An exotic group as limit of finite special linear groups
[Un groupe exotique comme limite de groupes linéaires spéciaux finis]
Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 257-273.

Nous étudions un groupe polonais obtenu comme complétion de la limite inductive de groupes linéaires spéciaux finis munis de la distance induite par le rang. Ce groupe polonais est topologiquement simple modulo son centre, extrêmement moyennable et n’a pas de représentations fortement continues non triviales sur un espace de Hilbert.

We consider the Polish group obtained as the rank-completion of an inductive limit of finite special linear groups. This Polish group is topologically simple modulo its center, it is extremely amenable and has no non-trivial strongly continuous unitary representation on a Hilbert space.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3160
Classification : 54H11, 16E50, 43A07, 43A65
Keywords: Polish groups, von Neumann regular rings, extreme amenability and representation theory
Mot clés : groupes polonais, anneaux réguliers de von Neumann, moyennabilité extrême, théorie des représentations.
Carderi, Alessandro 1 ; Thom, Andreas 1

1 Institut für Geometrie TU Dresden 01062 Dresden (Germany)
@article{AIF_2018__68_1_257_0,
     author = {Carderi, Alessandro and Thom, Andreas},
     title = {An exotic group as limit of finite special linear groups},
     journal = {Annales de l'Institut Fourier},
     pages = {257--273},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     doi = {10.5802/aif.3160},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.3160/}
}
TY  - JOUR
AU  - Carderi, Alessandro
AU  - Thom, Andreas
TI  - An exotic group as limit of finite special linear groups
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 257
EP  - 273
VL  - 68
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.3160/
DO  - 10.5802/aif.3160
LA  - en
ID  - AIF_2018__68_1_257_0
ER  - 
%0 Journal Article
%A Carderi, Alessandro
%A Thom, Andreas
%T An exotic group as limit of finite special linear groups
%J Annales de l'Institut Fourier
%D 2018
%P 257-273
%V 68
%N 1
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.3160/
%R 10.5802/aif.3160
%G en
%F AIF_2018__68_1_257_0
Carderi, Alessandro; Thom, Andreas. An exotic group as limit of finite special linear groups. Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 257-273. doi : 10.5802/aif.3160. http://archive.numdam.org/articles/10.5802/aif.3160/

[1] Ando, Hiroshi; Matsuzawa, Yasumichi On Polish groups of finite type, Publ. Res. Inst. Math. Sci., Volume 48 (2012) no. 2, pp. 389-408 | DOI | MR | Zbl

[2] Ando, Hiroshi; Matsuzawa, Yasumichi; Thom, Andreas; Törnquist, Asger Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type (2016) (https://arxiv.org/abs/1605.06909v2, submitted for publication)

[3] Banaszczyk, Wojciech Additive subgroups of topological vector spaces, Lecture Notes in Mathematics, 1466, Springer, 1991, vi+178 pages | DOI | MR | Zbl

[4] Bekka, Bachir; de la Harpe, Pierre; Valette, Alain Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008, xiv+472 pages | DOI | MR | Zbl

[5] Dowerk, Philip A.; Thom, Andreas Bounded Normal Generation and Invariant Automatic Continuity (2015) (https://arxiv.org/abs/1506.08549, submitted for publication)

[6] Dowerk, Philip A.; Thom, Andreas A new proof of extreme amenability of the unitary group of the hyperfinite II 1 factor, Bull. Belg. Math. Soc. Simon Stevin, Volume 22 (2015) no. 5, pp. 837-841 http://projecteuclid.org/euclid.bbms/1450389251 | MR | Zbl

[7] Dudko, Artem; Medynets, Konstantin On characters of inductive limits of symmetric groups, J. Funct. Anal., Volume 264 (2013) no. 7, pp. 1565-1598 | DOI | MR | Zbl

[8] Elek, Gábor The rank of finitely generated modules over group algebras, Proc. Am. Math. Soc., Volume 131 (2003) no. 11, pp. 3477-3485 | DOI | MR | Zbl

[9] Elek, Gábor Infinite dimensional representations of finite dimensional algebras and amenability (2015) (https://arxiv.org/abs/1512.03959v1)

[10] Elek, Gábor Convergence and limits of linear representations of finite groups, J. Algebra, Volume 450 (2016), pp. 588-615 | DOI | MR | Zbl

[11] Galindo, Jorge On unitary representability of topological groups, Math. Z., Volume 263 (2009) no. 1, pp. 211-220 | DOI | MR | Zbl

[12] Gao, Su Unitary group actions and Hilbertian Polish metric spaces, Logic and its applications (Contemporary Mathematics), Volume 380, American Mathematical Society, 2005, pp. 53-72 | DOI | MR | Zbl

[13] Giordano, Thierry; Pestov, Vladimir Some extremely amenable groups, C. R., Math., Acad. Sci. Paris, Volume 334 (2002) no. 4, pp. 273-278 | DOI | MR | Zbl

