  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 $(x,y) \mapsto x\cdot 2^y$. 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
essential steps is to compare two integers given by power circuits and this, in
general, is shown to be P-complete. The key observation is that the depth of
the occurring power circuits is logarithmic and such power circuits can be
compared in NC.
