Characterizing word problems of groups

The word problem of a finitely generated group is a fundamental notion in group theory; it can be defined as the set of all the words in the generators of the group that represent the identity element of the group. This definition allows us to consider a word problem as a formal language and a rich topic of research concerns the connection between the complexity of this language and the algebraic structure of the corresponding group. Another interesting problem is that of characterizing which languages are word problems of groups. There is a known necessary and sufficient criterion for a language to be a word problem of a group; however a natural question is what other characterizations there are. In this paper we investigate this question, using sentences expressed in first-order logic where the relations we consider are membership of the language in question and concatenation of words. We choose some natural conditions that apply to word problems and then characterize which sets of these conditions are sufficient to guarantee that the language in question really is the word problem of a group. We finish by investigating the decidability of these conditions for the families of regular and one-counter languages.