  This paper looks at a class of closed orientable 3-manifolds constructed from
a gluing of three handlebodies, such that the inclusion of each handlebody is
$\pi_1$-injective. This construction is the generalisation to handlebodies of
the condition for gluing three solid tori to produce non-Haken Seifert fibered
3-manifolds with infinite fundamental group. It is shown that there is an
efficient algorithm to decide if a gluing of handlebodies meets the
disk-condition. Also an outline for the construction of the characteristic
variety (JSJ decomposition) in such manifolds is given. Some non-Haken and
atoroidal examples are given.
