Holonomy theorem for finite semigroups

TL;DR

This paper presents a simplified proof of the Holonomy Theorem for finite semigroups using a novel Lyndon-Chiswell length function, revealing new elliptic actions on the Chiswell tree.

Contribution

It introduces a new Lyndon-Chiswell length function and demonstrates both left and right elliptic actions on the Chiswell tree, simplifying the proof of the Holonomy Theorem.

Findings

Simplified proof of the Holonomy Theorem

Introduction of a new Lyndon-Chiswell length function

Discovery of elliptic actions on the Chiswell tree

Abstract

We provide a simple proof of the Holonomy Theorem using a new Lyndon-Chiswell length function on the Karnofsky-Rhodes expansion of a semigroup. Unexpectedly, we have both a left and a right action on the Chiswell tree by elliptic maps.