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• W2963704829 abstract "There is a classical extension of Möbius automorphisms of the Riemann sphere into isometries of the hyperbolic space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H cubed> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {H}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is called the Poincaré extension. In this paper, we construct extensions of rational maps on the Riemann sphere over endomorphisms of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H cubed> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {H}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> exploiting the fact that any holomorphic covering between Riemann surfaces is Möbius for a suitable choice of coordinates. We show that these extensions define conformally natural homomorphisms on suitable subsemigroups of the semigroup of Blaschke maps. We extend the complex multiplication to a product in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper H cubed> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>H</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {H}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that allows us to construct an extension of any given rational map which is right equivariant with respect to the action of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper P upper S upper L left-parenthesis 2 comma double-struck upper C right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>S</mml:mi> <mml:mi>L</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>C</mml:mi> </mml:mrow> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>PSL(2,mathbb {C})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W2963704829 date "2015-07-29" @default.
• W2963704829 modified "2023-09-25" @default.
• W2963704829 title "On Poincaré extensions of rational maps" @default.
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• W2963704829 doi "https://doi.org/10.1090/ecgd/281" @default.
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