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 $q \geq 2$ 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.