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- W4320459055 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 $mathsf{BS}(1,q)$." @default.
- W4320459055 created "2023-02-14" @default.
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- W4320459055 date "2020-02-07" @default.
- W4320459055 modified "2023-09-23" @default.
- W4320459055 title "Knapsack and the power word problem in solvable Baumslag-Solitar groups" @default.
- W4320459055 doi "https://doi.org/10.48550/arxiv.2002.03837" @default.
- W4320459055 hasPublicationYear "2020" @default.
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