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On Poicarés Uniformization Theorem
Linköping University, Department of Mathematics.
2006 (English)Independent thesis Basic level (professional degree), 20 points / 30 hpStudent thesis
Abstract [en]

A compact Riemann surface can be realized as a quotient space $\mathcal{U}/\Gamma$, where $\mathcal{U}$ is the sphere $\Sigma$, the euclidian plane $\mathbb{C}$ or the hyperbolic plane $\mathcal{H}$ and $\Gamma$ is a discrete group of automorphisms. This induces a covering $p:\mathcal{U}\rightarrow\mathcal{U}/\Gamma$.

For each $\Gamma$ acting on $\mathcal{H}$ we have a polygon $P$ such that $\mathcal{H}$ is tesselated by $P$ under the actions of the elements of $\Gamma$. On the other hand if $P$ is a hyperbolic polygon with a side pairing satisfying certain conditions, then the group $\Gamma$ generated by the side pairing is discrete and $P$ tesselates $\mathcal{H}$ under $\Gamma$.

Place, publisher, year, edition, pages
Matematiska institutionen , 2006. , 59 p.
Keyword [en]
Hyperbolic plane, Fuchsian group, Riemann surface, uniformization, fundamental domain, Poincar\'es theorem
National Category
URN: urn:nbn:se:liu:diva-7968ISRN: LITH-MAT-EX--2006/14OAI: diva2:22858
2006-12-20, Glashuset, B-huset, Linköping, 13:15
Available from: 2007-01-11 Created: 2007-01-11

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