A correspondence between a class of monoids and self-similar group actions II

TL;DR

This paper explores a generalized construction of Levi monoids from bimodules, revealing their structure as equidivisible monoids with length functions and linking irreducible Rees monoids to HNN extensions.

Contribution

It extends the tensor monoid construction to arbitrary bimodules, characterizes Levi monoids, and connects irreducible Rees monoids with HNN extensions of their groups of units.

Findings

Levi monoids are precisely equidivisible monoids with length functions.

Irreducible Rees monoids are determined by partial automorphisms of their unit groups.

Universal groups of irreducible Rees monoids are HNN extensions of their unit groups.

Abstract

The first author showed in a previous paper that there is a correspondence between self-similar group actions and a class of left cancellative monoids called left Rees monoids. These monoids can be constructed either directly from the action using Zappa-Sz\'ep products, a construction that ultimately goes back to Perrot, or as left cancellative tensor monoids from the covering bimodule, utilizing a construction due to Nekrashevych, In this paper, we generalize the tensor monoid construction to arbitrary bimodules. We call the monoids that arise in this way Levi monoids and show that they are precisely the equidivisible monoids equipped with length functions. Left Rees monoids are then just the left cancellative Levi monoids. We single out the class of irreducible Levi monoids and prove that they are determined by an isomorphism between two divisors of its group of units. The irreducible Rees monoids are thereby shown to be determined by a partial automorphism of their group of units; this result turns out to be signficant since it connects irreducible Rees monoids directly with HNN extensions. In fact, the universal group of an irreducible Rees monoid is an HNN extension of the group of units by a single stable letter and every such HNN extension arises in this way.