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A proof is given in Example 8.10. of https://www.amazon.de/-/en/Johan-L-Dupont/dp/9810245084 . It is however not at all the elementary proof that you seem to be after. There is an exact sequence $$H_1 …

Properness is already needed to have a well-defined boundary at infinity, i.e., with a topology not depending on the chosen base point. This is Proposition III.3.7 in Bridson-Haefliger, which builds o …

Which hyperbolic $3$-manifolds are known to have quadratic cusp shape? Explanations: Cusps of hyperbolic $3$-manifolds are products torus x interval. They lift to horoballs in hyperbolic $3$-space, wh …

Kojima has proved that every finite group occurs as the symmetry group of some compact hyperbolic 3-manifold. Hence there is nothing like a hyperbolic 3-manifold of maximal symmetry. The same result …

The statement is only true if you restrict to geometrically finite hyperbolic metrics (possibly of infinite volume) and ignore parabolic elements, which basically means that you ignore the boundary …

The general statement of Mostow-Prasad rigidity cited from http://repository.ias.ac.in/36364/1/36364.pdf is as follows. Let $G$ (resp. $G^\prime$) be a semi-simple analytic group and $\Gamma$ (resp. …

The group of parabolic isometries fixing a point at infinity is isomorphic to ${\mathbb C}$. (Because it acts simply transitively on a horosphere $H$.) The discrete group $\Gamma$ intersects this stab …

This is a very special case of the Fock-Goncharov construction. Divide your ideal n-gon into n-2 ideal triangles. Given one cross ratio associated to each edge (i.e., to the 4 ideal vertices of the …

This means that the boundary is geodesic with cusps in the marked points. The easiest example is a disk with 3 marked points on its boundary. In this case the hyperbolic metric is given by identifica …

Think of $S^1$ as the ideal boundary of the hyperbolic plane, then every embedded $S^0\subset S^1$ determines a unique geodesic in the hyperbolic plane. The linking number of two embedded $S^0$s is de …

This is a Theorem of Yoshida, the reference is Yoshida, Tomoyoshi: ''The η-invariant of hyperbolic 3-manifolds.'' Invent. Math. 81, 473-514 (1985). http://mathlab.snu.ac.kr/~top/articles/Yoshida.pd …

For a hyperbolic surface $S$ and a homology class $h\in H_1(S)$ its stable norm is defined as $\lim_{n\to\infty}\frac{1}{n}l(nh)$, where $l(nh)$ means the minimal length among all closed geodesics rep …

About the relation to fractals: for Fuchsian groups of the first kind, the limit set has Hausdorff dimension 1, i.e., it is not fractal. However, for all other quasifuchsian groups of the first kind …

Too long for a comment and not sure whether this counts as a big picture, but anyway: There is a general formula for the covolume of S-arithmetic lattices in symmetric spaces, you find it in Prasad's …

2-bridge links $L(p/q)$ are described by the continuous fraction expansion $\frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right]$, where the $a_i$ are the numbers of twists in the boxes below: Looking at so …