The complexity of knapsack problems in wreath products
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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... u_d^k_d, where u_1, ..., u_d are words over the group generators and k_1, ..., k_d are binary encoded integers, evaluate to the group identity?) and knapsack problem (does a given equation u_1^x_1... u_d^x_d = v, where u_1, ..., u_d,v are words over the group generators and x_1,...,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 ≀ℤ with G nilpotent and iterated wreath products of free abelian groups belongs to 𝖳𝖢^0. As an application of the latter, the power word problem for free solvable groups is in 𝖳𝖢^0. On the other hand we show that for wreath products G ≀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 𝖼𝗈𝖭𝖯-hard. For the knapsack problem we show 𝖭𝖯-completeness for iterated wreath products of free abelian groups and hence free solvable groups. Moreover, the knapsack problem for every wreath product G ≀ℤ, where G is uniformly efficiently non-solvable, is Σ^2_p-hard.
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