Groups, the Theory of ends, and context-free languages

Abstract

This is the first of a series of three papers describing some surprising connections between group theory, the theory of ends, the theory of automata and formal languages, and second-order logic. This first paper discusses the interaction of group theory and formal language theory. Using the theory of ends and Rabin’s theorem that the monadic theory of the infinite binary tree is decidable, the second paper will establish the decidability of the monadic second-order theory of a very large class of graphs. The third paper will give give a new proof of Rabin’s theorem. The sketch ]9] is a detailed statement of the reults of the first and second papers. We begin our investigation with a question raised by Anisimov ] 11. A finitely generated group can be described by a presentation G = (X; R) in terms of generators and defining relations. The word problem W(G) is the set of all words on the generators and their inverses which represent the identity element of G. Anisimov asked, “If W(G) is a context-free language in the usual sense of formal language theory, what can one say about the algebraic structure of the group G?” Although the set W(G) depends on the presentation, an easy lemma shows that if W(G) is a context-free language for one presentation of G, then W(G) is a context-free laguage for every finitely generated presentation of G. Thus we shall simply say that a finitely generated group is context-free if the word problem is a context-free language for finitely generated presentations of G. A group is virtually free if it has a free subgroup of finite index. We were led to conjecture that a finitely generated group is