The universal cover of 3-manifolds built from injective handlebodies is ${\mathbb{R}}^3$

This paper gives a proof that the universal cover of a closed 3-manifold built from three ${\pi }_1$-injective handlebodies is homeomorphic to $\mathbb R^3$. This construction is an extension to handlebodies of the conditions for gluing of three solid tori to produce non-Haken Seifert fibered manifolds with infinite fundamental group. This class of manifolds has been shown to contain non-Haken non-Seifert fibered manifolds.