Hyperbolic Surfaces

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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$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; ... </wikitex> == Construction and examples == <wikitex>; Any hyperbolic metric on a closed, orientable surface $S_g$ of genus $g\ge 2$ is obtained by the following construction: choose a geodesic g$-gon in the hyperbolic plane ${\Bbb H}^2$ with area (g-1)\pi$. (This implies that the sum of interior angles is \pi$.) Then choose orientation-preserving isometries $I_1,J_1,\ldots,I_g,J_g$ which realise the gluing pattern of $S_g$: for $j=1,\ldots,g$ we require that $I_j$ maps $a_j$ to $\overline{a}_j$, $J_j$ maps $b_j$ to $\overline{b}_j$. Let $\Gamma\subset Isom^+\left({\Bbb H}^2\right)$ be the subgroup generated by $I_1,J_1,\ldots,I_g,J_g$. Then $\Gamma$ is a discrete subgroup of $Isom^+\left({\Bbb H}^2\right)$ and $\Gamma\backslash{\Bbb H}^2$ is a hyperbolic surface diffeomorphic to ${\Bbb H}^2$. The moduli space of hyperbolic metrics on the closed, orientable surface $S_g$ is $\left(6g-6\right)$-dimensional. </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> <!-- == Acknowledgments == ... == Footnotes == <references/> --> == References == {{#RefList:}} <!-- == External links == * The Wikipedia page about [[Wikipedia:Page_name|link text]]. --> <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]]S_g$ of genus $g\ge 2$$g\ge 2$ is obtained by the following construction: choose a geodesic $4g$$4g$-gon in the hyperbolic plane ${\Bbb H}^2$${\Bbb H}^2$ with area $4(g-1)\pi$$4(g-1)\pi$. (This implies that the sum of interior angles is $2\pi$$2\pi$.) Then choose orientation-preserving isometries $I_1,J_1,\ldots,I_g,J_g$$I_1,J_1,\ldots,I_g,J_g$ which realise the gluing pattern of $S_g$$S_g$: for $j=1,\ldots,g$$j=1,\ldots,g$ we require that $I_j$$I_j$ maps $a_j$$a_j$ to $\overline{a}_j$$\overline{a}_j$, $J_j$$J_j$ maps $b_j$$b_j$ to $\overline{b}_j$$\overline{b}_j$. Let $\Gamma\subset Isom^+\left({\Bbb H}^2\right)$$\Gamma\subset Isom^+\left({\Bbb H}^2\right)$ be the subgroup generated by $I_1,J_1,\ldots,I_g,J_g$$I_1,J_1,\ldots,I_g,J_g$. Then $\Gamma$$\Gamma$ is a discrete subgroup of $Isom^+\left({\Bbb H}^2\right)$$Isom^+\left({\Bbb H}^2\right)$ and $\Gamma\backslash{\Bbb H}^2$$\Gamma\backslash{\Bbb H}^2$ is a hyperbolic surface diffeomorphic to ${\Bbb H}^2$${\Bbb H}^2$.

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