Quantifying lawlessness in finitely generated groups

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