Combings of groups and the grammar of reparameterization

A new construction of combings is used to distinguish between several previously indistinguishableclasses of groups associated to the theory of automatic groupsand non-positive curvature in group theory. We construct synchronously bounded combings for a class of groups that are neither bicombable nor automatic.The linguistic complexity of these combings isanalysed: in many cases the language of wordsin the combing is an indexed language.