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.
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.
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Accepted:
Published online:
DOI: 10.5802/aif.3160
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.
@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, Volume 68 (2018) no. 1, pp. 257-273. doi : 10.5802/aif.3160. http://archive.numdam.org/articles/10.5802/aif.3160/
[1] On Polish groups of finite type, Publ. Res. Inst. Math. Sci., Volume 48 (2012) no. 2, pp. 389-408 | DOI | MR | Zbl
[2] Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type (2016) (https://arxiv.org/abs/1605.06909v2, submitted for publication)
[3] Additive subgroups of topological vector spaces, Lecture Notes in Mathematics, 1466, Springer, 1991, vi+178 pages | DOI | MR | Zbl
[4] Kazhdan’s property (T), New Mathematical Monographs, 11, Cambridge University Press, 2008, xiv+472 pages | DOI | MR | Zbl
[5] Bounded Normal Generation and Invariant Automatic Continuity (2015) (https://arxiv.org/abs/1506.08549, submitted for publication)
[6] A new proof of extreme amenability of the unitary group of the hyperfinite II factor, Bull. Belg. Math. Soc. Simon Stevin, Volume 22 (2015) no. 5, pp. 837-841 http://projecteuclid.org/euclid.bbms/1450389251 | MR | Zbl
[7] On characters of inductive limits of symmetric groups, J. Funct. Anal., Volume 264 (2013) no. 7, pp. 1565-1598 | DOI | MR | Zbl
[8] 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] Infinite dimensional representations of finite dimensional algebras and amenability (2015) (https://arxiv.org/abs/1512.03959v1)
[10] Convergence and limits of linear representations of finite groups, J. Algebra, Volume 450 (2016), pp. 588-615 | DOI | MR | Zbl
[11] On unitary representability of topological groups, Math. Z., Volume 263 (2009) no. 1, pp. 211-220 | DOI | MR | Zbl
[12] 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] Some extremely amenable groups, C. R., Math., Acad. Sci. Paris, Volume 334 (2002) no. 4, pp. 273-278 | DOI | MR | Zbl
[14] 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] Sharper character value estimates for groups of Lie type, J. Algebra, Volume 174 (1995) no. 1, pp. 229-266 | DOI | MR | Zbl
[16] Free products of units in algebras. I. Quaternion algebras, J. Algebra, Volume 214 (1999) no. 1, pp. 301-316 | DOI | MR | Zbl
[17] 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] Von Neumann’s manuscript on inductive limits of regular rings, Canad. J. Math., Volume 20 (1968), pp. 477-483 | DOI | MR | Zbl
[19] 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] 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] The concentration of measure phenomenon, Mathematical Surveys and Monographs, 89, American Mathematical Society, 2001, x+181 pages | MR | Zbl
[22] Diameters of finite simple groups: sharp bounds and applications, Ann. Math., Volume 154 (2001) no. 2, pp. 383-406 | DOI | MR | Zbl
[23] 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] Reflexively but not unitarily representable topological groups, Topol. Proc., Volume 25 (2000), pp. 615-625 | Zbl
[25] Every semitopological semigroup compactification of the group is trivial, Semigroup Forum, Volume 63 (2001) no. 3, pp. 357-370 | DOI | MR | Zbl
[26] Continuous geometry, Princeton Landmarks in Mathematics, Princeton University Press, 1998, xiv+299 pages (With a foreword by Israel Halperin) | DOI | MR | Zbl
[27] Entropy and isomorphism theorems for actions of amenable groups, J. Anal. Math., Volume 48 (1987), pp. 1-141 | DOI | MR | Zbl
[28] 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] Character rigidity for special linear groups, J. Reine Angew. Math., Volume 716 (2016), pp. 207-228 | DOI | MR | Zbl
[30] 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] On Følner sets in topological groups (2016) (https://arxiv.org/abs/1608.08185, submitted for publication)
[32] Topological matchings and amenability, Fundam. Math., Volume 238 (2017) no. 2, pp. 167-200 | DOI | MR | Zbl
[33] Rings of quotients. An introduction to methods of ring theory, Die Grundlehren der mathematischen Wissenschaften, 217, Springer, 1975, viii+309 pages | MR | Zbl
[34] 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] 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] Metric ultraproducts of finite simple groups, C. R., Math., Acad. Sci. Paris, Volume 352 (2014) no. 6, pp. 463-466 | DOI | MR | Zbl
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