A guide to constructing free transitive actions on median spaces

TL;DR

This paper develops methods to construct large families of groups with free transitive actions on median spaces, including real trees and their products, by introducing the concept of an ore and extracting groups with median structures.

Contribution

It introduces the notion of an ore, a new algebraic structure, and demonstrates how to extract groups with median actions from it, expanding the class of known median space actions.

Findings

Constructed groups acting freely and transitively on the universal real tree.

Realized subgroups of the additive reals as stabilizers of axes.

Produced $2^{2^{ ext{aleph}_0}}$ non-isomorphic groups with these actions.

Abstract

We construct large families of groups admitting free transitive actions on median spaces. In particular, we construct groups which act freely and transitively on the complete universal real tree with continuum valence such that any subgroup of the additive reals is realised as the stabiliser of an axis. We prove a more precise version of this, which implies that there are $2^{2^{\aleph_0}}$ pairwise non-isomorphic groups which admit a free transitive action on this real tree. We also construct free transitive actions on products of complete real trees such that any subgroup of $\mathbb{R}^n$ is realised as the stabiliser of a maximal flat, and an irreducible action on the product of two complete real trees. To construct each of these groups, we introduce the notion of an \textit{ore}: a set equipped with the structure of a meet semilattice and a cancellative monoid with involution, which verifies some additional axioms. We show that one can \textit{extract} a group from an ore and equip this group with a left-invariant median structure.