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Vol. 46 No. 4 (2020)
Serdica Mathematical Journal

Articles

Examples of group amalgamations with nontrivial quasi-kernels

Published 01-10-2021

  • Nikolay A. Ivanov

Nikolay A. Ivanov
St. Kl. Ohridski University of Sofia

Abstract

We introduce some examples of group amalgamations motivated by the problems of \(C^*\)-simplicity and unique trace property. Moreover, we prove that our examples are not inner amenable and identify a relatively large, simple, normal subgroup in each one.

2020 Mathematics Subject Classification:

22D25, 20E06, 46L05, 43A07, 20E08

Keywords

\(C^*\)-simplicity, free product of groups with amalgamation, inner amenability

Full text

Author Details

Nikolay A. Ivanov

Faculty of Mathematics and Informatics
University of Sofia
5, James Bourchier Blvd
1164 Sofia, Bulgaria
e-mail: nivanov@fmi.uni-sofia.bg

References

  1. G. Baumslag. Topics in Combinatorial Group Theory. Lectures in Mathematics, ETH Zurich. Basel, Birkhauser Verlag, 1993.
  2. E. Bédos, P. de la Harpe. Inner amenability of groups: definitions and examples (Moyennabilité intérieure des groupes: définitions et exemples). Enseign. Math (2) 32, 1-2 (1986), 139–157.
  3. E. Bédos, P. de la Harpe. Erratum pour “Moyennabilité intérieure des groupes: définitions et exemples”. Enseign. Math. (2) 62, 1–2 (2016), 1–2.
  4. E. Breuillard, M. Kalantar, M. Kennedy, N. Ozawa. C∗-simplicity and the unique trace property for discrete groups. Publ. Math., Inst. Hautes Étud. Sci. 126 (2017), 35–71.
  5. R. S. Bryder, N. A. Ivanov, T. Omland. C∗-simplicity of HNN-extensions and groups acting on trees. Ann. Inst. Fourier 70, 4 (2021), 1497–1543.
  6. D. Cohen. Combinatorial Group Theory: a Topological Approach. London Mathematical Society Student Texts, vol. 14. Cambridge, Cambridge University Press, 1989.
  7. K. J. Dykema. Exactness of reduced amalgamated free product C∗-algebras. Forum Math. 16, 2 (2004), 161–180.
  8. E. Effros. Property Γ and inner amenability. Proc. Am. Math. Soc. 47 (1975), 483–486.
  9. P. Fima, F. Le Maître, S. Moon, Y. Stalder. A characterization of high transitivity for groups acting on trees, 2020. https://arxiv.org/abs/2003.11116.
  10. P. de la Harpe. On simplicity of reduced C∗-algebras of groups. Bull. Lond. Math. Soc. 39, 1 (2007), 1–26.
  11. P. de la Harpe, J.-P. Préaux. C∗-simple groups: amalgamated free products, HNN-extensions, and fundamental groups of 3-manifolds. J. Topol. Anal. 3, 4 (2011), 451–489.
  12. N. A. Ivanov, T. Omland. C∗-simplicity of free products with amalgamation and radical classes of groups. J. Funct. Anal. 272, 9 (2017), 3712–3741.
  13. M. Kalantar, M. Kennedy. Boundaries of reduced C∗-algebras of discrete groups. J. Reine Angew. Math. 727 (2017), 247–267.
  14. A. Le Boudec. Groups acting on trees with almost prescribed local action. Comment. Math. Helv. 91, 2 (2016), 253–293.
  15. A. Le Boudec. C∗-simplicity and the amenable radical. Invent. Math. 209, 1 (2017), 159–174.
  16. A. Le Boudec. Tripple transitivity and non-free actions in dimension one, 2019. https://arxiv.org/abs/1906.05744v2.
  17. N. Monod, Y. Shalom. Cocycle superrigidity and bounded cohomology for negatively curved spaces. J. Differ. Geom. 67, 3 (2004), 395–455.
  18. G. J. Murphy. Uniqueness of the trace and simplicity. Proc. Amer. Math. Soc. 128, 12 (2000), 3563–3570.
  19. F. J. Murray, J. von Neumann. On rings of operators. IV. Ann. of Math. (2) 44 (1943), 716–808.
  20. J.-P. Serre. Trees. (Transl. from the French by John Stillwell. Corrected 2nd printing of the 1980 original). Springer Monographs in Mathematics.
  21. Berlin, Springer, 2003.
  22. Y. Stalder. Inner amenability and HNN extensions (Moyennabilité intérieure et extensions HNN). Ann. Inst. Fourier 56, 2 (2006), 309–323.
  23. S. Vaes. An inner amenable group whose von Neumann algebra does not have property Gamma. Acta Math. 208, 2 (2012), 389–394.