# Tag Info

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Yes, this is indeed a form of duality. The following outline follows the approach from Perspectives on Projective Geometry by J. Richter-Gebert. Cayley-Klein metric Probably the best way to understand this is probably using Cayley-Klein metrics. That's a very general way to measure distances and lengths in projective geometry. You start by fixing one ...

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The usefulness of the Poincaré model (or of Klein's) is that it shows that if Euclidean geometry is consistent, then also hyperbolic geometry is. This is because the hyperbolic axioms are true in the model, so a contradiction in hyperbolic geometry would yield a contradiction in Euclidean geometry. It would be wrong to prove theorems in hyperbolic geometry ...

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No. There is an infinite-dimensional space of homeomorphisms of the circle; the space of Mobius transformations is the 3-dimensional $PSL_2(\Bbb R)$. But more pointedly, the restriction of a Mobius transformation to the ideal boundary can have at most two fixed points if it's not the identity; but any closed subset of $S^1$ is the set of fixed points of some ...

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As proposed in the comments, I posted this question on MathOverflow and it got an answer there. Here is the link: The question on MathOverflow (answered) Thanks to everyone who read the question and thought about it.

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That is not true. A simple example is the following one: consider the hyperbolic isometry \begin{equation*} \phi:= \left[ \begin{array}{l l} 4 &0\\ 0 &\frac{1}{4}\\ \end{array} \right] \in\text{PSL}(2,\mathbb{R})=\text{Isom}^+\left(\mathbb{H}^2\right) \end{equation*} and the finite order elliptic element \begin{equation*} \rho:= \left[ \begin{...

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