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

TL;DR

The paper proves that the Diophantine problem is undecidable in iterated restricted wreath products of non-trivial free abelian groups of finite rank, for any number of factors $k \ge 2$. It achieves this by employing e-interpretability to reduce Diophantine questions from the integers to these wreath products, showing that the base subgroup $N$ is definable and the quotient $G/N$ is interpretable, thus transferring undecidability. A detailed construction demonstrates an undecidability result for the concrete case $G = {\mathbb{Z}}^m \;\mathrm{wr}\; {\mathbb{Z}}^n$, followed by an inductive argument that extends the result to all right iterated wreath products $IWP_R(A_k,\dots, A_1)$. The work connects to Magnus embeddings and existing undecidability results for metabelian and solvable-like groups, underscoring the broad reach of Diophantine undecidability in algebraic structures.

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 $G$.

Theorems & Definitions (27)

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• Lemma 1
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• Definition 1
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• Lemma 4: GMO