Introduction

The aim of this article is to survey some connections between formal language theory and group theory with particular emphasis on the word problem for groups and the consequence on the algebraic structure of a group of its word problem belonging to a certain class of formal languages. We define our terms in Section 2 and then consider the structure of groups whose word problem is regular or context-free in Section 3. In Section 4 we look at groups whose word-problem is a one-counter language, and we move up the Chomsky hierarchy to briefly consider what happens above context-free in Section 5. In Section 6, we see what happens if we consider languages lying in certain complexity classes. For general background material on group theory we refer the reader to, and for formal language theory to.

Word problems and decidability

In this section we set up the basic notation we shall be using and introduce the notions of “word problems” and “decidability”.

In order to consider the word problem of a group as a formal language, we need to introduce some terminology. Throughout, if Σ is a finite set (normally referred to in this context as an alphabet) then we let Σ* denote the set of all finite words of symbols from Σ, including the empty word ∈, and Σ+ denote the set of all nonempty finite words of symbols from Σ.