Approximate systoles for a hyperbolic surface

[siddhartha-gadgil]

• Hyperbolic surfaces are given by pairs of pants glued with twists, with pairs of pants given by right-angled hexagons.
• For each of these, we determine if there is an approximate systole with a large number of curves, where the systole is the set of geodesics of minimal length.
• Goal is to try to determine, by compact enumeration, the largest systole for a surface of genus 3.

[siddhartha-gadgil]

• The existence of a systole implies that of an epsilon-aprroximate systole, and hence we can get an upper bound on the size of a systole.
• By compactness, this is also a lower bound for epsilon small enough, but we need this to be effective, or have an alternative argument using, for example, implicit function theorem or real algerbraic geometry.