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In hyperbolic geometry, any line can have infinite "parallel" lines. So, it is possible to have a quadrilateral similar to that depicted in the figure, where the upper and lower sides are actually parallel hyperbolic lines.

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This is the metric that results in the Poincaré half-plane model of the hyperbolic plane.

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There is another common model where the points of the projective plane consist of the points of the Euclidean plane and equivalence classes of lines for the equivalence relation "is parallel to". The mental image is that the latter types of points are to be thought of as the "point at infinity" that the class of parallel lines intersects at. (and the lines ...

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The group $\Gamma = \langle \gamma_1,\gamma_2\rangle$ is not discrete. To prove this, consider the "height map" $\mathbb{H}^2 \to \mathbb{R}$ defined by $f(x,y) = \ln(y)$. The inverse images of points under this map are the horocycles $y = \text{(constant)}$ all based at $\infty$, each of which is preserved by both $\gamma_1$ and $\gamma_2$, and therefore ...

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