Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 052, 7 pages      arXiv:2206.06704      https://doi.org/10.3842/SIGMA.2022.052
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

Maximal Discrete Subgroups in Unitary Groups of Operator Algebras

Vadim Alekseev and Andreas Thom
TU Dresden, Institut für Geometrie, 01062 Dresden, Germany

Received February 22, 2022, in final form June 29, 2022; Published online July 09, 2022

Abstract
We show that if a group $G$ is mixed-identity-free, then the projective unitary group of its group von Neumann algebra contains a maximal discrete subgroup containing $G$. The proofs are elementary and make use of free probability theory. In addition, we clarify the situation for $C^*$-algebras.

Key words: maximal discrete subgroups; unitary groups; operator algebras.

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References

1. Allan N.D., Maximality of some arithmetic groups, An. Acad. Brasil. Ci. 38 (1966), 223-227.
2. Belolipetsky M., Lubotzky A., Finite groups and hyperbolic manifolds, Invent. Math. 162 (2005), 459-472, arXiv:math.GR/0406607.
3. Connes A., Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not=1$, Ann. of Math. 104 (1976), 73-115.
4. Gutnik L.A., Pjateckiǐ-Šapiro I.I., Maximal discrete subgroups of a unimodular group, Trudy Moskov. Mat. Obvšč. 15 (1966), 279-295.
5. Helling H., Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe, Math. Z. 92 (1966), 269-280.
6. Hull M., Osin D., Transitivity degrees of countable groups and acylindrical hyperbolicity, Israel J. Math. 216 (2016), 307-353, arXiv:1501.04182.
7. Jacobson B., A mixed identity-free elementary amenable group, Comm. Algebra 49 (2021), 235-241, arXiv:1912.06685.
8. Kuranishi M., On everywhere dense imbedding of free groups in Lie groups, Nagoya Math. J. 2 (1951), 63-71.
9. Popa S., Free-independent sequences in type ${\rm II}_1$ factors and related problems, Astérisque 232 (1995), 187-202.
10. Ramanathan K.G., Discontinuous groups. II, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1964 (1964), 145-164.
11. Toyama H., On discrete subgroups of a Lie group, Kōdai Math. Sem. Rep. 1 (1949), 36-37.
12. Voiculescu D.V., Dykema K.J., Nica A., Free random variables, CRM Monograph Series, Vol. 1, Amer. Math. Soc., Providence, RI, 1992.
13. Wang S.P., On subgroups with property $P$ and maximal discrete subgroups, Amer. J. Math. 97 (1975), 404-414.
14. Zassenhaus H., Beweis eines satzes über diskrete gruppen, Abh. Math. Sem. Univ. Hamburg 12 (1937), 289-312.