Two remarks on the first-order theories of Baumslag-Solitar groups

We characterize all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS(m, 1). It turns out that a finitely generated group G is elementarily equivalent to BS(m, 1) if and only if G is isomorphic to BS(m, 1). Furthermore, we show that two Baumslag-Solitar groups are existentially (universally) equivalent if and only if they are elementarily equivalent if and only if they are isomorphic.