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 $\mathrm{BS}(m,1)$. It turns out that a finitely generated group $G$ is elementarily equivalent to $\mathrm{BS}(m,1)$ if and only if $G$ is isomorphic to $\mathrm{BS}(m,1)$. 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.