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The (0,1,∞) Theorem
If L = (α,β,γ) is a list of distinct points of E+,
then there is a unique t in I+(2) which maps L to (0,1,∞).
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Proof
Suppose first that none of α, β and γ is ∞.
By Theorem I1, any element t of I+(2) is a Mobius transformation.
Then t maps α to 0 if and only if the numerator has a factor (z-α),
and t maps γ to ∞ if and only if the numerator has a factor (z-γ).
To achieve both, t must have the form t(z) = κ(z-α)/(z-γ), with κ ≠ 0.
| Such a t maps β to 1 if and only if |
1 = t(β) = κ(β-α)/(β-γ) |
| i.e. |
κ = (β-γ)/(β-α). |
Thus, there is a unique t in this case.
Now suppose that one of the points is ∞.
We can choose δ not equal to α, β or γ (so, in particular, not ∞).
Let r be the Mobius transformation r(z) = 1/(z-δ).
As r(δ) = ∞, r maps α, β and γ to non-infinte points
α', β' and γ'.
By the first part, there is a transformation s mapping (α',β',γ') to (0,1,∞).
Then t = sor maps (α,β,γ) to (0,1,∞), as required.
If u has the same effect on (α,β,γ), the uot-1 maps
(0,1,∞) to itself.
By the first part, there is a unique such map, obviously the identity.
Thus u = t, so t is unique.
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