Power circuits have been introduced in 2012 by Myasnikov, Ushakov and Won as a data structure for non-elementarily compressed integers supporting the arithmetic operations addition and (x, y) → x • 2 y . The same authors applied power circuits to give a polynomial-time solution to the word problem of the Baumslag group, which has a non-elementary Dehn function.In this work, we examine power circuits and the word problem of the Baumslag group under parallel complexity aspects. In particular, we establish that the word problem of the Baumslag group can be solved in NC -even though one of the essentials steps is to compare two integers given by power circuits and this problem, in general, is P-complete. The key observation here is that the depth of the occurring power circuits is logarithmic and such power circuits, indeed, can be compared in NC.

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 generalized Baumslag groups G1,q for q ≥ 2 and we established that the word problem of the Baumslag group G1,2 can be solved in TC 2 . In this work we further improve upon both previous results by showing that the word problem of all the generalized Baumslag groups G1,q can be solved in TC 1 -even for negative q. Our result is based on using refined operations on reduced power circuits. Moreover, we prove that the conjugacy problem in G1,q is strongly generically in TC 1 (meaning that for "most" inputs it is in TC 1 ). Finally, for every fixed g ∈ G1,q conjugacy to g can be decided in TC 1 for all inputs.

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