We discuss a family of arithmetic cross-sections for geodesic flow on quotients of the hyperbolic plane by cocompact torsion-free Fuchsian groups. In joint work with Svetlana Katok, we study generalized Bowen-Series boundary maps, whose natural extensions have global attractors with finite rectangular structure. The associated arithmetic cross-section is parametrized by the attractor, and this construction allows us to represent the geodesic flow as a special flow over a symbolic system of coding sequences. In special cases where the "cycle ends" are discontinuity points of the boundary maps, the resulting symbolic system is sofic. This extends pioneering result of Adler-Flatto and recent work of Katok-Ugarcovici.