Hostname: page-component-7b7567c8c-tnzcf Total loading time: 0 Render date: 2026-11-12T08:07:34.370Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergent Sequences in Discrete Groups

Published online by Cambridge University Press:  20 November 2018

Andreas Thom
Andreas Thom*
Affiliation:
Universit¨at Leipzig, Germany e-mail: thom@math.uni-leipzig.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We prove that a finitely generated group contains a sequence of non-trivial elements that converge to the identity in every compact homomorphic image if and only if the group is not virtually abelian. As a consequence of the methods used, we show that a finitely generated group satisfies Chu duality if and only if it is virtually abelian.

Keywords

54H11Chu dualityBohr topology

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013, pp. 424 - 433
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Alperin, R. C., An elementary account of Selberg's lemma. Enseign. Math. (2) 33(1987), no. 3–4. 269–273. Google Scholar
[2] Chu, H., Compactification and duality of topological groups. Trans. Amer. Math. Soc. 123(1966), 310–324. http://dx.doi.org/10.1090/S0002-9947-1966-0195988-5 Google Scholar
[3] Comfort, W.W., S. Hernández, Remus, D., and Trigos-Arrieta, F. J., Some open questions on topological groups. In: Nuclear groups and Lie groups (Madrid, 1999), Res. Exp. Math., 24, Heldermann, Lemgo, 2001, pp. 57–76. Google Scholar
[4] Gamburd, A., Jakobson, D., and Sarnak, P., Spectra of elements in the group ring of SU(2). J. Eur. Math. Soc. (JEMS) 1(1999), no. 1, 51–85. http://dx.doi.org/10.1007/PL00011157 Google Scholar
[5] Gerstenhaber, M. and Rothaus, O. S., The solution of sets of equations in groups. Proc. Nat. Acad. Sci. U.S.A. 48(1962), 1531–1533. http://dx.doi.org/10.1073/pnas.48.9.1531 Google Scholar
[6] Glicksberg, I., Uniform boundedness for groups. Canad. J. Math. 14(1962), 269–276. http://dx.doi.org/10.4153/CJM-1962-017-3 Google Scholar
[7] Hausdorff, F., Bemerkung ¨uber den Inhalt von Punktmengen. Math. Ann. 75(1914), no. 3, 428–433. http://dx.doi.org/10.1007/BF01563735 Google Scholar
[8] Hernández, S., The Bohr topology of discrete nonabelian groups. J. Lie Theory 18(2008), no. 3, 733–746. Google Scholar
[9] Hernández, S. and T.-S.Wu, Some new results on the Chu duality of discrete groups. Monatsh. Math. 149(2006), no. 3, 215–232. http://dx.doi.org/10.1007/s00605-006-0382-z Google Scholar
[10] Hewitt, E. and Ross, K. A., Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups. Die Grundlehren der mathematischen Wissenschaften, 152, Springer-Verlag, New York, 1970.Google Scholar
[11] Heyer, H., Groups with Chu duality. In: Probability and information theory, II, Lecture Notes in Math., 296, Springer, Berlin, 1973, pp. 181–215. Google Scholar
[12] Jordan, C., Mémoire sur les équations différentielles linéaires a intégrale algébrique. J. Reine angew. Math. 1878(1878), no. 84, 89–215. Google Scholar
[13] Kaloshin, V. and Rodnianski, I., Diophantine properties of elements of SO(3). Geom. Funct. Anal. 11(2001), no. 5, 953–970. http://dx.doi.org/10.1007/s00039-001-8222-8 Google Scholar
[14] Leptin, H., Abelsche Gruppen mit kompakten Charaktergruppen und Dualit¨atstheorie gewisser linear topologischer abelscher Gruppen. Abh. Math. Sem. Univ. Hamburg 19(1955), 244–263. http://dx.doi.org/10.1007/BF02988875 Google Scholar
[15] Tits, J., Free subgroups in linear groups. J. Algebra 20(1972), 250–270. http://dx.doi.org/10.1016/0021-8693(72)90058-0 Google Scholar