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 {R } 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 {R } 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 {R } 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 {R } 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 {R } 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.
The limit set of GΓ is connected ⇐⇒ Γ is 2-connected. Define the Nielsen map NΓ of GΓ as [ ˆ NΓ : Di → C, z 7→ ρi (z), if z ∈ Di , i
where ρi is reflection in the circle ∂Di . NΓ and GΓ have the same grand orbits. d If Γ is a “(d + 1)-gon”, then N | 1 and z | 1 are topologically d Γd S S conjugate.
Nielsen/Bowen-Series Map, and a Fundamental Link
For a connected, simple, planar graph Γ, let {Di } be the (disjoint) round disks bounded by the circles of the packing PΓ. NΓ and GΓ have the same grand orbits. d If Γ is a “(d + 1)-gon”, then N | 1 and z | 1 are topologically d Γd S S conjugate.
Nielsen/Bowen-Series Map, and a Fundamental Link
For a connected, simple, planar graph Γ, let {Di } be the (disjoint) round disks bounded by the circles of the packing PΓ. Define the Nielsen map NΓ of GΓ as [ ˆ NΓ : Di → C, z 7→ ρi (z), if z ∈ Di , i
where ρi is reflection in the circle ∂Di . d If Γ is a “(d + 1)-gon”, then N | 1 and z | 1 are topologically d Γd S S conjugate.
Nielsen/Bowen-Series Map, and a Fundamental Link
For a connected, simple, planar graph Γ, let {Di } be the (disjoint) round disks bounded by the circles of the packing PΓ. Define the Nielsen map NΓ of GΓ as [ ˆ NΓ : Di → C, z 7→ ρi (z), if z ∈ Di , i
where ρi is reflection in the circle ∂Di . NΓ and GΓ have the same grand orbits. Nielsen/Bowen-Series Map, and a Fundamental Link
For a connected, simple, planar graph Γ, let {Di } be the (disjoint) round disks bounded by the circles of the packing PΓ. Define the Nielsen map NΓ of GΓ as [ ˆ NΓ : Di → C, z 7→ ρi (z), if z ∈ Di , i
where ρi is reflection in the circle ∂Di . NΓ and GΓ have the same grand orbits. d If Γ is a “(d + 1)-gon”, then N | 1 and z | 1 are topologically d Γd S S conjugate. Nielsen/Bowen-Series Map, and a Fundamental Link
For a connected, simple, planar graph Γ, let {Di } be the (disjoint) round disks bounded by the circles of the packing PΓ. Define the Nielsen map NΓ of GΓ as [ ˆ NΓ : Di → C, z 7→ ρi (z), if z ∈ Di , i
where ρi is reflection in the circle ∂Di . NΓ and GΓ have the same grand orbits. d If Γ is a “(d + 1)-gon”, then N | 1 and z | 1 are topologically d Γd S S conjugate. When Γ is reduced (no separating triangles), then Λ(GΓ) and J (RΓ) have isomorphic quasisymmetry groups.
New Lines in The Dictionary (Lodge, Lyubich, Merenkov, M)
If Γ is 3-regular, then there exists a critically fixed anti-rational map
RΓ such that NΓ|Λ(GΓ) and RΓ|J (RΓ) are topologically conjugate. When Γ is reduced (no separating triangles), then Λ(GΓ) and J (RΓ) have isomorphic quasisymmetry groups.
New Lines in The Dictionary (Lodge, Lyubich, Merenkov, M)
If Γ is 3-regular, then there exists a critically fixed anti-rational map
RΓ such that NΓ|Λ(GΓ) and RΓ|J (RΓ) are topologically conjugate. New Lines in The Dictionary (Lodge, Lyubich, Merenkov, M)
If Γ is 3-regular, then there exists a critically fixed anti-rational map
RΓ such that NΓ|Λ(GΓ) and RΓ|J (RΓ) are topologically conjugate.
When Γ is reduced (no separating triangles), then Λ(GΓ) and J (RΓ) have isomorphic quasisymmetry groups. Moreover,
NΓ : Λ(GΓ) → Λ(GΓ) and RΓ : J (RΓ) → J (RΓ)
are topologically conjugate.
