Quantifying lawlessness in finitely generated groups

Abstract

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

Theorem 1.2 has been significantly strengthened; other minor corrections