Closing Large Hyperbolic 2g-gons

Poincaré's uniformisation theorem says that any Riemann surface is conformally equivalent to a unique (up to isometry) surface of constant Gauss curvature 0, 1 or –1. The (topologically) richest of these three worlds is for curvature –1 formed of hyperbolic surfaces, in which we are interested. Hyperbolic surfaces can all be seen as a quotient H/G of Poincaré's upper half-plane H by a subgroup G of Isom(H), the group of isometries of H, where G contains no elliptic elements. Now for a subgroup G of PSL2(R) generated by given hyperbolic elements, it is a very difficult question to determine if G is discrete or not and if so get geometric information on the quotient H/G. The classical approach to this problem is to consider orientation preserving isometries of H as the action of PSL2(R) on H (as Möbius transformations) and describe groups as generated by hyperbolic elements (usually labelled α, β, . . .). In this thesis, the hyperbolic elements will be seen as generated by pairs of half-turns or points that will be considered in SL2(R). This distinction will prove to be useful, as keeping track of the sign gives additional information on the position of the points. The main result of this thesis is to give a sufficient criteria for the group generated by half-turn pairs around a given set of points to describe a hyperelliptic surface (closed or with 1 or 2 half-cylinders), in terms of trace of products of points. Moreover, any closed hyperelliptic surface (of genus g ≥ 2) can be described in such a way, meaning that this gives a parametrisation of all hyperelliptic surfaces in Teichmüller space (different from more classical coordinates, like Fenchel-Nielsen). More specifically, any closed hyperelliptic surface can be obtained by closing a large hyperbolic n-gon P = p1, . . . , pn, where large means there exists n non-intersecting geodesic lines l1, . . . , ln each li passing through pi ∈ P, in a way that each li separates H in two half-planes, such that one of them contains all the other points and lines. Geometric interpretation of different trace inequalities will lead to trace criterions for largeness, for example we will show that a given polygon is large and positively oriented if and only if trace(pk . . . p1) < –2 ∀k ≥ 2.