The complexity of knapsack problems in wreath products

TL;DR

The paper analyzes the complexity of power word and knapsack problems in wreath products, focusing on the wreath product $G \wr \mathbb{Z}$ and iterated constructions. It proves that $ extsc{PowerWP}(G \wr \mathbb{Z}) \in \mathsf{TC}^0$ when $G$ is finitely generated nilpotent, and extends to iterated wreath products of free abelian groups, with applications to free solvable groups. It also establishes hardness results, showing $ extsc{PowerWP}(G \wr \mathbb{Z})$ is $ extsf{coNP}$-hard for uniformly SENS groups, and that $ extsc{Knapsack}$ and $ extsc{ExpEq}$ exhibit $ extsf{NP}$- and $ ext{$\Sigma^p_2$}$-hardness in various wreath-product settings; among corollaries are $ extsf{coNP}$- and $ ext{$\Sigma^p_2$}$-hardness results for Thompson's group $F$. These results extend the landscape of algorithmic properties for wreath products and free solvable groups, and provide sharp complexity boundaries for these classical decision problems.

Abstract

We prove new complexity results for computational problems in certain wreath products of groups and (as an application) for free solvable group. For a finitely generated group we study the so-called power word problem (does a given expression $u_1^{k_1} \ldots u_d^{k_d}$, where $u_1, \ldots, u_d$ are words over the group generators and $k_1, \ldots, k_d$ are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation $u_1^{x_1} \ldots u_d^{x_d} = v$, where $u_1, \ldots, u_d,v$ are words over the group generators and $x_1,\ldots,x_d$ are variables, has a solution in the natural numbers). We prove that the power word problem for wreath products of the form $G \wr \mathbb{Z}$ with $G$ nilpotent and iterated wreath products of free abelian groups belongs to $\mathsf{TC}^0$. As an application of the latter, the power word problem for free solvable groups is in $\mathsf{TC}^0$. On the other hand we show that for wreath products $G \wr \mathbb{Z}$, where $G$ is a so called uniformly strongly efficiently non-solvable group (which form a large subclass of non-solvable groups), the power word problem is $\mathsf{coNP}$-hard. For the knapsack problem we show $\mathsf{NP}$-completeness for iterated wreath products of free abelian groups and hence free solvable groups. Moreover, the knapsack problem for every wreath product $G \wr \mathbb{Z}$, where $G$ is uniformly efficiently non-solvable, is $Σ^2_p$-hard.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Lemma 7
• Lemma 8: HaaseZ19
• Lemma 9
• Lemma 10