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ISphere conformal map of a rational Cantor Kleinian group:

a = {{0, 3}, {1, 2*I}}
b = {{1, 0}, {I, 2}}
A = {{1/3, 2/3}, {0, 1}}
B = {{1/3, 0}, {0, 1}}
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Half plane to disk of the middle half rational Cantor:
a = {{0, 2}, {1, I}}
b = {{1, 0}, {I, 1}}
A = {{1/2, 1/2}, {0, 1}}
B = {{1/2, 0}, {0, 1}}
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