Bratteli diagrams, translation flows and their $C^*$-algebras

In [LT16], Kathryn Lindsey and the second author constructed a translation surface from a bi-infinite Bratteli diagram. We continue an investigation into these surfaces. The construction given in [LT16] was essentially combinatorial. Here, we provide explicit links between the path space of the Bratteli diagram and the surface, including various intermediate topological spaces. This allows us to relate the $C^{*}$-algebras associated with tail equivalence on the Bratteli diagram and the foliation of the surface, under some mild hypotheses. This also allows us to relate the K-theory of the $C^{*}$-algebras involved. We also treat the case of finite genus surfaces in some detail, where the process of Rauzy-Veech induction (and its inverse) provide an explicit construction of the Bratteli diagrams involved.