  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 left and right tail
equivalence on the Bratteli diagram and the vertical and horizontal foliations
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.
