Systolic Geometry - Systolic Hyperbolic Geometry

Systolic Hyperbolic Geometry

The study of the asymptotic behavior for large genus g of the systole of hyperbolic surfaces reveals some interesting constants. Thus, Hurwitz surfaces Σ_g defined by a tower of principal congruence subgroups of the (2,3,7) hyperbolic triangle group satisfy the bound

and a similar bound holds for more general arithmetic Fuchsian groups. This 2007 result by Katz, Schaps, and Vishne generalizes the results of Peter Sarnak and Peter Buser in the case of arithmetic groups defined over Q, from their seminal 1994 paper (see below).

A bibliography for systoles in hyperbolic geometry currently numbers forty articles. Interesting examples are provided by the Bolza surface, Klein quartic, Macbeath surface, First Hurwitz triplet.