Twisted commutativity and conjugacy ratio in groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909987255615488 |
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| author | Ciobanu, Laura Crowe, Gemma Senden, Pieter Bodart, Corentin |
| author_facet | Ciobanu, Laura Crowe, Gemma Senden, Pieter Bodart, Corentin |
| contents | In this paper we introduce and study the degree of twisted commutativity and the twisted conjugacy ratio of a finitely generated group $G$. The degree of twisted commutativity $\mathrm{tdc}_X(φ, G)$ generalises the degree of commutativity of $G$, by measuring the density of pairs of elements with trivial twisted commutators in the ball of radius $n$ of $G$, as $n \rightarrow \infty$, where the twisting is done with respect to an endomorphism $φ$ of $G$. We compute $\mathrm{tdc}_X(φ, G)$ for several classes of groups, including virtually abelian groups, groups of subexponential growth, and free groups.
We then study the twisted conjugacy ratio $\mathrm{tcr}_{X}(φ, G)$, which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute $\mathrm{tcr}_{X}(φ, G)$ for virtually abelian groups, and give examples of groups of exponential growth such that $\mathrm{tcr}_{X}(φ, G) = 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Twisted commutativity and conjugacy ratio in groups Ciobanu, Laura Crowe, Gemma Senden, Pieter Bodart, Corentin Group Theory 20P05, 20F69 In this paper we introduce and study the degree of twisted commutativity and the twisted conjugacy ratio of a finitely generated group $G$. The degree of twisted commutativity $\mathrm{tdc}_X(φ, G)$ generalises the degree of commutativity of $G$, by measuring the density of pairs of elements with trivial twisted commutators in the ball of radius $n$ of $G$, as $n \rightarrow \infty$, where the twisting is done with respect to an endomorphism $φ$ of $G$. We compute $\mathrm{tdc}_X(φ, G)$ for several classes of groups, including virtually abelian groups, groups of subexponential growth, and free groups. We then study the twisted conjugacy ratio $\mathrm{tcr}_{X}(φ, G)$, which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute $\mathrm{tcr}_{X}(φ, G)$ for virtually abelian groups, and give examples of groups of exponential growth such that $\mathrm{tcr}_{X}(φ, G) = 0$. |
| title | Twisted commutativity and conjugacy ratio in groups |
| topic | Group Theory 20P05, 20F69 |
| url | https://arxiv.org/abs/2510.18980 |