023 (https://arxiv.org/abs/2308.00863)

[12] Schafhauser, Christopher Finite dimensional approximations of certain amalgamated free products of groups, 2023 (https://arxiv.org/abs/2306.02498)

[13] Simpson, William A.; Frame, J. Sutherland The character tables for SL (3,q), SU (3,q 2 ), PSL (3,q), PSU (3,q 2 ), Can. J. Math., Volume 25 (1973), pp. 486-494 | DOI | MR | Zbl

[14] Voiculescu, Dan Around quasidiagonal operators, Integral Equations Oper. Theory, Volume 17 (1993) no. 1, pp. 137-149 | DOI | MR | Zbl

Cité par Sources :