Formal languages and the word problem for groups

Abstract

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 wordproblem 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, and we summarize the situation in Section 7. For general background material on group theory we refer the reader to [25, 26], and for formal language theory to [7, 16, 20].