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

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 $\mathrm{BS}(1,n)$.