To avoid discriminatory uses of their data, organizations can learn to map them into a representation that filters out information related to sensitive attributes. However, all existing methods in fair representation learning generate a fairness-information trade-off. To achieve different points on the fairness-information plane, one must train different models. In this paper, we first demonstrate that fairness-information trade-offs are fully characterized by rate-distortion trade-offs. Then, we use this key result and propose SoFaiR, a single shot fair representation learning method that generates with one trained model many points on the fairness-information plane. Besides its computational saving, our single-shot approach is, to the extent of our knowledge, the first fair representation learning method that explains what information is affected by changes in the fairness / distortion properties of the representation. Empirically, we find on three datasets that SoFaiR achieves similar fairness information trade-offs as its multi-shot counterparts.