An ordered abelian group is called a dimension group if it is the inductive limit of a sequence of direct sums of copies of Z. Dimension groups are of interest in the study of operator algebras because they are the K0-groups of AF C*-algebras.

If, in addition, a dimension group admits such an inductive limit representation in which the maps are injective, then it is called an ultrasimplicial group. The question then arises: exactly which dimension groups are ultrasimplicial?

There have been positive and negative results on this subject. Elliott showed that every totally ordered (countable) group is ultrasimplicial, and Riedel showed that a free simple dimension group of finite rank with a unique state is ultrasimplicial. Much later, Marra showed that every lattice ordered abelian group is ultrasimplicial. On the other hand, Elliott produced an example of a simple dimension group that is not ultrasimplicial, and later Riedel produced a collection of simple free dimension groups that are not ultrasimplicial. I will discuss the history of this subject and go through some calculations in detail.