The Power Word Problem in Graph Products

The power word problem of a group $G$ asks whether an expression $p_1^{x_1} \dots p_n^{x_n}$, where the $p_i$ are words and the $x_i$ binary encoded integers, is equal to the identity of $G$. We show that the power word problem in a fixed graph product is $\mathsf{AC}^0$-Turing-reducible to the word problem of the free group $F_2$ and the power word problem of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups $\mathcal{C}$, the uniform power word problem in a graph product can be solved in $\mathsf{AC^0(C_=L^{PowWP\mathcal{C}})}$. As a consequence of our results, the uniform knapsack problem in graph groups is $\mathsf{NP}$-complete