Hyperbolic Surfaces
From Manifold Atlas
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Contents |
[edit] 1 Introduction
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[edit] 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.
[edit] 3 Invariants
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[edit] 4 Classification/Characterization
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[edit] 5 Further discussion
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