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It is straightforward to verify that the curve of shortest hyperbolic length connecting $0$ to $s > 0$ is the line segment $[0,s]$. One computes $$\rho(0,s) = \int_0^s \frac{2}{1-t^2}\,dt = \log \frac{1+s}{1-s}.$$ Since holomorphic automorphisms of the unit disk leave the hyperbolic length invariant, given two distinct $z,w \in \mathbb{D}$ we find ...


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When dealing with manifolds, volume forms are often conflated with measures. Here, the volume form on the upper half plane is $|y|^{-2}dxdy$, so the measure of a Borel set is its integral with respect to that volume form. The group $SL(2,\mathbb{R})$ acts on the upper half plane by fractional linear transformations, $$\begin{pmatrix} a & b \\ c & ...


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For the first part: it would maybe be more correct to call $\Delta$ a semi-ideal triangle; two of its three vertices are at infinity. On the other hand, the union of $\Delta$ and $\bar{\Delta}$ does form the ideal triangle $\langle1, i, -i\rangle$, and the tiling generated by $a$ and $b$ induces a tiling of these ideal triangles, the uniform $\{\infty, 3\}$ ...


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Here are two sketches (that should at least get you started even if they're not exactly what you're seeking): (Geometric) Let $c$ be a non-zero vector at $x$ that bisects the hyperbolic angle between $v$ and $w$, and let $C$ be the hyperbolic line tangent to $c$ at $x$. Euclidean inversion in $C$ (i.e., hyperbolic reflection across $C$) exchanges $v$ and ...


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Does there exist a conformal bijection/Mobius transformation from the open unit disk to the whole complex plane? No. The inverse of such a map would be a bounded function holomorphic in $\mathbb C$. Such a function must be a constant by Liouville's theorem. Does there exist a conformal bijection/Mobius transformation from the annulus to the ...


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You're using your Euclidean intuition of distance to reach a clearly not clearly false conclusion! That angles are preserved only means that the geodesic between $u$ and $v$ is perpendicular to any horocycle tangent to $u$ or $v$. It doesn't mean that being "cut in half" is preserved, and in fact, there's not even an unambiguous notion of "half" for a ...



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