Classical Geometry - Lecture Notes by Calegari D.

By Calegari D.

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It turns out that any orbifold whose underlying surface is not the sphere, but a surface of higher genus, is a good orbifold, and is obtained as a quotient X/Γ for X one of S2 , E2 , H2 and Γ a properly discontinuous group of isometries. 4. Discrete subgroups of P SL(2, R). 1. Glueing hyperbolic polygons. Gluing up hyperbolic polygons to make a closed hyperbolic surface is not essentially different from glueing up spherical or flat polygons. If Σg denotes the unique (up to homeomorphism) closed orientable surface of genus g > 1, then we can decompose Σg (nonuniquely) into pairs of pants.

T1 ψ(C) intersects C in one of finitely many possibilities. After twisting some more in elements of C, we can assume the image of C is one of finitely many possibilities, which can be explicitly identified. In short, ψ can be written as a product of Dehn twists. Now, for each such twist tc , we can replace tc by td ttd (c) t−1 d where each d, td (c) intersect C more simply than c. In this way, each tc can be expanded as a product of Dehn twists in curves which intersect C very simply. After twisting in C, it follows that these involve only finitely many possibilities, which can be explicitly enumerated.

This is a simple example of what is known as a category; in particular, it is a category with one object X. 2. More generally, a category can be thought of as a collection of objects (denoted O) and a collection of morphisms or admissible maps between objects (denoted M). Every morphism m has a source object s(m) and a target object t(m), which might be the same object. e. 1x m = m for any m with t(m) = x m1x = m for any m with s(m) = x The composition of two morphisms is another morphism, and this composition is associative; composition can be expressed as a function c : M × M → M.

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