On one-relator inverse monoids and one-relator groups

It is known that the word problem for one-relator groups and for one-relator monoids of the form Mon〈A||w=1〉 is decidable. However, the question of decidability of the word problem for general one-relation monoids of the form M=Mon〈A||u=v〉 where u and v are arbitrary (positive) words in A remains open. The present paper is concerned with one-relator inverse monoids with a presentation of the form M=Inv〈A||w=1〉 where w is some word in A∪A^-1. We show that a positive solution to the word problem for such monoids for all reduced words w would imply a positive solution to the word problem for all one-relation monoids. We prove a conjecture of Margolis, Meakin and Stephen by showing that every inverse monoid of the form M=Inv〈A||w=1〉, where w is cyclically reduced, must be E-unitary. As a consequence the word problem for such an inverse monoid is reduced to the membership problem for the submonoid of the corresponding one-relator group G=Gp〈A||w=1〉 generated by the prefixes of the cyclically reduced word w. This enables us to solve the word problem for inverse monoids of this type in certain cases.