  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 $(x,y) \mapsto x2^y$. 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.
