Groups and simple languages
Robert H. Haring-Smith · Transactions of the American Mathematical Society · 1983
With any finitely generated group presentation, one can associate a formal language (called the reduced word problem) consisting of those words on the generators and their inverses which are equal to the identity but which have no proper prefix equal to the identity. We show that the reduced word problem is a simple language if and only if each vertex of the presentation’s Cayley diagram has only a finite number of simple closed paths passing through it. Furthermore, if the reduced word problem is simple, then the group is a free product of a free group of finite rank and a finite number of finite groups.