Groups with Context-Free Co-Word Problem and Embeddings into Thompson's Group $V$

Abstract

Let G be a finitely generated group, and let Σ be a finite subset that generates G as a monoid. The word problem of G with respect to Σ consists of all words in the free monoid Σ * that are equal to the identity in G. The co-word problem of G with respect to Σ is the complement in Σ * of the word problem. We say that a group G is coCF if its co-word problem with respect to some (equivalently, any) finite generating set Σ is a context-free language.We describe a generalized Thompson group V (G,θ) for each finite group G and homomorphism θ: G → G. Our group is constructed using the cloning systems introduced by Witzel and Zaremsky. We prove that V (G,θ) is coCF for any homomorphism θ and finite group G by constructing a pushdown automaton and showing that the co-word problem of V (G,θ) is the cyclic shift of the language accepted by our automaton.A version of a conjecture due to Lehnert says that a group has context-free co-word problem exactly if it is a finitely generated subgroup of V. The groups V (G,θ) where θ is not the identity homomorphism do not appear to have obvious embeddings into V, and may therefore be considered possible counterexamples to the conjecture.Demonstrative subgroups of V , which were introduced by Bleak and Salazar-Diaz, can be used to construct embeddings of certain wreath products and amalgamated free products into V . We extend the class of known finitely generated demonstrative subgroups of V to include all virtually cyclic groups.2010 Mathematics Subject Classification. 20F10, 20E06.