3-manifolds built from injective handlebodies

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.