On one-relator groups and units of special one-relation inverse monoids

Abstract

This paper investigates and clarifies some connections between the theory of one-relator groups and special one-relation inverse monoids, i.e. those inverse monoids with a presentation of the form [Formula: see text]. We show that every one-relator group admits a special one-relation inverse monoid presentation. We subsequently consider the classes [Formula: see text] of one-relator groups which can be defined by special one-relation inverse monoid presentations in which the defining word is arbitrary; reduced; cyclically reduced; or positive, respectively. We show that the inclusions [Formula: see text] are all strict. Conditional on a natural conjecture, we prove [Formula: see text]. Following this, we use the Benois algorithm recently devised by Gray and Ruškuc to produce an infinite family of special one-relation inverse monoids which exhibit similar pathological behavior (which we term O’Haresque) to the O’Hare monoid with respect to computing the minimal invertible pieces of the defining word. Finally, we provide a counterexample to a conjecture by Gray and Ruškuc that the Benois algorithm always correctly computes the minimal invertible pieces of a special one-relation inverse monoid.