We study the maximal subgroups of free idempotent generated semigroups on a biordered set by topological methods. These subgroups are realized as the fundamental groups of a number of 2-complexes naturally associated to the biorder structure of the set of idempotents. We use this to construct the first example of a free idempotent generated semigroup containing a non-free subgroup.

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

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 MWe 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.

No abstract

No abstract