A characterization of wreath products where knapsack is decidable

TL;DR

This paper characterizes when the knapsack problem is decidable for wreath products of groups, linking it to the decidability properties of the individual factors, and applies this to specific groups like the Heisenberg and Baumslag-Solitar groups.

Contribution

It provides a complete characterization of decidability of the knapsack problem for wreath products based on properties of the factors, introducing new decision problems.

Findings

Knapsack problem undecidable for G wreath H_3(Z) if G ≠ 1.

Decidability for G wreath BS(1,q) depends on exponent equations in G.

Decidability for G wreath H depends on H being virtually abelian and G's exponent system solvability.

Abstract

The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group $G$ and takes as input group elements $g_1,\ldots,g_n,g\in G$ and asks whether there are $x_1,\ldots,x_n\ge 0$ with $g_1^{x_1}\cdots g_n^{x_n}=g$. We study the knapsack problem for wreath products $G\wr H$ of groups $G$ and $H$. Our main result is a characterization of those wreath products $G\wr H$ for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors $G$ and $H$. To this end, we introduce two decision problems, the intersection knapsack problem and its restriction, the positive intersection knapsack problem. Moreover, we apply our main result to $H_3(\mathbb{Z})$, the discrete Heisenberg group, and to Baumslag-Solitar groups $\mathsf{BS}(1,q)$ for $q\ge 1$. First, we show that the knapsack problem is undecidable for $G\wr H_3(\mathbb{Z})$ for any $G\ne 1$. This implies that for $G\ne 1$ and for infinite and virtually nilpotent groups $H$, the knapsack problem for $G\wr H$ is decidable if and only if $H$ is virtually abelian and solvability of systems of exponent equations is decidable for $G$. Second, we show that the knapsack problem is decidable for $G\wr\mathsf{BS}(1,q)$ if and only if solvability of systems of exponent equations is decidable for $G$.