  We investigate an extended version of the stable Andrews-Curtis
transformations, referred to as EAC transformations, and compare it with a
notion of equivalence in a family of $3$-manifolds with boundary, called the
{\emph{simple balanced $3$-manifolds}}. A simple balanced $3$-manifold is a
$3$-manifold with boundary, such that every connected component $N$ of it has
unique positive and negative boundary components $\partial^+N$ and
$\partial^-N$, such that $\pi_1(N)$ is the normalizer of the image of
$\pi_1(\partial^\pm N)$ in $\pi_1(N)$. Associated with every simple balanced
$3$-manifold $N$ is the EAC equivalence class of a balanced presentation of the
trivial group, denoted by $P_N$, which remains unchanged as long as $N$ remains
in a fixed equivalence class of simple balanced $3$-manifolds. In particular,
the isomorphism class of the corresponding group is unchanged. Motivated by the
Andrews-Curtis conjecture, we study the equivalence class of a trivial balanced
$3$-manifold (obtained as the product of a closed oriented surface with the
unit interval). We show that every balanced $3$-manifold in the trivial
equivalence class admits a {\emph{simplifier}} to a trivial balanced
$3$-manifold.
