A Riemann surface of genus g has at most 84(g − 1) automorphisms. A Hurwitz surface is one for which this maximum is attained; the corresponding group of automorphisms is called a Hurwitz group. By uniformization, the surface admits a hyperbolic structure wherein the automorphisms act by isometry. Such isometries descend from the (2,3,7) triangle group T acting on the universal cover H2.
We develop a combinatorial approach which leads to a classification of the conjugacy classes of hyperbolic elements of T, arranged by length. This allows us to study the closed geodesics of Hurwitz surfaces by performing calculations in the
corresponding Hurwitz groups.
We identify the systoles and other short curves on most of the Hurwitz surfaces of genus less than 10,000. We also determine which of these surfaces are chiral and which are amphichiral. In addition, we show that certain families of closed geodesics are simple on every Hurwitz surface.
