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 $\Sigma^2_p$-hard.

翻译：我们证明了某些群环积（以及作为应用的自由可解群）中计算问题的新复杂性结果。对于有限生成群，我们研究了所谓的幂词问题（给定表达式 $u_1^{k_1} \ldots u_d^{k_d}$，其中 $u_1, \ldots, u_d$ 是群生成元上的词，$k_1, \ldots, k_d$ 是二进制编码整数，其值是否等于群单位元？）和背包问题（给定方程 $u_1^{x_1} \ldots u_d^{x_d} = v$，其中 $u_1, \ldots, u_d,v$ 是群生成元上的词，$x_1,\ldots,x_d$ 是变量，该方程在自然数中是否有解？）。我们证明了对于形式为 $G \wr \mathbb{Z}$（其中 $G$ 是幂零群）的环积以及自由阿贝尔群的迭代环积，其幂词问题属于 $\mathsf{TC}^0$。作为后者的一个应用，自由可解群的幂词问题也在 $\mathsf{TC}^0$ 中。另一方面，我们证明了对于环积 $G \wr \mathbb{Z}$（其中 $G$ 是所谓的一致强效非可解群，构成非可解群的一个大类），其幂词问题是 $\mathsf{coNP}$-困难的。对于背包问题，我们证明了自由阿贝尔群的迭代环积（从而自由可解群）的背包问题是 $\mathsf{NP}$-完全的。此外，对于每个环积 $G \wr \mathbb{Z}$（其中 $G$ 是一致高效非可解群），其背包问题是 $\Sigma^2_p$-困难的。