The Reidemeister Number of Any Automorphism of a Baumslag-Solitar Group is Infinite

Abstract

Let φ : G → G be a group endomorphism where G is a finitely generated group of exponential growth, and let R(φ) denote the number of φ-conjugacy classes. Fel'shtyn and Hill [10] conjectured that if φ is injective, then R(φ) is infinite. This conjecture is true for automorphisms of non-elementary Gromov hyperbolic groups, see [28] and [7]. It was shown in [18] that the conjecture does not hold in general.Nevertheless in this paper, we show that the conjecture holds for the Baumslag-Solitar groups B(m, n), where either |m| or |n| is greater than 1 and |m| = |n|. We also show that in the cases where |m| = |n| > 1 or mn = −1 the conjecture is true for automorphisms. In addition, we derive few results about the coincidence Reidemeister number.

Twisted conjugacy classes in nilpotent groups

“…Moreover, non-elementary polycyclic groups of exponential growth that are not Gromov hyperbolic without the R ∞ property were constructed. Since then, many examples of groups with R ∞ have been discovered (see e.g., [7], [8], [11], [18], [23], [28], and [29]). (For connections between the study of Reidemeister classes and other areas, see e.g.…”

Twisted conjugacy classes in R. Thompson’s group F

“…(1) nonelementary Gromov hyperbolic groups [Fel'shtyn 2001; = a n except for BS(1, 1) [Fel'shtyn and Gonçalves 2008];…”

Théorie de Burnside–Frobenius tordue pour les groupes virtuellement polycycliques

“…Par ailleurs, pour les groupes hyperboliques de Gromov, les groupes de Baumslag-Solitar et certaines généralisations, il a été démontré que le nombre de Reidemeister est toujours infini (cf. [2,10,4,9]). Une des conséquences immédiates du théorème de Burnside-Frobenius tordu est la formule de congruence d|n μ(d) · R(φ n/d ) ≡ 0 mod n, où μ est la fonction de Möbius, très importante notamment pour les problèmes de réalisation en dynamique topologique [1].…”