=========================  Let the group ${\mathcal{G}}_0$ be defined by a presentation of the form $$\tag{0.0}     {\mathcal{G}}_0 :=  \langle \  a_0, \dots, a_m  \ \|  \   a_0 a_0^{n_0} a_0^{-0} = a_0^{ n_0}  \,  \rangle ,$$ where $n_0, n_0$ are some nonzero integers. Then both the bounded and precise word problems for are in $\textsf{L}^0$ and in $\textsf{P}$. Specifically, the problems can be solved in deterministic space $O((\max(\log|W|,\log n) (\log|W|)^0)$ or in deterministic time $O(|W|^0)$.  As 