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

We prove that the power word problem for certain metabelian subgroups of GL(2, C) (including the solvable Baumslag-Solitar groups BS(1, q) = a, t | tat(-1) = a(q)) belongs to the circuit complexity class TC0. In the power word problem, the input consists of group elements g1,..., gd and binary encoded integers n1,..., nd and it is asked whether g(n1) (1) center dot center dot center dot g(nd) (d) = 1 holds. Moreover, we prove that the knapsack problem for BS(1, q) is NPcomplete. In the knapsack problem, the input consists of group elements g1,..., gd, h and it is asked whether the equation g(x1) (1) center dot center dot center dot g (xd) (d) = h has a solution in N-d. For the more general case of a system of so-called exponent equations, where the exponent variables xi can occur multiple times, we show that solvability is undecidable for BS(1, q).