On plausible counterexamples to Lehnert's conjecture

Abstract

A group whose co-word problem is a context free language is called co𝐶𝐹 . Lehnert's conjecture states that a group 𝐺 is co𝐶𝐹 if and only if 𝐺 embeds as a finitely generated subgroup of R. Thompson's group V . In this thesis we explore a class of groups, Faug, proposed by Berns-Zieze, Fry, Gillings, Hoganson, and Mathews to contain potential counterexamples to Lehnert's conjecture. We create infinite and finite presentations for such groups and go on to prove that a certain subclass of 𝓕𝑎𝑢𝑔 consists of groups that do embed into 𝑉.

By Anisimov a group has regular word problem if and only if it is finite. It is also known that a group 𝐺 is finite if and only if there exists an embedding of 𝐺 into 𝑉 such that its natural action on 𝕮₂:= {0, 1}[super]𝜔 is free on the whole space. We show that the class of groups with a context free word problem, the class of 𝐶𝐹 groups, is precisely the class of finitely generated demonstrable groups for 𝑉 . A demonstrable group for V is a group 𝐺 which is isomorphic to a subgroup in 𝑉 whose natural action on 𝕮₂ acts freely on an open subset. Thus our result extends the correspondence between language theoretic properties of groups and dynamical properties of subgroups of V . Additionally, our result also shows that the final condition of the four known closure properties of the class of co𝐶𝐹 groups also holds for the set of finitely generated subgroups of 𝑉.