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 + τ Z , ( Z + τ Z ) ∩ 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.