| Title:
|
Metric groups, unitary representations and continuous logic (English) |
| Author:
|
Ivanov, Aleksander |
| Language:
|
English |
| Journal:
|
Communications in Mathematics |
| ISSN:
|
1804-1388 (print) |
| ISSN:
|
2336-1298 (online) |
| Volume:
|
29 |
| Issue:
|
1 |
| Year:
|
2021 |
| Pages:
|
35-48 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find $L_{\omega _1 \omega }$-axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan's property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures. (English) |
| Keyword:
|
Continuous logic |
| Keyword:
|
metric groups |
| Keyword:
|
unitary representations |
| Keyword:
|
amenable groups. |
| MSC:
|
03C52 |
| MSC:
|
22F05 |
| idZBL:
|
Zbl 07413356 |
| idMR:
|
MR4251310 |
| . |
| Date available:
|
2021-07-09T12:24:43Z |
| Last updated:
|
2021-11-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148990 |
| . |
| Reference:
|
[1] Bekka, B., Harpe, P. de la, Valette, P.: Kazhdan's Property (T).New Mathematical Monographs, 11, Cambridge University Press, Cambridge, 2008, MR 2415834 |
| Reference:
|
[2] Yaacov, I. Ben, Berenstein, A., Henson, W., Usvyatsov, A.: Model theory for metric structures.in: Model theory with Applications to Algebra and Analysis, v.2, Z. Chatzidakis, H.D. Macpherson, A. Pillay, A. Wilkie (eds.), London Math. Soc. Lecture Notes, v. 350, Cambridge University Press, 2008, 315 - 427, MR 2436146 |
| Reference:
|
[3] Yaacov, I. Ben, Iovino, J.: Model theoretic forcing in analysis.Annals of Pure and Appl. Logic, 158, 2009, 163 - 174, MR 2500090, 10.1016/j.apal.2007.10.011 |
| Reference:
|
[4] Bunina, E.I., Mikhalev, A.V.: Elementary properties of linear groups and related problems.J. Math. Sci. (N. Y.), 123, 2004, 3921 - 3985, MR 2096270, 10.1023/B:JOTH.0000036655.73484.a8 |
| Reference:
|
[5] Doucha, M.: Generic norms and metrics on countable abelian groups.Monatsh. Mathematik, 189, 2020, 51 - 74, MR 3948286, 10.1007/s00605-019-01277-7 |
| Reference:
|
[6] Ershov, M., Jaikin-Zapirain, A.: Property (T) for noncommutative universal lattices.Invent. Math., 179, 2010, 303 - 347, MR 2570119, 10.1007/s00222-009-0218-2 |
| Reference:
|
[7] Goldbring, I.: Enforceable operator algebras.Journal of the Institute of Mathematics of Jussieu, 2019, 1 - 33, MR 4205777 |
| Reference:
|
[8] Goldbring, I.: On the nonexistence of Folner sets.arXiv:1901.02445, 2019, |
| Reference:
|
[9] Grigorchuk, R., Harpe, P. de la: Amenability and ergodic properties of topological groups: from Bogolyubov onwards.in: Groups, graphs and random walks, London Math. Soc. Lecture Note Ser., 436, Cambridge Univ. Press, Cambridge, 2017, 215 - 249, MR 3644011 |
| Reference:
|
[10] Hernandes, S., Hofmann, K.H., Morris, S.A.: The weights of closed subgroups of a locally compact subgroup.J. Group Theory, 15, 2012, 613 - 630, MR 2982605 |
| Reference:
|
[11] Ivanov, A.: Locally compact groups which are separably categorical structures.Arch. Math. Logic, 56, 2017, 67 - 78, MR 3598797, 10.1007/s00153-016-0516-5 |
| Reference:
|
[12] Ivanov, A.: Actions of metric groups and continuous logic.arXiv:1706.04157, 2017, MR 4251310 |
| Reference:
|
[13] Pestov, V.: Amenability versus property (T) for non locally compact topological groups.Trans. Amer. Math. Soc., 370, 2018, 7417 - 7436, MR 3841853, 10.1090/tran/7256 |
| Reference:
|
[14] Schneider, F.M., Thom, A.: On Folner sets in topological groups.Compositio Mathematica, 154, 2018, 1333 - 1361, MR 3809992, 10.1112/S0010437X1800708X |
| . |
