  In \cite{HigherGnk}, the author has constructed natural maps from fundamental
groups of topological spaces (restricted configuration spaces) to the groups
$G_{n}^{k}$. In the present paper, we show that in the case of $n=k+1$, the
group $G_{k+1}^{k}$ is isomorphic to the fundamental group of some (quotient
space of) some configuration space. In particular, this leads to the solution
of word and conjugacy problems in $G_{4}^{3}$ and sheds light on $G_{k+1}^{k}$
for higher $k$.
