Topological Methods in Nonlinear Analysis

The $R_\infty$ property for abelian groups

Karel Dekimpe and Daciberg Lima Gonçalves

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

It is well known there is no finitely generated abelian group which has the $R_\infty$ property. We will show that also many non-finitely generated abelian groups do not have the $R_\infty$ property, but this does not hold for all of them! In fact we construct an uncountable number of infinite countable abelian groups which do have the $R_{\infty}$ property. We also construct an abelian group such that the cardinality of the Reidemeister classes is uncountable for any automorphism of that group.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 2 (2015), 773-784.

Dates
First available in Project Euclid: 21 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1458588661

Digital Object Identifier
doi:10.12775/TMNA.2015.066

Mathematical Reviews number (MathSciNet)
MR3494970

Zentralblatt MATH identifier
1372.20051

Citation

Dekimpe, Karel; Gonçalves, Daciberg Lima. The $R_\infty$ property for abelian groups. Topol. Methods Nonlinear Anal. 46 (2015), no. 2, 773--784. doi:10.12775/TMNA.2015.066. https://projecteuclid.org/euclid.tmna/1458588661


Export citation

References

  • K. Dekimpe and D. Gonçalves, The $R_\infty$ property for free groups, free nilpotent groups and free solvable groups, Bull. London Math. Soc. 2014, doi:10.1112/blms/bdu029, 10 p.
  • A. Fel'shtyn, Dynamical zeta functions, Nielsen theory and Reidemeister torsion, Mem. Amer. Math. Soc. 147 (2000), no. 699, xii+146 pp.
  • ––––, New directions in Nielsen–Reidemeister theory, Topology Appl. 157 (2010), no. 10–11, 1724–1735.
  • A. Fel'shtyn, Y. Leonov and E. Troitsky, Twisted conjugacy classes in saturated weakly branch groups, Geom. Dedicata 134 (2008), 61–73.
  • L. Fuchs Infinite Abelian Groups I, Academic Press, New York and London (1970).
  • ––––, Infinite Abelian Groups II, Academic Press, New York and London (1970).
  • B. Jiang, A primer of Nielsen fixed point theory, Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005, 617–645.
  • I. Kaplansky, Infinite Abelian Groups, The University of Michigan Press, Ann Arbor, (1970).
  • T. Mubeena and P. Sankaran, Twisted conjugacy classes in lattices in semisimple Lie groups, Transform. Groups 19 (2014), no. 1, 159–169.
  • ––––, Twisted conjugacy classes in abelian extensions of certain linear groups, Canad. Math. Bull. 57 (2014), no. 1, 132–140.
  • T.R. Nasybullov, Twisted conjugacy classes in general and special linear groups, Algebra Logic 51 (2012), no. 3, 220–231.
  • P. Wong, Fixed point theory for homogeneous spaces – a brief survey, Handbook of Topological Fixed Point Theory, Springer, Dordrecht, 2005, 265–283.