Semigroup Actions, Covering Spaces and Schutzenberger Groups

TL;DR

This paper constructs a 2-complex from semigroup actions and presentations, providing topological insights into Schützenberger groups and characterizing polynomial growth in finitely generated regular semigroups.

Contribution

It introduces a new topological framework linking semigroup actions, covering spaces, and Schützenberger groups, with novel proofs and growth characterizations.

Findings

A 2-complex associated with semigroup actions can be simply connected under certain conditions.

Topological proofs for properties of Schützenberger groups are established.

Finitely generated regular semigroups have polynomial growth iff their maximal subgroups are virtually nilpotent.

Abstract

We associate a 2-complex to the following data: a presentation of a semigroup $S$ and a transitive action of $S$ on a set $V$ by partial transformations. The automorphism group of the action acts properly discontinuously on this 2-complex. A sufficient condition is given for the 2-complex to be simply connected. As a consequence we obtain simple topological proofs of results on presentations of Sch\"utzenberger groups. We also give a geometric proof that a finitely generated regular semigroup with finitely many idempotents has polynomial growth if and only if all its maximal subgroups are virtually nilpotent.