The Diophantine problem in iterated wreath products of free abelian groups is undecidable

TL;DR

This paper proves that the problem of determining solutions to systems of equations in certain complex group structures called iterated wreath products of free abelian groups is fundamentally undecidable, meaning no algorithm can solve all cases.

Contribution

It establishes the undecidability of the Diophantine problem for a broad class of iterated wreath products of free abelian groups, extending previous results in group theory.

Findings

Diophantine problem is undecidable in these groups

No algorithm can solve all systems of equations in the specified groups

Results apply to iterated wreath products of arbitrary non-trivial free abelian groups

Abstract

In this paper we prove that the Diophantine problem in iterated restricted wreath products $G$ of arbitrary non-trivial free abelian groups $A_1,\ldots, A_k$, $k>1$ of finite ranks is undecidable, i.e., there is no algorithm that given a finite system of group equations with coefficients in $G$ decides whether or not the system has a solution in $G$.