Equations in wreath products

TL;DR

This paper surveys the solvability of equations in wreath products of groups, demonstrating that quadratic diophantine problems are solvable in certain classes of wreath products and metabelian groups.

Contribution

It proves the solvability of quadratic diophantine problems in wreath products of Abelian groups and in Baumslag's finitely presented metabelian group, advancing understanding of equations in these groups.

Findings

Quadratic diophantine problem is solvable in wreath products of Abelian groups.

Quadratic diophantine problem is solvable in Baumslag's finitely presented metabelian group.

The paper provides a survey of equations solvability in wreath products.

Abstract

We survey solvability of equations in wreath products of groups, and prove that the quadratic diophantine problem is solvable in wreath products of Abelian groups. We consider the related question of determining commutator width, and prove that the quadratic diophantine problem is also solvable in Baumslag's finitely presented metabelian group. This text is a short version of an extensive article by the first-named authors.