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In this paper we further study properties of Magnus embedding, give a precisereducibility diagram for Dehn problems in groups of the form $F/N^{(d)}$, andprovide a detailed answer to Problem 12.98 in Kourovka notebook. We also showthat most of the reductions are polynomial time reductions and can be used inpractical computation.

Magnus embedding and algorithmic properties of groups $F/N^{(d)}$