[14] Giordano, Thierry; Pestov, Vladimir Some extremely amenable groups related to operator algebras and ergodic theory, J. Inst. Math. Jussieu, Volume 6 (2007) no. 2, pp. 279-315 | DOI | MR | Zbl

[15] Gluck, David Sharper character value estimates for groups of Lie type, J. Algebra, Volume 174 (1995) no. 1, pp. 229-266 | DOI | MR | Zbl

[16] Gonçalves, Jairo Z.; Mandel, Arnaldo; Shirvani, Mazi Free products of units in algebras. I. Quaternion algebras, J. Algebra, Volume 214 (1999) no. 1, pp. 301-316 | DOI | MR | Zbl

[17] Gromov, Mikhael Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) (London Mathematical Society Lecture Note Series), Volume 182, Cambridge University Press, 1993, pp. 1-295 | MR | Zbl

[18] Halperin, Israel Von Neumann’s manuscript on inductive limits of regular rings, Canad. J. Math., Volume 20 (1968), pp. 477-483 | DOI | MR | Zbl

[19] Herer, Wojchiech; Christensen, Jens Peter Reus On the existence of pathological submeasures and the construction of exotic topological groups, Math. Ann., Volume 213 (1975), pp. 203-210 | DOI | MR | Zbl

[20] Kirillov, Alexandre Aleksandrovich Positive-definite functions on a group of matrices with elements from a discrete field, Dokl. Akad. Nauk SSSR, Volume 162 (1965), pp. 503-505 | MR | Zbl

[21] Ledoux, Michel The concentration of measure phenomenon, Mathematical Surveys and Monographs, 89, American Mathematical Society, 2001, x+181 pages | MR | Zbl

[22] Liebeck, Martin W.; Shalev, Aner Diameters of finite simple groups: sharp bounds and applications, Ann. Math., Volume 154 (2001) no. 2, pp. 383-406 | DOI | MR | Zbl

[23] Linnell, Peter A. Noncommutative localization in group rings, Non-commutative localization in algebra and topology (London Mathematical Society Lecture Note Series), Volume 330, Cambridge University Press, 2006, pp. 40-59 | DOI | MR | Zbl

[24] Megrelishvili, Michael G. Reflexively but not unitarily representable topological groups, Topol. Proc., Volume 25 (2000), pp. 615-625 | Zbl

[25] Megrelishvili, Michael G. Every semitopological semigroup compactification of the group H + [0,1] is trivial, Semigroup Forum, Volume 63 (2001) no. 3, pp. 357-370 | DOI | MR | Zbl

[26] von Neumann, John Continuous geometry, Princeton Landmarks in Mathematics, Princeton University Press, 1998, xiv+299 pages (With a foreword by Israel Halperin) | DOI | MR | Zbl

[27] Ornstein, Donald S.; Weiss, Benjamin Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., Volume 48 (1987), pp. 1-141 | DOI | MR | Zbl

[28] Pestov, Vladimir Dynamics of infinite-dimensional groups. The Ramsey-Dvoretzky-Milman phenomenon, University Lecture Series, 40, American Mathematical Society, 2006, vii+192 pages | DOI | MR | Zbl

[29] Peterson, Jesse; Thom, Andreas Character rigidity for special linear groups, J. Reine Angew. Math., Volume 716 (2016), pp. 207-228 | DOI | MR | Zbl

[30] Popa, Sorin; Takesaki, Masamichi The topological structure of the unitary and automorphism groups of a factor, Commun. Math. Phys., Volume 155 (1993) no. 1, pp. 93-101 http://projecteuclid.org/euclid.cmp/1104253201 | DOI | MR | Zbl

[31] Schneider, Friedrich Martin; Thom, Andreas On Følner sets in topological groups (2016) (https://arxiv.org/abs/1608.08185, submitted for publication)

[32] Schneider, Friedrich Martin; Thom, Andreas Topological matchings and amenability, Fundam. Math., Volume 238 (2017) no. 2, pp. 167-200 | DOI | MR | Zbl

[33] Stenström, Bo Rings of quotients. An introduction to methods of ring theory, Die Grundlehren der mathematischen Wissenschaften, 217, Springer, 1975, viii+309 pages | MR | Zbl

[34] Stolz, Abel; Thom, Andreas On the lattice of normal subgroups in ultraproducts of compact simple groups, Proc. Lond. Math. Soc., Volume 108 (2014) no. 1, pp. 73-102 | DOI | MR | Zbl

[35] Tamari, Dov A refined classification of semi-groups leading to generalized polynomial rings with a generalized degree concept (Proceedings of the ICM), Volume 3 (1954), pp. 439-440

[36] Thom, Andreas; Wilson, John S. Metric ultraproducts of finite simple groups, C. R., Math., Acad. Sci. Paris, Volume 352 (2014) no. 6, pp. 463-466 | DOI | MR | Zbl

Cité par Sources :