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Fundamental Polygons for Coverings of the Double-Torus

Dooper, J. (2011) Fundamental Polygons for Coverings of the Double-Torus. Bachelor's Thesis, Mathematics.

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This text is concerned with the construction of fundamental polygons for coverings of finite multiplicity for the double torus, or the orientable surface of genus 2. We will consider the double torus as a geometric surface of constant curvature. We will see that this means that it is locally isometric to the hyperbolic plane and is therefor called a hyperbolic surface. The surfaces we consider are assumed to be connected, compact and orientable. By the classication of compact surfaces, such a surface is always homeomporphic to a sphere with n > 0 handles. A covering of a surface S, is a map from another surface, called the coverings surface, onto S. This map is usually required to be a local homeomorphism such that every point of S is evenly covered. The coverings we are considering are local isometries. A hyperbolic surface can be given as the quotient of the hyperbolic plane by the action of a discontinuous group of isometries. The points of the surface become the orbit of a point under the group action. A surface can thus be given as a discontinuous group of isometries, and we will see that subgroups of this group correspond to covering surfaces for the original surfaces. Such a covering is automatically a local isometry. Another way of describing a surface is by identifying the edges of a polygon. This amounts to labeling the edges of the polygon in a particular way, such that edges of the polygon become identified in pairs. The identification of sides automatically identifies the vertices. If this polygon is constructed in the right geometry, the resulting identifocation space becomes a geometric surface. Constructing a covering for an identication space can be done identifying the edges of seperate polygons. The methods of describing a surface by a discontinuous group and as an identifiction space come together via the notion of fundamental polygon. For a discontinous group, a fundamental polygon is a polygon containing in it's interior precisely one representative for each orbit. The isometries mapping an edge of the fundamental polygon onto another are called side-pairings and they are seen to generate the group and to yield an edge-labeling for the polygon. On the other hand, if a polygon for the identication space is given in the right geometric setting, the edge-identifications can be realized by isometries of that geometry and the polygon becomes a fundamental domain for the group generated by the side-pairings. The double torus can be constructed by identifying the edges of a hyperbolic octagon, and can be given as the group generated by the side-pairings of this octagon. First we discuss hyperbolic geometry, then we discuss the construction of surfaces as an identication space of a polygon and finally we discuss the construction via discontinuous groups. The last part considers some constructions for covering spaces of the double torus. The example of the torus will be used as a guide

Item Type: Thesis (Bachelor's Thesis)
Degree programme: Mathematics
Thesis type: Bachelor's Thesis
Language: English
Date Deposited: 15 Feb 2018 07:47
Last Modified: 15 Feb 2018 07:47

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