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 ∗ 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 Neuhoffer, 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 FSS of groups that are determined by finite similarity structures. An 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 FSS groups have context-free co-word problem under a minimal addi- tional hypothesis. As a result, we can specify a subfamily of FSS groups that are potential counterexamples to the conjecture.