Knapsack and the Power Word Problem in Solvable Baumslag-Solitar Groups

We prove that the power word problem for the solvable Baumslag-Solitar groups $\mathsf{BS}(1,q) = \langle a,t \mid t a t^{-1} = a^q \rangle$ can be solved in $\mathsf{TC}^0$. In the power word problem, the input consists of group elements $g_1, \ldots, g_d$ and binary encoded integers $n_1, \ldots, n_d$ and it is asked whether $g_1^{n_1} \cdots g_d^{n_d} = 1$ holds. Moreover, we prove that the knapsack problem for $\mathsf{BS}(1,q)$ is $\mathsf{NP}$-complete. In the knapsack problem, the input consists of group elements $g_1, \ldots, g_d,h$ and it is asked whether the equation $g_1^{x_1} \cdots g_d^{x_d} = h$ has a solution in $\mathbb{N}^d$.