New Lines in The Dictionary (Lodge, Luo, M)
Contact graph Kissing reflection group Critically fixed anti-rational map
2-connected Connected limit set Critically fixed anti-rational map
3-connected/ Gasket limit set/ Gasket Julia set/ Polyhedral Acylindrical/ Bounded QC space “Bounded deformation space”
2-connected + Outerplanar Function kissing reflection group Critically fixed anti-polynomial
Hamiltonian Closure of QuasiFuchsian space Mating of two anti-polynomials
A pair of non-parallel anti-polynomial laminations A marked Hamiltonian cycle A pair of non-parallel multicurves (Examples of shared matings) New Lines in The Dictionary (Lodge, Luo, M)
Contact graph Kissing reflection group Critically fixed anti-rational map
2-connected Connected limit set Critically fixed anti-rational map
3-connected/ Gasket limit set/ Gasket Julia set/ Polyhedral Acylindrical/ Bounded QC space “Bounded deformation space”
2-connected + Outerplanar Function kissing reflection group Critically fixed anti-polynomial
Hamiltonian Closure of QuasiFuchsian space Mating of two anti-polynomials
A pair of non-parallel anti-polynomial laminations A marked Hamiltonian cycle A pair of non-parallel multicurves (Examples of shared matings)
Moreover,
NΓ : Λ(GΓ) → Λ(GΓ) and RΓ : J (RΓ) → J (RΓ)
are topologically conjugate. The action of NΓ on each “invariant component” of Ω(GΓ) is conjugate to the Nielsen map of an ideal polygon group. k Topologically modify NΓ by gluing in the dynamics of z in each of 2 the these components. This produces a branched covering of S . Thurston’s realization theorem combined with “expansiveness” of this branched cover allows us to realize it as an anti-rational map RΓ, and get the desired conjugacy. The Tischler graph of RΓ is the planar dual of Γ. That this yields a bijection follows from combinatorial classification of critically fixed anti-rational maps.
Constructing RΓ from GΓ
Γ GΓ RΓ k Topologically modify NΓ by gluing in the dynamics of z in each of 2 the these components. This produces a branched covering of S . Thurston’s realization theorem combined with “expansiveness” of this branched cover allows us to realize it as an anti-rational map RΓ, and get the desired conjugacy. The Tischler graph of RΓ is the planar dual of Γ. That this yields a bijection follows from combinatorial classification of critically fixed anti-rational maps.
Constructing RΓ from GΓ
Γ GΓ RΓ
The action of NΓ on each “invariant component” of Ω(GΓ) is conjugate to the Nielsen map of an ideal polygon group. Thurston’s realization theorem combined with “expansiveness” of this branched cover allows us to realize it as an anti-rational map RΓ, and get the desired conjugacy. The Tischler graph of RΓ is the planar dual of Γ. That this yields a bijection follows from combinatorial classification of critically fixed anti-rational maps.
Constructing RΓ from GΓ
Γ GΓ RΓ
The action of NΓ on each “invariant component” of Ω(GΓ) is conjugate to the Nielsen map of an ideal polygon group. k Topologically modify NΓ by gluing in the dynamics of z in each of 2 the these components. This produces a branched covering of S . The Tischler graph of RΓ is the planar dual of Γ. That this yields a bijection follows from combinatorial classification of critically fixed anti-rational maps.
Constructing RΓ from GΓ
Γ GΓ RΓ
The action of NΓ on each “invariant component” of Ω(GΓ) is conjugate to the Nielsen map of an ideal polygon group. k Topologically modify NΓ by gluing in the dynamics of z in each of 2 the these components. This produces a branched covering of S . Thurston’s realization theorem combined with “expansiveness” of this branched cover allows us to realize it as an anti-rational map RΓ, and get the desired conjugacy. That this yields a bijection follows from combinatorial classification of critically fixed anti-rational maps.
