In the 1970s, Elliott gave a complete classification of AF C*-algebras using the K_0 functor. The ordered groups in the range of this invariant are precisely the countable members of the class of dimension groups. One question that has remained unanswered since the early days of this theory is how to characterize those countable dimension groups that are ultrasimplicial, meaning that they can be written as inductive limits of simplicial groups in which the maps are injective.

I will present some positive and negative results for rank two simple dimension groups with unique state, the image of which has rank one.