The R_\infty property for abelian groups
DOI:
https://doi.org/10.12775/TMNA.2015.066Keywords
Reidemeister number, twisted conjugacy classes, Reidemeister classes, $R_\infty$ property, Abelian groupAbstract
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 groupsdo 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.
References
K. Dekimpe and D. Goncalves, The R1 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.
A. Fel'shtyn, 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).
L. Fuchs, 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.
T. Mubeena and P. Sankaran, 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.
Published
How to Cite
Issue
Section
Stats
Number of views and downloads: 497
Number of citations: 7
