Groups of arbitrary lawlessness growth

Abstract

For a finitely generated lawless group $\hat{I}“$ and $n\in \mathbb{N}$, let ${\mathcal{A}}_{\hat{I}}“(n)$ 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 $k\ge 2$, there exists $\overline{g}\in \hat{I}{“}^k$ of word-length at most $M_n$ with $w(\overline{g})\ne e$. For any unbounded nondecreasing function $f:\mathbb{N}\to \mathbb{N}$ satisfying some mild assumptions, we construct $\hat{I}“$ such that the function ${\mathcal{A}}_{\hat{I}}“$ is equivalent to $f$. Our result generalizes both a Theorem of the first named author, who constructed groups for which ${\mathcal{A}}_{\hat{I}}“$ is unbounded but grows more slowly than any prescribed function $f$, and a result of Petschick, who constructed lawless groups for which ${\mathcal{A}}_{\hat{I}}“$ grows faster than any tower of exponential functions.

13 pages; improved formatting