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In general given discrete subgroups $\Gamma_2 < \Gamma_1 < PSL(2,\mathbb{R}) = \text{Isom}(\mathbb{H}^2)$, there is an equation $$\text{Area}(\mathbb{H}^2 / \Gamma_2) = [\Gamma_1:\Gamma_2] \cdot \text{Area}(\mathbb{H}^2 / \Gamma_1) $$ So if $\text{Area}(\mathbb{H}^2 / \Gamma_1)$ is finite, as is the case when $\Gamma_1 = PSL(2,\mathbb{Z})$, then ...


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Yes, the reference is Jim Cannon's article in Bedford, Keane, Series @book{bedford1991ergodic, title={Ergodic theory, symbolic dynamics, and hyperbolic spaces}, author={Bedford, Tim and Michael (Michael S.) Keane and Series, Caroline}, year={1991}, publisher={Oxford University Press} }


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$\def\Z{\mathbb{Z}}$ $\def\union{\cup}$ If $\Gamma$ is of index $n$ in $SL_2(\Z)$ (where we allow $n = \infty$) and $D$ denotes the fundamental domain for $SL_2(\Z)$, then a fundamental domain for $\Gamma$ is given by $$ \gamma_1 D \union \gamma_2 D \union \ldots \union \gamma_n D $$ where the $\gamma_i$ are a choice of coset representatives. As the action ...


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$xy = d^2 \Leftrightarrow y = \frac{d^2}{x}$. Tangent line equation to $f(x)$ in point $x=a$ is $g(x) = f(a) + f'(a)(x-a)$. $f'(x) = -\frac{d^2}{x^2}$, so $g(x) = \frac{d^2}{a} - \frac{d^2}{a^2}(x-a) = -\frac{d^2}{a^2}x + \frac{2d^2}{a}$, hence $m = -\frac{d^2}{a^2}, c = \frac{2d^2}{a}$ and $m = -\frac{c^2}{4d^2}$.


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An alternative method, we know the quadratic equation $\alpha r^2+\beta r+ \gamma=0$ has only one(repeated) root iff $\beta^2=4\alpha\gamma$. Plug in $y=mx+c$ into $xy=d^2$ you get $mx^2+cx-d^2=0$, because tangent line intersects the curve at only 1 point we conclude $c^2=-4md^2$.


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Read carefully: We could repeat the calculation for much smaller parallaxes, for example 0",1, and we would find $k$ to be greater than a million times the diameter of the earth's orbit. Thus, if the Euclidean Geometry and the Fifth Postulate are to hold in actual space, $k$ must be infinitely great. That is to say, there must be stars whose ...


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More generally, let $X$ be a $n$-dimensional closed hyperbolic manifold and let $G$ denote its fundamental group. It is a standard theorem in Riemannian geometry that the universal cover of such a manifold is isometric to the hyperbolic space $\mathbb{H}^n$. Therefore, $G$ acts geometrically on $\mathbb{H}^n$ and we deduce from Milnor-Svarc lemma that $G$ ...


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The usual sine and cosine are the solutions to the differential equation $f''(x)=-f(x)$ that satisfy $\sin(0)=0$, $\sin'(0)=1$ and $\cos(0)=1$, $\cos'(0)=0$. The hyperbolic functions are solutions to $f''(x)=f(x)$ with the same boundary conditions. These characterizations work with either $\mathbb R$ or $\mathbb C$ as the domain.


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Lets start at the beginning There is no 3 dimensional Euclidean surface that represents the complete hyperbolic plane (Hilberts theorem ) There are Surfaces of revolution that have (outside some boundaries ) a constant negative curvature $K$ . All surfacesof revolution revolving around the $x_3 $ axis can be formulated as: $ x_1 = \phi(v) \cos u $ , $ ...


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EDIT: essay questions: what is the exponential map in the ordinary plane? What is the exponential map on the ordinary unit sphere, say at point $(a,b,c)$ with $a^2 + b^2 + c^2 = 1,$ in tangent direction $(d,e,f),$ so that $ad+be+cf=0?$ ORIGINAL: There are two types of geodesics, vertical lines and semicircles with center on the real axis. All you need for ...



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