Groups of arbitrary lawlessness growth
Henry Bradford, Jacob Willis
TL;DR
The paper investigates quantitative lawlessness by introducing and analyzing the lawlessness growth function $\mathcal{A}_{\Gamma}$. It develops sparse wreath product techniques to realize a wide spectrum of growth behaviours for $\mathcal{A}_{\Gamma}$ in finitely generated, elementary amenable groups, including both very slow and arbitrarily fast growth, while also producing non-residually finite examples. The core method involves encoding word-map behavior through carefully chosen data $(L,p,q)$ and base groups $\Delta$, with the construction robust enough to realize any unbounded target growth $f$ under mild assumptions. The results extend prior work by Brad and Petschick, demonstrating a rich landscape of possible lawlessness growth types and highlighting the sparseness mechanism as a flexible tool for engineering quantitative group-theoretic phenomena.
Abstract
For a finitely generated lawless group $Γ$ and $n \in \mathbb{N}$, let $\mathcal{A}_Γ (n)$ be the minimal positive integer $M_n$ such that for all nontrivial reduced words $w$ of length at most $n$ in the free group of fixed rank $k \geq 2$, there exists $\overline{g} \in Γ^k$ of word-length at most $M_n$ with $w(\overline{g}) \neq e$. For any unbounded nondecreasing function $f : \mathbb{N} \rightarrow \mathbb{N}$ satisfying some mild assumptions, we construct $Γ$ such that the function $\mathcal{A}_Γ$ is equivalent to $f$. Our result generalizes both a Theorem of the first named author, who constructed groups for which $\mathcal{A}_Γ$ is unbounded but grows more slowly than any prescribed function $f$, and a result of Petschick, who constructed lawless groups for which $\mathcal{A}_Γ$ grows faster than any tower of exponential functions.