Quantifying lawlessness in finitely generated groups

Abstract

Abstract We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the lawlessness growth function A Γ : N → N \mathcal{A}_{\Gamma}\colon\mathbb{N}\to\mathbb{N} . We show that A Γ \mathcal{A}_{\Gamma} 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 f\colon\mathbb{N}\to\mathbb{N} , an elementary amenable lawless group for which A Γ \mathcal{A}_{\Gamma} grows more slowly than 𝑓. We produce torsion lawless groups for which A Γ \mathcal{A}_{\Gamma} is at least linear using Golod–Shafarevich theory and give some upper bounds on A Γ \mathcal{A}_{\Gamma} for Grigorchuk’s group and Thompson’s group 𝐅. We note some connections between A Γ \mathcal{A}_{\Gamma} and quantitative versions of residual finiteness. Finally, we also describe a function M Γ \mathcal{M}_{\Gamma} quantifying the property of Γ having no mixed identities and give bounds for non-abelian free groups. By contrast with A Γ \mathcal{A}_{\Gamma} , there are no groups for which M Γ \mathcal{M}_{\Gamma} is bounded: we prove a universal lower bound on M Γ ⁢ ( n ) \mathcal{M}_{\Gamma}(n) of the order of log ⁡ ( n ) \log(n) .