  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 $q \geq 2$ and we
established that the word problem of the Baumslag group $G_{1, 2}$ can be
solved in $\mathsf{TC}^2$.
  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 $|p|,|q| \geq 1$ can be solved in $\mathsf{uTC}^1$. Moreover,
we prove that the conjugacy problem in $G_{p, pq}$ is strongly generically in
$\mathsf{uTC}^1$ (meaning that for "most" inputs it is in $\mathsf{uTC}^1$).
Finally, for every fixed $g \in G_{1, q}$ (case $p=1$) conjugacy to $g$ can be
decided in $\mathsf{uTC}^1$ for all inputs.
  We further show that the word problem of the Baumslag-Solitar groups $BS_{p,
pq}$ is in $\mathsf{uAC}^0(F_2)$ 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.
