Local similarity groups with context-free co-word problem

Abstract

Let $G$ be a group, and let $S$ be a finite subset of $G$ that generates $G$ as a monoid. The co-word problem is the collection of words in the free monoid $S^{\ast}$ that represent non-trivial elements of $G$. A current conjecture, based originally on a conjecture of Lehnert and modified into its current form by Bleak, Matucci, and Neuh\"{o}ffer, says that Thompson's group $V$ is a universal group with context-free co-word problem. In other words, it is conjectured that a group has a context-free co-word problem exactly if it is a finitely generated subgroup of $V$. Hughes introduced the class $\mathcal{FSS}$ of groups that are determined by finite similarity structures. An $\mathcal{FSS}$ group acts by local similarities on a compact ultrametric space. Thompson's group $V$ is a representative example, but there are many others. We show that $\mathcal{FSS}$ groups have context-free co-word problem under a minimal additional hypothesis. As a result, we can specify a subfamily of $\mathcal{FSS}$ groups that are potential counterexamples to the conjecture.