$3$–manifolds built from injective handlebodies

Abstract

This paper studies 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 construction for gluing three solid tori to produce non-Haken Seifert fibred $3$–manifolds with infinite fundamental group. It is shown that there is an efficient algorithm to decide if a gluing of handlebodies satisfies 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.