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- Title
Quantifying lawlessness in finitely generated groups.
- Authors
Bradford, Henry
- Abstract
We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the lawlessness growth function A Γ : N → N . We show that A Γ is bounded if and only if Γ has a non-abelian free subgroup. By contrast, we construct, for any non-decreasing unbounded function f : N → N , an elementary amenable lawless group for which A Γ grows more slowly than 푓. We produce torsion lawless groups for which A Γ is at least linear using Golod–Shafarevich theory and give some upper bounds on A Γ for Grigorchuk's group and Thompson's group 퐅. We note some connections between A Γ and quantitative versions of residual finiteness. Finally, we also describe a function M Γ quantifying the property of Γ having no mixed identities and give bounds for non-abelian free groups. By contrast with A Γ , there are no groups for which M Γ is bounded: we prove a universal lower bound on M Γ (n) of the order of log (n) .
- Subjects
NONABELIAN groups; FREE groups; TORSION
- Publication
Journal of Group Theory, 2024, Vol 27, Issue 1, p31
- ISSN
1433-5883
- Publication type
Article
- DOI
10.1515/jgth-2022-0113