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In [LT16], Kathryn Lindsey and the second author constructed a translationsurface from a bi-infinite Bratteli diagram. We continue an investigation intothese surfaces. The construction given in [LT16] was essentially combinatorial.Here, we provide explicit links between the path space of the Bratteli diagramand the surface, including various intermediate topological spaces. This allowsus to relate the $C^{*}$-algebras associated with left and right tailequivalence on the Bratteli diagram and the vertical and horizontal foliationsof the surface, under some mild hypotheses. This also allows us to relate theK-theory of the $C^{*}$-algebras involved. We also treat the case of finitegenus surfaces in some detail, where the process of Rauzy-Veech induction (andits inverse) provide an explicit construction of the Bratteli diagramsinvolved.

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