Hyperbolic Surfaces
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Contents |
[edit] 1 Introduction
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[edit] 2 Construction and examples
<a href="/File:Polygon_construction.png" class="internal" title="Enlarge"> <img src="/skins/common/images/magnify-clip.png" width="15" height="11" alt="" /> </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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