Two Remarks on First-Order Theories of Baumslag-Solitar Groups

TL;DR

This paper characterizes the elementary equivalence of finitely generated groups to solvable Baumslag-Solitar groups, establishing that such groups are isomorphic to the original group, and explores the equivalence relations among Baumslag-Solitar groups.

Contribution

It proves that finitely generated groups elementarily equivalent to BS(1,n) are exactly those isomorphic to BS(1,n), and shows the equivalence of various logical relations among Baumslag-Solitar groups.

Findings

Finitely generated groups elementarily equivalent to BS(1,n) are isomorphic to BS(1,n).

Two Baumslag-Solitar groups are existentially and universally equivalent if and only if they are isomorphic.

Elementary equivalence coincides with isomorphism for these groups.

Abstract

In this note we characterise all finitely generated groups elementarily equivalent to a solvable Baumslag-Solitar group BS$(1,n)$. It turns out that a finitely generated group $G$ is elementarily equivalent to BS$(1,n)$ if and only if $G$ is isomorphic to BS$(1,n)$. 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.