Embedding rank one simple groups into rank one simple Riesz groups

We give a method for embedding a large family of partially ordered simple groups of rank one into simple Riesz groups of rank one. In particular, we answer in the affirmative a question of Wehrung, by constructing a torsion-free, simple Riesz group G of rank one containing an interval D not subset of or equal to G(+) such that 2D = G(+). We sketch some potential applications of this result in the context of monoids of intervals and K-Theory of rings.