Hyperbolic Surfaces
From Manifold Atlas
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
...
2 Construction and examples
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
...
4 Classification/Characterization
...
5 Further discussion
...