Hyperbolic Surfaces

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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>

<a href="/File:Polygon_construction.png" class="internal" title="Enlarge"> <img src="/skins/common/images/magnify-clip.png" width="15" height="11" alt="" /> </a>

The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.

Any hyperbolic metric on a closed, orientable surface S_g of genus g\ge 2 is obtained by the following construction: choose a geodesic 4g-gon in the hyperbolic plane {\Bbb H}^2 with area 4(g-1)\pi. (This implies that the sum of interior angles is 2\pi.) Then choose orientation-preserving isometries I_1,J_1,\ldots,I_g,J_g which realise the gluing pattern of S_g: for j=1,\ldots,g we require that I_j maps a_j to \overline{a}_j, J_j maps b_j to \overline{b}_j. Let \Gamma\subset Isom^+\left({\Bbb H}^2\right) be the subgroup generated by I_1,J_1,\ldots,I_g,J_g. Then \Gamma is a discrete subgroup of Isom^+\left({\Bbb H}^2\right) and \Gamma\backslash{\Bbb H}^2 is a hyperbolic surface diffeomorphic to {\Bbb H}^2.

The moduli space of hyperbolic metrics on the closed, orientable surface S_g is \left(6g-6\right)-dimensional.

3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References

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