Hyperbolic Surfaces
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== Construction and examples == | == Construction and examples == | ||
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+ | <div class="thumb tright"><div class="thumbinner" style="width:214px;"><a href="/File:Polygon_construction.png" class="image"><img alt="" src="/images/5/5e/Polygon_construction.png" width="212" height="207" class="thumbimage" /></a><div class="thumbcaption"> | ||
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+ | </div>The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon. | ||
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Any hyperbolic metric on a closed, orientable surface $S_g$ of genus $g\ge 2$ is obtained by the following | Any hyperbolic metric on a closed, orientable surface $S_g$ of genus $g\ge 2$ is obtained by the following |
Revision as of 10:10, 29 October 2011
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
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2 Construction and examples
<a href="/File:Polygon_construction.png" class="image"><img alt="" src="/images/5/5e/Polygon_construction.png" width="212" height="207" class="thumbimage" /></a>
Any hyperbolic metric on a closed, orientable surface of genus is obtained by the following construction: choose a geodesic -gon in the hyperbolic plane with area . (This implies that the sum of interior angles is .) Then choose orientation-preserving isometries which realise the gluing pattern of : for we require that maps to , maps to . Let be the subgroup generated by . Then is a discrete subgroup of and is a hyperbolic surface diffeomorphic to .
The moduli space of hyperbolic metrics on the closed, orientable surface is -dimensional.
3 Invariants
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4 Classification/Characterization
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5 Further discussion
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