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EDITED: One "canonical" choice for the basepoint for the Voronoi decomposition, also called the Dirichlet decomposition, is a point of of maximal injectivity radius. For example, when displaying the …

Here are some pointers in the right direction. For 1), I would suggest that you read Peter Scott's classic article on the eight Thurston geometries. The article can be found on his webpage and it in …

To define an ideal triangulation one typically starts with a hyperbolic metric. Suppose that $T^2 = S^1 \times S^1$ is the two-torus and $X = T - D$ is the torus minus an closed embedded disk. There …

Casson and Gordon's argument goes through for minimal genus splittings of manifolds with boundary. See Moriah's paper "On boundary primitive manifolds and a theorem of Casson–Gordon". Now we follow …

This is an expansion of Misha's comment. Since $g : (S,\rho) \to N$ is a pleating map we have $g$ is $1$-Lipschitz. That is, for any $x, y \in S$ we have $d_N(g(x), g(y)) \leq d_\rho(x, y)$. In fac …

For the pants, yes. In general, no. To prove this for the pants, classify all geodesic arcs and just observe the result. There are many ways to find a "no" example in the general case; the first on …

You are describing the "grafting construction". I am not an expert: however if you google "grafting a Riemann surface" there are many references available. If you take $h$ to be non-negative you c …

Consider the ideal triangle with vertices at infinity, zero, and one. Let $C$ be the semicircle perpendicular to the vertical line $[0, \infty]$ and meeting $1/2 + i/2$ (ie the midpoint of the semici …

There is a topological criterion due to Thurston. Using the JSJ machine (and work of many others) this criterion can also be phrased algebraically. I'll essay these below. Please note that the situ …

EDIT - Maxime has answer the question in the comments. The desired $h$ is not a function of the lengths $\ell_1, \ell_2, \ell_3$, but rather lies in an open interval. The minimum (resp. maximum) of …

Following Gordon's corollary "For every problem you can't solve, there is an easier problem you also can't solve" let me suggest two easier problems. Suppose that P is the hyperbolic paraboloid solvin …

A convenient definition of the cross-ratio, in hyperbolic geometry, is as follows. We work in the upper half plane model; see here, for example. Suppose that $a$, $b$, $c$, and $d$ are points on the …

A pair of optimal pants decompositions $A, B$ need not have disjoint shortest curves. Here is an example. Let $S$ be the genus two hyperbolic surface built from four equilateral right-angled hexago …

There are numerous algorithms to decide this question. At bottom all of these are based on the "monogon" and "bigon" condition: If $\alpha$ is a closed loop on a surface then we can homotope $\alpha …

[See Peter Scott's Bulletin article for more information.] Typically, we say an orbifold $Q$ is hyperbolic if it comes to us as a quotient of hyperbolic space $H^n$ by the action of a discrete group …