Embedding theorems for actions on generalized trees, I

TL;DR

This paper explores how free actions on median sets can be extended and characterizes the relationship between median groups and free actions, using deformations of simplicial trees and duality theory.

Contribution

It introduces a method to extend free actions on median sets to transitive actions and establishes the categorical relationship between median groups and free actions.

Findings

Every free action on a median set can be extended to a free and transitive action.

The category of median groups is a reflective full subcategory of free actions on pointed median sets.

Abstract

Using suitable deformations of simplicial trees and the duality theory for median sets, we show that every free action on a median set can be extended to a free and transitive one. We also prove that the category of median groups is a reflective full subcategory of the category of free actions on pointed median sets.