I want to that any tetrahedron in $\Bbb{H}^3$ can be transformed into a tetrahedron that has $0, 1, \infty, z$ as vertices. Also I need to show that the above defined tetrahedron has maximum volume if $z=e^{\frac{2\pi i}{6}}$ and I need to find the maximum volume. Please provide some reference or give me some hints. I have a very limited exposure to hyperbolic geometry.
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Please also refer some good undergraduate text of three dimensional hyperbolic geometry. – toric_actions Mar 04 '19 at 21:05
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First of all, what do you mean by “0, 1, ∞, z”? I'm not aware of a convention that uses complex scalars as coordinates in a 3space. – Anton Sherwood Mar 05 '19 at 00:58
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The upper half space model of three dimensional hyperbolic geometry is the complex plane plus the point at infinity and the region above the complex plane in Euclidean three space. – toric_actions Mar 05 '19 at 22:24
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1Ah. It's good to specify which model you mean. – Anton Sherwood Mar 05 '19 at 22:26
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Why do you need to find the maximal volume? I don't think this is an easy exercise, or there is a very simple way to see it, especially if you are new to hyperbolic geometry. You can look at chapter 7 of Thurstons notes. – Mar 27 '19 at 16:39