Equivalence relation groupoids associated with certain linearly ordered dimension groups

We consider linearly ordered, Archimedean dimension groups (G,G+,u) for which the group G/〈u〉 is torsion-free. It will be shown that if, in addition, G/〈u〉 is generated by a single element (i.e., G/〈u〉≅Z), then (G,G+,u) is isomorphic to (Z+τZ,(Z+τZ)∩R+,1) for some irrational number τ∈(0,1). This amounts to an extension of related results where dimension groups for which G/〈u〉 is torsion were considered. We will prove, in the case of the Fibonacci dimension group, that these results can be used to directly construct an equivalence relation groupoid whose C∗-algebra is the Fibonacci C∗-algebra.