Local similarity groups with context-free co-word problem

2014/06/18 by Daniel Farley, Farley, Daniel · 2 citations
Mathematics · #03D40 #20F10 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:03D40 #msc:20F10

paper · pdf · doi:10.48550/arxiv.1406.4590

17 pages, no figures

arxiv created 2014/06/18 · arxiv updated 2014/06/19

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 Neuhöffer, 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 additional hypothesis. As a result, we can specify a subfamily of FSS groups that are potential counterexamples to the conjecture.

Cited by

• On Similarity Structure Groups and their W^* and C^*-Algebras
• A graph-theoretical characterisation of subgroups of Thompson's group V

Related

• Groups with Context-Free Co-Word Problem and Embeddings into Thompson's Group V
• Context-Freeness of Higman-Thompson group's co-word problem
• Finiteness properties of some groups of local similarities
• A note on Context Sensitive languages and Word Problems
• Non-finitely Generated Maximal Subgroups of Context-free Monoids