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• W2997231012 abstract "According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose limit set is a generalized Apollonian gasket <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Lamda Subscript upper H> <mml:semantics> <mml:msub> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We design a surgery that relates <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a rational map <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding=application/x-tex>g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose Julia set <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper J Subscript g> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>J</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal {J}_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is (non-quasiconformally) homeomorphic to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Lamda Subscript upper H> <mml:semantics> <mml:msub> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show for a large class of triangulations, however, the groups of quasisymmetries of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Lamda Subscript upper H> <mml:semantics> <mml:msub> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper J Subscript g> <mml:semantics> <mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>J</mml:mi> </mml:mrow> <mml:mi>g</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>mathcal {J}_g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, this group is equal to the group of Möbius symmetries of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Lamda Subscript upper H> <mml:semantics> <mml:msub> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, which is the semi-direct product of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Lamda Subscript upper H> <mml:semantics> <mml:msub> <mml:mi mathvariant=normal>Λ<!-- Λ --></mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>Lambda _H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the classical Apollonian gasket), we give a quasiregular model for the above actions which is quasiconformally equivalent to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=g> <mml:semantics> <mml:mi>g</mml:mi> <mml:annotation encoding=application/x-tex>g</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and produces <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding=application/x-tex>H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle." @default.
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• W2997231012 date "2023-02-01" @default.
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• W2997231012 title "On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections" @default.
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