Twisted conjugacy classes of Automorphisms of Baumslag-Solitar groups
Let \(\phi:G \to G\) be a group endomorphism where \(G\) is a finitely generated group of exponential growth, and denote by \(R(\phi)\) the number of twisted \(\phi\)-conjugacy classes. Fel'shtyn and Hill [7] conjectured that if \(\phi\) is injective, then \(R(\phi)\) is infinite. This conjectu...
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| Date: | 2018 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2018
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/896 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| Summary: | Let \(\phi:G \to G\) be a group endomorphism where \(G\) is a finitely generated group of exponential growth, and denote by \(R(\phi)\) the number of twisted \(\phi\)-conjugacy classes. Fel'shtyn and Hill [7] conjectured that if \(\phi\) is injective, then \(R(\phi)\) is infinite. This conjecture is true for automorphisms of non-elementary Gromov hyperbolic groups, see [17] and [6]. It was showed in [12] that the conjecture does not hold in general. Nevertheless in this paper, we show that the conjecture holds for injective homomorphisms for the family of the Baumslag-Solitar groups \(B(m,n)\) where \(m\ne n\) and either \(m\) or \(n\) is greater than 1, and for automorphisms for the case \(m=n>1\). family of the Baumslag-Solitar groups \(B(m,n)\) where \(m\ne n\). |
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