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We describe a family of arithmetic cross-sections for geodesic flow oncompact surfaces of constant negative curvature based on the study ofgeneralized Bowen-Series boundary maps associated to cocompact torsion-freeFuchsian groups and their natural extensions, introduced in Katok-Ugarcovici's2017 paper. If the boundary map satisfies the short cycle property, i.e., theforward orbits at each discontinuity point coincide after one step, the naturalextension map has a global attractor with finite rectangular structure, and theassociated arithmetic cross-section is parametrized by the attractor. Thisconstruction allows us to represent the geodesic flow as a special flow over asymbolic system of coding sequences. In special cases where the "cycle ends"are discontinuity points of the boundary maps, the resulting symbolic system issofic. Thus we extend and in some ways simplify several results ofAdler-Flatto's 1991 paper. We also compute the measure-theoretic entropy of theboundary maps.

Adler and Flatto revisited: cross-sections for geodesic flow on compact surfaces of constant negative curvature