Algorithmic Aspects of Left-Orderings of Solvable Baumslag–Solitar Groups via its Dynamical Realization

We answer a question of Calderoni and Clay [4] by showing that the conjugation equivalence relation of left orderings of the Baumslag-Solitar groups \({{\,\textrm{BS}\,}}(1,n)\) is hyperfinite for any n. Our proof relies on a classification of \({{\,\textrm{BS}\,}}(1,n)\) ’s left-orderings via its one-dimensional dynamical realizations. We furthermore use the effectiveness of the dynamical realizations of \({{\,\textrm{BS}\,}}(1,n)\) to study algorithmic properties of the left-orderings on \({{\,\textrm{BS}\,}}(1,n)\) .