On the geometry of simple closed geodesics

The leitmotif of this dissertation is the search for length formulas and sharp constants in relation with simple closed geodesics on hyperbolic compact Riemann surfaces. The main tools used are those of hyperbolic trigonometry, topological properties of simple closed geodesics and the convexity of geodesic length functions under the influence of twist transformations. The first question addressed is the question of their explicit density on a compact surface. The existence of a positive number ρ such that a simple closed geodesic on a given surface never crosses all disks of radius ρ is proved. A sharp bound on ρ is calculated, and this bound depends neither on the genus, nor on the choice of a surface. The second part concerns distances between boundary geodesics on surfaces of signature (1,2) and signature (0,4). These surfaces are natural building blocks for Riemann surfaces and lengths on such surfaces gives explicit information concerning twist parameters, contrary to surfaces of signature (0,3). Within the set of all surfaces with given boundary length and systole, maximal surfaces for distance between boundary geodesics are found. The last part is based on the search for surfaces that reach an upper bound among all surfaces of same genus and same systole for minimal length canonical homology bases. This direction has brought new light to the search for maximal surfaces for other problems as well, including properties that a surface incarnating Bers' partition constant would have.