Reflection Groups, Anti-rational Maps, and Univalent Functions Sabya Mukherjee Tata Institute of Fundamental Research, Mumbai Quasiworld, July 2020 Based on joint works with Kirill Lazebnik (Caltech), Seung-Yeop Lee (South Florida), Russell Lodge (Indiana State), Yusheng Luo (Stony Brook), Mikhail Lyubich (Stony Brook), Nikolai Makarov (Caltech), Sergiy Merenkov (CUNY), and Dimitrios Ntalampekos (Stony Brook). A Kleinian group is a discrete subgroup of Γ of PSL2(C). The Fatou set F(R) of a rational map R is the maximal open subset ◦n of Cb on which the iterates fR g form a normal family. Its complement is the Julia set J (R). The domain of discontinuity Ω(Γ) of a Kleinian group Γ is the maximal open subset of Cb on which Γ acts properly discontinuously. Its complement is the limit set Λ(Γ). Same definitions for anti-holomorphic rational maps and Kleinian reflection groups. Rational Maps and Kleinian Groups p(z) A rational map R : Cb ! Cb is a map of the form R(z) = q(z) , where p(z); q(z) are polynomials. The Fatou set F(R) of a rational map R is the maximal open subset ◦n of Cb on which the iterates fR g form a normal family. Its complement is the Julia set J (R). The domain of discontinuity Ω(Γ) of a Kleinian group Γ is the maximal open subset of Cb on which Γ acts properly discontinuously. Its complement is the limit set Λ(Γ). Same definitions for anti-holomorphic rational maps and Kleinian reflection groups. Rational Maps and Kleinian Groups p(z) A rational map R : Cb ! Cb is a map of the form R(z) = q(z) , where p(z); q(z) are polynomials. A Kleinian group is a discrete subgroup of Γ of PSL2(C). The domain of discontinuity Ω(Γ) of a Kleinian group Γ is the maximal open subset of Cb on which Γ acts properly discontinuously. Its complement is the limit set Λ(Γ). Same definitions for anti-holomorphic rational maps and Kleinian reflection groups. Rational Maps and Kleinian Groups p(z) A rational map R : Cb ! Cb is a map of the form R(z) = q(z) , where p(z); q(z) are polynomials. A Kleinian group is a discrete subgroup of Γ of PSL2(C). The Fatou set F(R) of a rational map R is the maximal open subset ◦n of Cb on which the iterates fR g form a normal family. Its complement is the Julia set J (R). Same definitions for anti-holomorphic rational maps and Kleinian reflection groups. Rational Maps and Kleinian Groups p(z) A rational map R : Cb ! Cb is a map of the form R(z) = q(z) , where p(z); q(z) are polynomials. A Kleinian group is a discrete subgroup of Γ of PSL2(C). The Fatou set F(R) of a rational map R is the maximal open subset ◦n of Cb on which the iterates fR g form a normal family. Its complement is the Julia set J (R). The domain of discontinuity Ω(Γ) of a Kleinian group Γ is the maximal open subset of Cb on which Γ acts properly discontinuously. Its complement is the limit set Λ(Γ). Rational Maps and Kleinian Groups p(z) A rational map R : Cb ! Cb is a map of the form R(z) = q(z) , where p(z); q(z) are polynomials. A Kleinian group is a discrete subgroup of Γ of PSL2(C). The Fatou set F(R) of a rational map R is the maximal open subset ◦n of Cb on which the iterates fR g form a normal family. Its complement is the Julia set J (R). The domain of discontinuity Ω(Γ) of a Kleinian group Γ is the maximal open subset of Cb on which Γ acts properly discontinuously. Its complement is the limit set Λ(Γ). Same definitions for anti-holomorphic rational maps and Kleinian reflection groups. QC deformation techniques: Ahlfors Finiteness Theorem and Sullivan’s No Wandering Domain Theorem. Thurston’s Realization Theorem. Teichmüller space of conformal dynamical systems (Sullivan-McMullen). Mating modular group with quadratic maps (Bullett-Penrose, Bullett-Lomonaco). Laminations for rational maps as analogues of hyperbolic 3-orbifold quotients of Kleinian groups (Lyubich-Minsky). Renormalization and 3-manifolds fibering over the circle (McMullen). Canonical Decomposition Theorem for Thurston maps as an analogue of Torus Decomposition Theorem for 3-manifolds (Pilgrim). Fatou-Sullivan Dictionary Similarities between limit and Julia sets (Fatou). Thurston’s Realization Theorem. Teichmüller space of conformal dynamical systems (Sullivan-McMullen). Mating modular group with quadratic maps (Bullett-Penrose, Bullett-Lomonaco). Laminations for rational maps as analogues of hyperbolic 3-orbifold quotients of Kleinian groups (Lyubich-Minsky). Renormalization and 3-manifolds fibering over the circle (McMullen). Canonical Decomposition Theorem for Thurston maps as an analogue of Torus Decomposition Theorem for 3-manifolds (Pilgrim). Fatou-Sullivan Dictionary Similarities between limit and Julia sets (Fatou). QC deformation techniques: Ahlfors Finiteness Theorem and Sullivan’s No Wandering Domain Theorem. Teichmüller space of conformal dynamical systems (Sullivan-McMullen). Mating modular group with quadratic maps (Bullett-Penrose, Bullett-Lomonaco). Laminations for rational maps as analogues of hyperbolic 3-orbifold quotients of Kleinian groups (Lyubich-Minsky). Renormalization and 3-manifolds fibering over the circle (McMullen). Canonical Decomposition Theorem for Thurston