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 BS (1, n) is hyperfinite for any n. Our proof relies on a classification of BS (1, n) 's left-orderings via its one-dimensional dynamical realizations. We furthermore use the effectiveness of the dynamical realizations of BS (1, n) to study algorithmic properties of the left-orderings on BS (1, n).