Groups elementarily equivalent to metabelian Baumslag$-$Solitar groups and regular bi-interpretability

TL;DR

This paper shows that certain metabelian Baumslag–Solitar groups are strongly regularly bi-interpretable with the integers and characterizes all groups elementarily equivalent to these groups.

Contribution

It establishes a bi-interpretability between $BS(1,k)$ groups and the ring of integers, and describes all groups elementarily equivalent to these groups.

Findings

$BS(1,k)$ is strongly regularly bi-interpretable with $ ext{Z}$

All groups elementarily equivalent to $BS(1,k)$ are characterized algebraically

The paper provides a complete algebraic description of these groups

Abstract

We prove that metabelian Baumslag$-$Solitar group $BS(1,k)$, $k>1$, is (strongly) regularly bi-interpretable with the ring of integers $\mathbb{Z}$, and describe in algebraic terms all groups that are elementarily equivalent to $BS(1,k)$.