maps as an analogue of Torus Decomposition Theorem for 3-manifolds (Pilgrim). Fatou-Sullivan Dictionary Similarities between limit and Julia sets (Fatou). QC deformation techniques: Ahlfors Finiteness Theorem and Sullivan’s No Wandering Domain Theorem. Thurston’s Realization Theorem. Mating modular group with quadratic maps (Bullett-Penrose, Bullett-Lomonaco). Laminations for rational maps as analogues of hyperbolic 3-orbifold quotients of Kleinian groups (Lyubich-Minsky). Renormalization and 3-manifolds fibering over the circle (McMullen). Canonical Decomposition Theorem for Thurston maps as an analogue of Torus Decomposition Theorem for 3-manifolds (Pilgrim). Fatou-Sullivan Dictionary Similarities between limit and Julia sets (Fatou). QC deformation techniques: Ahlfors Finiteness Theorem and Sullivan’s No Wandering Domain Theorem. Thurston’s Realization Theorem. Teichmüller space of conformal dynamical systems (Sullivan-McMullen). Laminations for rational maps as analogues of hyperbolic 3-orbifold quotients of Kleinian groups (Lyubich-Minsky). Renormalization and 3-manifolds fibering over the circle (McMullen). Canonical Decomposition Theorem for Thurston maps as an analogue of Torus Decomposition Theorem for 3-manifolds (Pilgrim). Fatou-Sullivan Dictionary Similarities between limit and Julia sets (Fatou). QC deformation techniques: Ahlfors Finiteness Theorem and Sullivan’s No Wandering Domain Theorem. Thurston’s Realization Theorem. Teichmüller space of conformal dynamical systems (Sullivan-McMullen). Mating modular group with quadratic maps (Bullett-Penrose, Bullett-Lomonaco). Renormalization and 3-manifolds fibering over the circle (McMullen). Canonical Decomposition Theorem for Thurston maps as an analogue of Torus Decomposition Theorem for 3-manifolds (Pilgrim). Fatou-Sullivan Dictionary Similarities between limit and Julia sets (Fatou). QC deformation techniques: Ahlfors Finiteness Theorem and Sullivan’s No Wandering Domain Theorem. Thurston’s Realization Theorem. Teichmüller space of conformal dynamical systems (Sullivan-McMullen). Mating modular group with quadratic maps (Bullett-Penrose, Bullett-Lomonaco). Laminations for rational maps as analogues of hyperbolic 3-orbifold quotients of Kleinian groups (Lyubich-Minsky). Canonical Decomposition Theorem for Thurston maps as an analogue of Torus Decomposition Theorem for 3-manifolds (Pilgrim). Fatou-Sullivan Dictionary Similarities between limit and Julia sets (Fatou). QC deformation techniques: Ahlfors Finiteness Theorem and Sullivan’s No Wandering Domain Theorem. Thurston’s Realization Theorem. Teichmüller space of conformal dynamical systems (Sullivan-McMullen). Mating modular group with quadratic maps (Bullett-Penrose, Bullett-Lomonaco). Laminations for rational maps as analogues of hyperbolic 3-orbifold quotients of Kleinian groups (Lyubich-Minsky). Renormalization and 3-manifolds fibering over the circle (McMullen). Fatou-Sullivan Dictionary Similarities between limit and Julia sets (Fatou). QC deformation techniques: Ahlfors Finiteness Theorem and Sullivan’s No Wandering Domain Theorem. Thurston’s Realization Theorem. Teichmüller space of conformal dynamical systems (Sullivan-McMullen). Mating modular group with quadratic maps (Bullett-Penrose, Bullett-Lomonaco). Laminations for rational maps as analogues of hyperbolic 3-orbifold quotients of Kleinian groups (Lyubich-Minsky). Renormalization and 3-manifolds fibering over the circle (McMullen). Canonical Decomposition Theorem for Thurston maps as an analogue of Torus Decomposition Theorem for 3-manifolds (Pilgrim). Relations between limit sets and Julia sets What is The Connection between these Limit and Julia Sets? Define a kissing (Kleinian) reflection group GΓ as the group generated by the reflections in the circles in PΓ. The limit set of GΓ is connected () Γ is 2-connected. Circle Packings, and Kleinian Reflection Groups Circle Packing Theorem: For every connected, simple, planar graph Γ, there is a (finite) circle packing PΓ in the plane whose contact graph is (isomorphic to) Γ. The limit set of GΓ is connected () Γ is 2-connected. Circle Packings, and Kleinian Reflection Groups Circle Packing Theorem: For every connected, simple, planar graph Γ, there is a (finite) circle packing PΓ in the plane whose contact graph is (isomorphic to) Γ. Define a kissing (Kleinian) reflection group GΓ as the group generated by the reflections in the circles in PΓ. Circle Packings, and Kleinian Reflection Groups Circle Packing Theorem: For every connected, simple, planar graph Γ, there is a (finite) circle packing PΓ in the plane whose contact graph is (isomorphic to) Γ. Define a kissing (Kleinian) reflection group GΓ as the group generated by the reflections in the circles in PΓ. The limit set of GΓ is connected () Γ is 2-connected. Circle Packings, and Kleinian Reflection Groups Circle Packing Theorem: For every connected, simple, planar graph Γ, there is a (finite) circle packing PΓ in the plane whose contact graph is (isomorphic to) Γ. Define a kissing (Kleinian) reflection group GΓ as the group generated by the reflections in the circles in PΓ.