Irreducible word problems in groups

Abstract

Abstract In this paper we consider irreducible word problems in groups. In particular, we look at results concerning groups whose irreducible word problem lies in some given class of languages (such as the class of finite languages or the class of context-free languages). Introduction In this paper we look at irreducible word problems in groups; see Section 4 below for the definition. We are particularly interested in connections with formal language theory; to be more specific, we consider which types of group can have their irreducible word problem lying in some given class of languages (such as the class of finite languages or the class of context-free languages). We summarize what we need from formal language theory in Section 2. The general question of the connection between irreducible word problems and classes of languages follows on from the analogous question concerning the links between word problems and classes of languages, and we look at some relevant information in Section 3. We come to reduced and irreducible word problems in Section 4, and we talk there about groups with a finite irreducible word problem. We mention some general results about irreducible word problems and languages in Section 5, and then, in Section 6, concentrate on groups whose irreducible word problem is context-free. We finish with some further comments in Section 7.