Residual Finiteness Growth in Virtually Nilpotent Groups
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| Format: | Preprint |
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2025
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| _version_ | 1866914420806909952 |
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| author | Deré, Jonas Matthys, Joren Vandeputte, Lukas |
| author_facet | Deré, Jonas Matthys, Joren Vandeputte, Lukas |
| contents | The residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ of a finitely generated group $G$ is a function that gives the smallest value of the index $[G:N]$ with $N$ a normal subgroup not containing a non-trivial element $g$, in function of the word norm of that element $g$. It has been studied for several classes of finitely generated groups, including free groups, linear groups and virtually abelian groups. This paper shows that if $G$ is virtually nilpotent, then $\text{RF}_G = \log^δ$ for some $δ\in \mathbb{N}\cup\{0\}$, with moreover an explicit formula for $δ$ in terms of Lie algebras. This implies in particular that it is an invariant of the complex Mal'cev completion, leading to the application that residual finiteness growth is a profinite invariant for virtually nilpotent groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_16585 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Residual Finiteness Growth in Virtually Nilpotent Groups Deré, Jonas Matthys, Joren Vandeputte, Lukas Group Theory 20E26 The residual finiteness growth $\text{RF}_G: \mathbb{N} \to \mathbb{N}$ of a finitely generated group $G$ is a function that gives the smallest value of the index $[G:N]$ with $N$ a normal subgroup not containing a non-trivial element $g$, in function of the word norm of that element $g$. It has been studied for several classes of finitely generated groups, including free groups, linear groups and virtually abelian groups. This paper shows that if $G$ is virtually nilpotent, then $\text{RF}_G = \log^δ$ for some $δ\in \mathbb{N}\cup\{0\}$, with moreover an explicit formula for $δ$ in terms of Lie algebras. This implies in particular that it is an invariant of the complex Mal'cev completion, leading to the application that residual finiteness growth is a profinite invariant for virtually nilpotent groups. |
| title | Residual Finiteness Growth in Virtually Nilpotent Groups |
| topic | Group Theory 20E26 |
| url | https://arxiv.org/abs/2512.16585 |