Knapsack and the power word problem in solvable Baumslag-Solitar groups

Abstract

We prove that the power word problem for certain metabelian subgroups of $\mathsf{GL}(2,\mathbb{C})$ (including the solvable Baumslag-Solitar groups $\mathsf{BS}(1,q) = \langle a,t \mid t a t^{-1} = a^q \rangle$) belongs to the circuit complexity class $\mathsf{TC}^0$. 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 $\mathbb{N}^d$. For the more general case of a system of so-called exponent equations, where the exponent variables $x_i$ can occur multiple times, we show that solvability is undecidable for $\mathsf{BS}(1,q)$.