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

This note 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 $\operatorname{Inv}\langle A \mid w=1 \rangle$. We show that every one-relator group admits a special one-relation inverse monoid presentation. We subsequently consider the classes ${\rm {\small ANY}}, {\rm {\small RED}}, {\rm {\small CRED}},$ and ${\rm {\small POS}}$ 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 ${\rm {\small ANY}} \supset {\rm {\small CRED}} \supset {\rm {\small POS}}$ are all strict. Conditional on a natural conjecture, we prove ${\rm {\small ANY}} \supset {\rm {\small RED}}$. Following this, we use the Benois algorithm recently devised by Gray&Ruskuc to produce an infinite family of special one-relation inverse monoids which exhibit similar pathological behaviour (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&Ruskuc that the Benois algorithm always correctly computes the minimal invertible pieces of a special one-relation inverse monoid.