The Conjugacy Ratio of Abelian-by-Cyclic Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911418929905664 |
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| author | Guo, David |
| author_facet | Guo, David |
| contents | Let $G = K \rtimes \langle t \rangle $ be a finitely generated group where $K$ is abelian and $\langle t\rangle$ is the infinite cyclic group. Let $ R $ be a finite symmetric subset of $K$ such that $S = \{ (r,1),(0,t^{\pm 1}) \mid r \in R \}$ is a generating set of $G$. We prove that the spherical conjugacy ratio, and hence the conjugacy ratio, of $G$ with respect to $S$ is $0$ unless $G$ is virtually abelian, confirming a conjecture of Ciobanu, Cox and Martino in this case. We also show that the Baumslag--Solitar group $\mathrm{BS}(1,2)$ has a one-sided Følner sequence $F$ such that the conjugacy ratio with respect to $F$ is non-zero, even though $\mathrm{BS}(1,2)$ is not virtually abelian. This is in contrast to two-sided Følner sequences, where Tointon showed that the conjugacy ratio with respect to a two-sided Følner sequence is positive if and only if the group is virtually abelian. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23034 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Conjugacy Ratio of Abelian-by-Cyclic Groups Guo, David Group Theory Let $G = K \rtimes \langle t \rangle $ be a finitely generated group where $K$ is abelian and $\langle t\rangle$ is the infinite cyclic group. Let $ R $ be a finite symmetric subset of $K$ such that $S = \{ (r,1),(0,t^{\pm 1}) \mid r \in R \}$ is a generating set of $G$. We prove that the spherical conjugacy ratio, and hence the conjugacy ratio, of $G$ with respect to $S$ is $0$ unless $G$ is virtually abelian, confirming a conjecture of Ciobanu, Cox and Martino in this case. We also show that the Baumslag--Solitar group $\mathrm{BS}(1,2)$ has a one-sided Følner sequence $F$ such that the conjugacy ratio with respect to $F$ is non-zero, even though $\mathrm{BS}(1,2)$ is not virtually abelian. This is in contrast to two-sided Følner sequences, where Tointon showed that the conjugacy ratio with respect to a two-sided Følner sequence is positive if and only if the group is virtually abelian. |
| title | The Conjugacy Ratio of Abelian-by-Cyclic Groups |
| topic | Group Theory |
| url | https://arxiv.org/abs/2410.23034 |