Constructing RΓ from GΓ
Γ GΓ RΓ
The action of NΓ on each “invariant component” of Ω(GΓ) is conjugate to the Nielsen map of an ideal polygon group. k Topologically modify NΓ by gluing in the dynamics of z in each of 2 the these components. This produces a branched covering of S . Thurston’s realization theorem combined with “expansiveness” of this branched cover allows us to realize it as an anti-rational map RΓ, and get the desired conjugacy. The Tischler graph of RΓ is the planar dual of Γ. Constructing RΓ from GΓ
Γ GΓ RΓ
The action of NΓ on each “invariant component” of Ω(GΓ) is conjugate to the Nielsen map of an ideal polygon group. k Topologically modify NΓ by gluing in the dynamics of z in each of 2 the these components. This produces a branched covering of S . Thurston’s realization theorem combined with “expansiveness” of this branched cover allows us to realize it as an anti-rational map RΓ, and get the desired conjugacy. The Tischler graph of RΓ is the planar dual of Γ. That this yields a bijection follows from combinatorial classification of critically fixed anti-rational maps. 3-connected Graph/ Gasket/ Acylindrical 3-connected Graph/ Gasket/ Acylindrical Outerplanar Graph/ Function Group/ Anti-polynomials • The blue sub-graph corresponds to:
γ1 • Pinch γ2 γ1 and γ2: GΓ
• The anti-rational map RΓ is the mating of the two anti-polynomials corresponding to the outerplanar graphs:
Hamiltonicity/ QuasiFuchsian Closure/ Mating
• Hamiltonian Γ: γ1 • Pinch γ2 γ1 and γ2: GΓ
• The anti-rational map RΓ is the mating of the two anti-polynomials corresponding to the outerplanar graphs:
Hamiltonicity/ QuasiFuchsian Closure/ Mating
• Hamiltonian Γ:
• The blue sub-graph corresponds to: GΓ
• The anti-rational map RΓ is the mating of the two anti-polynomials corresponding to the outerplanar graphs:
Hamiltonicity/ QuasiFuchsian Closure/ Mating
• Hamiltonian Γ:
• The blue sub-graph corresponds to:
γ1 • Pinch γ2 γ1 and γ2: • The anti-rational map RΓ is the mating of the two anti-polynomials corresponding to the outerplanar graphs:
Hamiltonicity/ QuasiFuchsian Closure/ Mating
• Hamiltonian Γ:
• The blue sub-graph corresponds to:
γ1 • Pinch γ2 γ1 and γ2: GΓ Hamiltonicity/ QuasiFuchsian Closure/ Mating
• Hamiltonian Γ:
• The blue sub-graph corresponds to:
γ1 • Pinch γ2 γ1 and γ2: GΓ
• The anti-rational map RΓ is the mating of the two anti-polynomials corresponding to the outerplanar graphs: Hamiltonicity/ QuasiFuchsian Closure/ Mating
• Hamiltonian Γ:
• The blue sub-graph corresponds to:
γ1 • Pinch γ2 γ1 and γ2: GΓ
• The anti-rational map RΓ is the mating of the two anti-polynomials corresponding to the outerplanar graphs: To do so, one needs to replace zk -dynamics on invariant Fatou components by Nielsen maps of ideal polygon groups, and uniformize.
Theorem (Lyubich-Merenkov-Ntalampekos-M) d The topological conjugacy between N | 1 and z | 1 extends as a Γd S S David homeomorphism of D. There exists a David homeomorphism H : Cb → Cb such that H(J (RΓ)) = Λ(GΓ).
Corollary
If Γ is 2-connected and outerplanar, then Λ(GΓ) is conformally removable.
Γ is 2-connected and outerplanar =⇒ J (RΓ) is the boundary of a 1,1 John domain =⇒ J (RΓ) is W -removable. W 1,1-removability is preserved under global David homeomorphisms.
Anti-rational Maps to Groups: Removability of Limit Sets
The group GΓ can also be directly constructed from RΓ. Theorem (Lyubich-Merenkov-Ntalampekos-M) d The topological conjugacy between N | 1 and z | 1 extends as a Γd S S David homeomorphism of D. There exists a David homeomorphism H : Cb → Cb such that H(J (RΓ)) = Λ(GΓ).
Corollary
If Γ is 2-connected and outerplanar, then Λ(GΓ) is conformally removable.
