# Diophantine Problem in Some Metabelian Groups

## TL;DR

This paper proves that the Diophantine problem for quadratic equations in certain metabelian groups, including Baumslag-Solitar groups and wreath products with finitely generated abelian groups, is decidable.

## Contribution

It establishes the decidability of quadratic Diophantine equations in specific classes of metabelian groups, expanding understanding of their algebraic properties.

## Key findings

- Decidability of quadratic Diophantine equations in Baumslag-Solitar groups $BS(1,k)$.
- Decidability of quadratic Diophantine equations in wreath products $A
times bZ$ with finitely generated abelian $A$.
- Extension of Diophantine problem results to broader classes of metabelian groups.

## Abstract

In this paper we show that Diophantine problem for quadratic equations in Baumslag-Solitar groups $BS(1,k)$ and in wreath products $A \wr \mathbb{Z}$, where $A$ is a finitely generated abelian group and $\mathbb{Z}$ is an infinite cyclic group, is decidable.