A note on the existence of the Reidemeister zeta function on groups
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910448738107392 |
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| author | Deré, Jonas |
| author_facet | Deré, Jonas |
| contents | Given an endomorphism $φ: G \to G$ on a group $G$, one can define the Reidemeister number $R(φ) \in \mathbb{N} \cup \{\infty\}$ as the number of twisted conjugacy classes. The corresponding Reidemeister zeta function $R_φ(z)$, by using the Reidemeister numbers $R(φ^n)$ of iterates $φ^n$ in order to define a power series, has been studied a lot in the literature, especially the question whether it is a rational function or not. For example, it has been shown that the answer is positive for finitely generated torsion-free virtually nilpotent groups, but negative in general for abelian groups that are not finitely generated.
However, in order to define the Reidemeister zeta function of an endomorphism $φ$, it is necessary that the Reidemeister numbers $R(φ^n)$ of all iterates $φ^n$ are finite. This puts restrictions, not only on the endomorphism $φ$, but also on the possible groups $G$ if $φ$ is assumed to be injective. In this note, we want to initiate the study of groups having a well-defined Reidemeister zeta function for a monomorphism $φ$, because of its importance for describing the behavior of Reidemeister zeta functions. As a motivational example, we show that the Reidemeister zeta function is indeed rational on torsion-free virtually polycyclic groups. Finally, we give some partial results about the existence in the special case of automorphisms on finitely generated torsion-free nilpotent groups, showing that it is a restrictive condition. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_06853 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A note on the existence of the Reidemeister zeta function on groups Deré, Jonas Group Theory 20F18, 37C25, 20E36, 37C30 Given an endomorphism $φ: G \to G$ on a group $G$, one can define the Reidemeister number $R(φ) \in \mathbb{N} \cup \{\infty\}$ as the number of twisted conjugacy classes. The corresponding Reidemeister zeta function $R_φ(z)$, by using the Reidemeister numbers $R(φ^n)$ of iterates $φ^n$ in order to define a power series, has been studied a lot in the literature, especially the question whether it is a rational function or not. For example, it has been shown that the answer is positive for finitely generated torsion-free virtually nilpotent groups, but negative in general for abelian groups that are not finitely generated. However, in order to define the Reidemeister zeta function of an endomorphism $φ$, it is necessary that the Reidemeister numbers $R(φ^n)$ of all iterates $φ^n$ are finite. This puts restrictions, not only on the endomorphism $φ$, but also on the possible groups $G$ if $φ$ is assumed to be injective. In this note, we want to initiate the study of groups having a well-defined Reidemeister zeta function for a monomorphism $φ$, because of its importance for describing the behavior of Reidemeister zeta functions. As a motivational example, we show that the Reidemeister zeta function is indeed rational on torsion-free virtually polycyclic groups. Finally, we give some partial results about the existence in the special case of automorphisms on finitely generated torsion-free nilpotent groups, showing that it is a restrictive condition. |
| title | A note on the existence of the Reidemeister zeta function on groups |
| topic | Group Theory 20F18, 37C25, 20E36, 37C30 |
| url | https://arxiv.org/abs/2311.06853 |