Quantifying lawlessness in finitely generated groups

TL;DR

This paper introduces a new quantitative measure of lawlessness in finitely generated groups, explores its properties, and constructs examples with various growth behaviors, connecting it to residual finiteness and other group properties.

Contribution

It defines the lawlessness growth function, analyzes its boundedness, constructs groups with prescribed growth, and relates it to residual finiteness and identities.

Findings

Bounded lawlessness growth iff the group has a nonabelian free subgroup.

Constructed elementary amenable lawless groups with arbitrarily slow growth.

Produced torsion lawless groups with at least linear growth.

Abstract

We introduce a quantitative notion of lawlessness for finitely generated groups, encoded by the "lawlessness growth function" $\mathcal{A}_{\Gamma} : \mathbb{N} \rightarrow \mathbb{N}$. We show that $\mathcal{A}_{\Gamma}$ is bounded iff $\Gamma$ has a nonabelian free subgroup. By contrast we construct, for any nondecreasing unbounded function $f: \mathbb{N} \rightarrow \mathbb{N}$, 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 $\log (n)$.