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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 wordsarXiv:2002.08086v1 fatcat:xcrnxqt4irftxjvfdl7nt5r7mi