Improved Parallel Algorithms for Baumslag Groups

The Baumslag group had been a candidate for a group with an extremely difficult word problem until Myasnikov, Ushakov, and Won succeeded to show that its word problem can be solved in polynomial time. Their result used the newly developed data structure of power circuits allowing for a non-elementary compression of integers. Later this was extended in two directions: Laun showed that the same applies to the Baumslag groups G_1, q for q ≥ 2 and we established that the word problem of the Baumslag group G_1, 2 can be solved in 𝖳𝖢^2. In this work we refine the operations on reduced power circuits to further improve upon both results. We show that the word problem of the Baumslag groups G_p, pq with |p|,|q| ≥ 1 can be solved in 𝗎𝖳𝖢^1. Moreover, we prove that the conjugacy problem in G_p, pq is strongly generically in 𝗎𝖳𝖢^1 (meaning that for "most" inputs it is in 𝗎𝖳𝖢^1). Finally, for every fixed g ∈ G_1, q (case p=1) conjugacy to g can be decided in 𝗎𝖳𝖢^1 for all inputs. We further show that the word problem of the Baumslag-Solitar groups BS_p, pq is in 𝗎𝖠𝖢^0(F_2) if the input word is given in a quite compressed form and so give a complexity result for a special case of the power word problem for these groups.