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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.23582 |
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| _version_ | 1866918439785857024 |
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| author | Bradford, Henry Willis, Jacob |
| author_facet | Bradford, Henry Willis, Jacob |
| contents | For a finitely generated lawless group $Γ$ and $n \in \mathbb{N}$, let $\mathcal{A}_Γ (n)$ be the minimal positive integer $M_n$ such that for all nontrivial reduced words $w$ of length at most $n$ in the free group of fixed rank $k \geq 2$, there exists $\overline{g} \in Γ^k$ of word-length at most $M_n$ with $w(\overline{g}) \neq e$. For any unbounded nondecreasing function $f : \mathbb{N} \rightarrow \mathbb{N}$ satisfying some mild assumptions, we construct $Γ$ such that the function $\mathcal{A}_Γ$ is equivalent to $f$. Our result generalizes both a Theorem of the first named author, who constructed groups for which $\mathcal{A}_Γ$ is unbounded but grows more slowly than any prescribed function $f$, and a result of Petschick, who constructed lawless groups for which $\mathcal{A}_Γ$ grows faster than any tower of exponential functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_23582 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Groups of arbitrary lawlessness growth Bradford, Henry Willis, Jacob Group Theory For a finitely generated lawless group $Γ$ and $n \in \mathbb{N}$, let $\mathcal{A}_Γ (n)$ be the minimal positive integer $M_n$ such that for all nontrivial reduced words $w$ of length at most $n$ in the free group of fixed rank $k \geq 2$, there exists $\overline{g} \in Γ^k$ of word-length at most $M_n$ with $w(\overline{g}) \neq e$. For any unbounded nondecreasing function $f : \mathbb{N} \rightarrow \mathbb{N}$ satisfying some mild assumptions, we construct $Γ$ such that the function $\mathcal{A}_Γ$ is equivalent to $f$. Our result generalizes both a Theorem of the first named author, who constructed groups for which $\mathcal{A}_Γ$ is unbounded but grows more slowly than any prescribed function $f$, and a result of Petschick, who constructed lawless groups for which $\mathcal{A}_Γ$ grows faster than any tower of exponential functions. |
| title | Groups of arbitrary lawlessness growth |
| topic | Group Theory |
| url | https://arxiv.org/abs/2503.23582 |