Groups of arbitrary lawlessness growth

[2503.23582] Groups of arbitrary lawlessness growth Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Group Theory arXiv:2503.23582 (math) [Submitted on 30 Mar 2025 ( v1 ), last revised 12 Apr 2026 (this version, v2)] Title: Groups of arbitrary lawlessness growth Authors: Henry Bradford , Jacob Willis View a PDF of the paper titled Groups of arbitrary lawlessness growth, by Henry Bradford and Jacob Willis View PDF HTML (experimental) Abstract: For a finitely generated lawless group $\Gamma$ and $n \in \mathbb{N}$, let $\mathcal{A}_{\Gamma} (n)$ 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 $\overline{g} \in \Gamma^k$ 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 $\Gamma$ such that the function $\mathcal{A}_{\Gamma}$ is equivalent to $f$. Our result generalizes both a Theorem of the first named author, who constructed groups for which $\mathcal{A}_{\Gamma}$ is unbounded but grows more slowly than any prescribed function $f$, and a result of Petschick, who constructed lawless groups for which $\mathcal{A}_{\Gamma}$ grows faster than any tower of exponential functions. Comments: 13 pages; improved formatting Subjects: Group Theory (math.GR) Cite as: arXiv:2503.23582 [math.GR] (or arXiv:2503.23582v2 [math.GR] for this version) https://doi.org/10.48550/arXiv.2503.23582 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Jacob Willis [ view email ] [v1] Sun, 30 Mar 2025 20:31:22 UTC (12 KB) [v2] Sun, 12 Apr 2026 01:27:55 UTC (12 KB) Full-text links: Access Paper: View a PDF of the paper titled Groups of arbitrary lawlessness growth, by Henry Bradford and Jacob Willis View PDF HTML (experimental) TeX Source view license Current browse context: math.GR < prev | next > new | recent | 2025-03 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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