Topological Methods in Nonlinear Analysis

The $R_\infty$ property for abelian groups

Karel Dekimpe and Daciberg Lima Gonçalves

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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.

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

First available in Project Euclid: 21 March 2016

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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.

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  • 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.