Algorithmic aspects of left-orderings of solvable Baumslag--Solitar groups via its dynamical realization

TL;DR

This paper classifies left-orderings of Baumslag-Solitar groups using dynamical realizations, proving the conjugation relation is hyperfinite and exploring algorithmic properties of these orderings.

Contribution

It provides a classification of left-orderings of BS(1,n) via dynamical realizations and establishes the hyperfiniteness of their conjugation relation.

Findings

Conjugation equivalence relation of left orderings is hyperfinite.

Classification of left-orderings via one-dimensional dynamical realizations.

Analysis of algorithmic properties of left-orderings.

Abstract

We answer a question of Calderoni and Clay by showing that the conjugation equivalence relation of left orderings of the Baumslag-Solitar groups $\mathrm{BS}(1,n)$ is hyperfinite for any $n$. Our proof relies on a classification of $\mathrm{BS}(1,n)$'s left-orderings via its one-dimensional dynamical realizations. We furthermore use the effectiveness of the dynamical realizations of $\mathrm{BS}(1,n)$ to study algorithmic properties of the left-orderings on $\mathrm{BS}(1,n)$.