Linear equations with monomial constraints and decision problems in abelian-by-cyclic groups

TL;DR

This paper proves the undecidability of certain linear equations with monomial constraints over Laurent polynomial rings and applies these results to demonstrate undecidability of various problems in computational group theory, especially in abelian-by-cyclic groups.

Contribution

It introduces new undecidability results for linear equations with monomial constraints and applies them to solve open problems in the computational theory of abelian-by-cyclic groups.

Findings

Undecidability of solutions in wreath products $ obreak \\mathbb{Z} \\wr \\mathbb{Z}$

Existence of finitely generated abelian-by-cyclic groups with undecidable quadratic equations

Existence of finitely generated abelian-by-cyclic groups with undecidable Knapsack Problem

Abstract

We show that it is undecidable whether a system of linear equations over the Laurent polynomial ring $\mathbb{Z}[X^{\pm}]$ admit solutions where a specified subset of variables take value in the set of monomials $\{X^z \mid z \in \mathbb{Z}\}$. In particular, we construct a finitely presented $\mathbb{Z}[X^{\pm}]$-module, where it is undecidable whether a linear equation $X^{z_1} \boldsymbol{f}_1 + \cdots + X^{z_n} \boldsymbol{f}_n = \boldsymbol{f}_0$ has solutions $z_1, \ldots, z_n \in \mathbb{Z}$. This contrasts the decidability of the case $n = 1$, which can be deduced from Noskov's Lemma. We apply this result to settle a number of problems in computational group theory. We show that it is undecidable whether a system of equations has solutions in the wreath product $\mathbb{Z} \wr \mathbb{Z}$, providing a negative answer to an open problem of Kharlampovich, L\'{o}pez and Miasnikov (2020). We show that there exists a finitely generated abelian-by-cyclic group in which the problem of solving a single quadratic equation is undecidable. We also construct a finitely generated abelian-by-cyclic group, different to that of Mishchenko and Treier (2017), in which the Knapsack Problem is undecidable. In contrast, we show that the problem of Coset Intersection is decidable in all finitely generated abelian-by-cyclic groups.