Simply connected 3-manifolds with a dense set of ends of specified genus

We show that for every sequence ▫$(n_i)$▫, where each ▫$n_i$▫ is either an integer greater than 1 or is ▫$infty$▫, there exists a simply connected open 3-manifold ▫$M$▫ with a countable dense set of ends ▫${e_i}$▫ so that, for every ▫$i$▫, the genus of end ▫$e_i$▫ is equal to ▫$n_i$▫. In addition, the genus of the ends not in the dense set is shown to be less than or equal to 2. These simply connected 3-manifolds are constructed as the complements of certain Cantor sets in ▫$S^3$▫. The methods used require careful analysis of the genera of ends and new techniques for dealing with infinite genus