The Power Word Problem in Graph Products
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The power word problem of a group G asks whether an expression p_1^x_1… p_n^x_n, where the p_i are words and the x_i binary encoded integers, is equal to the identity of G. We show that the power word problem in a fixed graph product is 𝖠𝖢^0-Turing-reducible to the word problem of the free group F_2 and the power word problem of the base groups. Furthermore, we look into the uniform power word problem in a graph product, where the dependence graph and the base groups are part of the input. Given a class of finitely generated groups 𝒞, the uniform power word problem in a graph product can be solved in 𝖠𝖢^0(𝖢_=𝖫^𝖯𝗈𝗐𝖶𝖯𝒞). As a consequence of our results, the uniform knapsack problem in graph groups is 𝖭𝖯-complete.
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