Maximal subgroups of finitely presented special inverse monoids

TL;DR

This paper characterizes the maximal subgroups of finitely presented special inverse monoids, showing they are exactly the recursively presented groups and exploring their complexity and diversity.

Contribution

It establishes that maximal subgroups are precisely the recursively presented groups and that the subgroup structure can be more complex than the group of units.

Findings

Maximal subgroups are exactly the recursively presented groups.

Every such maximal subgroup can also appear in the $E$-unitary case.

Finitely presented special inverse monoids can have infinitely many non-isomorphic maximal subgroups.

Abstract

We study the maximal subgroups (also known as group $\mathcal{H}$-classes) of finitely presented special inverse monoids. We show that the maximal subgroups which can arise in such monoids are exactly the recursively presented groups, and moreover every such maximal subgroup can also arise in the $E$-unitary case. We also prove that the possible groups of units are exactly the finitely generated recursively presented groups; this improves upon a result of, and answers a question of, the first author and Ru\v{s}kuc. These results give the first significant insight into the maximal subgroups of such monoids beyond the group of units, and the results together demonstrate that it is possible for the subgroup structure to have a complexity which significantly exceeds that of the group of units. We also observe that a finitely presented special inverse monoid (even an $E$-unitary one) may have infinitely many pairwise non-isomorphic maximal subgroups.