  We describe arithmetic cross-sections for geodesic flow on compact surfaces
of constant negative curvature using generalized Bowen-Series boundary maps and
their natural extensions associated to cocompact torsion-free Fuchsian groups.
If the boundary map parameters are extremal, that is, each is an endpoint of a
geodesic that extends a side of the fundamental polygon, then the natural
extension map has a domain with finite rectangular structure, and the
associated arithmetic cross-section is parametrized by this set. This
construction allows us to represent the geodesic flow as a special flow over a
symbolic system of coding sequences. Moreover, each extremal parameter has a
corresponding dual parameter choice such that the "past" of the arithmetic code
of a geodesic is the "future" for the code using the dual parameter. This
duality was observed for two classical parameter choices by Adler and Flatto;
here we show constructively that every extremal parameter set has a dual.
