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Given a hyperbolic $4n$-gon $P$ in the Poincaré disk, how can we construct explicitly the subgroup $G < \mathrm{Aut}{\left(\mathbb{D}\right)}$ which gives a tiling of $\mathbb D$ with fundamental domain $P$ ?

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How is the polygon given? If the vertices are given by points in $\mathbb D$ then it is a simple matter to write down the Möbius transforms matching the sides appropriately. These transforms will generate your group. –  t.b. Nov 13 '11 at 7:46

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