On the Reidemeister spectrum of an Abelian group

Abstract

The Reidemeister number of an automorphism ϕ of an Abelian group G is calculated by determining the cardinality of the quotient group {G/(\phi-1_{G})(G)} , and the Reidemeister spectrum of G is precisely the set of Reidemeister numbers of the automorphisms of G. In this work we determine the full spectrum of several types of group, paying particular attention to groups of torsion-free rank 1 and to direct sums and products. We show how to make use of strong realization results for Abelian groups to exhibit many groups where the Reidemeister number is infinite for all automorphisms; such groups then possess the so-called {R_{\infty}} -property. We also answer a query of Dekimpe and Gonçalves by exhibiting an Abelian 2-group which has the {R_{\infty}} -property.

The Reidemeister spectrum of finite abelian groups

“…The aim of this paper is to completely determine the Reidemeister spectrum of finite abelian groups. We would like to mention that Reidemeister spectra of infinite abelian groups, on the other hand, have already been studied, see [14, § 3],[3, 9]. To determine complete and explicit expressions for the Reidemeister spectrum of finite abelian groups, we start in § 2 with recalling the necessary results regarding Reidemeister numbers and reducing the problem to counting fixed points on finite abelian groups of prime power order.…”

The Reidemeister spectrum of finite abelian groups

“…The aim of this paper is to completely determine the Reidemeister spectrum of finite abelian groups. We would like to mention that Reidemeister spectra of infinite abelian groups, on the other hand, have already been studied, see [14, §3], [2,9]. This paper is organised as follows.…”