# Hyperbolic Surfaces

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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$$ {{Stub}} == Introduction == ; ... == Construction and examples == ;
The orientable surface of genus 2 can be obtained by identifying edges in a regular octagon.
Any hyperbolic metric on a closed, orientable surface S_g of genus g\ge 2 is obtained by the following construction: choose a geodesic g-gon in the hyperbolic plane {\Bbb H}^2 with area (g-1)\pi. (This implies that the sum of interior angles is \pi.) Then choose orientation-preserving isometries I_1,J_1,\ldots,I_g,J_g which realise the gluing pattern of S_g: for j=1,\ldots,g we require that I_j maps a_j to \overline{a}_j, J_j maps b_j to \overline{b}_j. Let \Gamma\subset Isom^+\left({\Bbb H}^2\right) be the subgroup generated by I_1,J_1,\ldots,I_g,J_g. Then \Gamma is a discrete subgroup of Isom^+\left({\Bbb H}^2\right) and \Gamma\backslash{\Bbb H}^2 is a hyperbolic surface diffeomorphic to {\Bbb H}^2. The moduli space of hyperbolic metrics on the closed, orientable surface S_g is \left(6g-6\right)-dimensional.
== Invariants == ; ... == Classification/Characterization == ; ... == Further discussion == ; ... == References == {{#RefList:}} [[Category:Manifolds]]S_g$
of genus $g\ge 2$$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.

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