On plausible counterexamples to Lehnert's conjecture

A group whose co-word problem is a context free language is called co . Lehnert's conjecture states that a group is co if and only if 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 consists of groups that do embed into . By Anisimov a group has regular word problem if and only if it is finite. It is also known that a group is finite if and only if there exists an embedding of into such that its natural action on ₂:= {0, 1}[super] is free on the whole space. We show that the class of groups with a context free word problem, the class of groups, is precisely the class of finitely generated demonstrable groups for . A demonstrable group for V is a group which is isomorphic to a subgroup in whose natural action on ₂ 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 co groups also holds for the set of finitely generated subgroups of