Γ is 2-connected and outerplanar =⇒ J (RΓ) is the boundary of a 1,1 John domain =⇒ J (RΓ) is W -removable. W 1,1-removability is preserved under global David homeomorphisms.
Anti-rational Maps to Groups: Removability of Limit Sets
The group GΓ can also be directly constructed from RΓ. To do so, one needs to replace zk -dynamics on invariant Fatou components by Nielsen maps of ideal polygon groups, and uniformize. There exists a David homeomorphism H : Cb → Cb such that H(J (RΓ)) = Λ(GΓ).
Corollary
If Γ is 2-connected and outerplanar, then Λ(GΓ) is conformally removable.
Γ is 2-connected and outerplanar =⇒ J (RΓ) is the boundary of a 1,1 John domain =⇒ J (RΓ) is W -removable. W 1,1-removability is preserved under global David homeomorphisms.
Anti-rational Maps to Groups: Removability of Limit Sets
The group GΓ can also be directly constructed from RΓ. To do so, one needs to replace zk -dynamics on invariant Fatou components by Nielsen maps of ideal polygon groups, and uniformize.
Theorem (Lyubich-Merenkov-Ntalampekos-M) d The topological conjugacy between N | 1 and z | 1 extends as a Γd S S David homeomorphism of D. Corollary
If Γ is 2-connected and outerplanar, then Λ(GΓ) is conformally removable.
Γ is 2-connected and outerplanar =⇒ J (RΓ) is the boundary of a 1,1 John domain =⇒ J (RΓ) is W -removable. W 1,1-removability is preserved under global David homeomorphisms.
Anti-rational Maps to Groups: Removability of Limit Sets
The group GΓ can also be directly constructed from RΓ. To do so, one needs to replace zk -dynamics on invariant Fatou components by Nielsen maps of ideal polygon groups, and uniformize.
Theorem (Lyubich-Merenkov-Ntalampekos-M) d The topological conjugacy between N | 1 and z | 1 extends as a Γd S S David homeomorphism of D. There exists a David homeomorphism H : Cb → Cb such that H(J (RΓ)) = Λ(GΓ). Γ is 2-connected and outerplanar =⇒ J (RΓ) is the boundary of a 1,1 John domain =⇒ J (RΓ) is W -removable. W 1,1-removability is preserved under global David homeomorphisms.
Anti-rational Maps to Groups: Removability of Limit Sets
The group GΓ can also be directly constructed from RΓ. To do so, one needs to replace zk -dynamics on invariant Fatou components by Nielsen maps of ideal polygon groups, and uniformize.
Theorem (Lyubich-Merenkov-Ntalampekos-M) d The topological conjugacy between N | 1 and z | 1 extends as a Γd S S David homeomorphism of D. There exists a David homeomorphism H : Cb → Cb such that H(J (RΓ)) = Λ(GΓ).
Corollary
If Γ is 2-connected and outerplanar, then Λ(GΓ) is conformally removable. W 1,1-removability is preserved under global David homeomorphisms.
Anti-rational Maps to Groups: Removability of Limit Sets
The group GΓ can also be directly constructed from RΓ. To do so, one needs to replace zk -dynamics on invariant Fatou components by Nielsen maps of ideal polygon groups, and uniformize.
Theorem (Lyubich-Merenkov-Ntalampekos-M) d The topological conjugacy between N | 1 and z | 1 extends as a Γd S S David homeomorphism of D. There exists a David homeomorphism H : Cb → Cb such that H(J (RΓ)) = Λ(GΓ).
Corollary
If Γ is 2-connected and outerplanar, then Λ(GΓ) is conformally removable.
Γ is 2-connected and outerplanar =⇒ J (RΓ) is the boundary of a 1,1 John domain =⇒ J (RΓ) is W -removable. Anti-rational Maps to Groups: Removability of Limit Sets
The group GΓ can also be directly constructed from RΓ. To do so, one needs to replace zk -dynamics on invariant Fatou components by Nielsen maps of ideal polygon groups, and uniformize.
Theorem (Lyubich-Merenkov-Ntalampekos-M) d The topological conjugacy between N | 1 and z | 1 extends as a Γd S S David homeomorphism of D. There exists a David homeomorphism H : Cb → Cb such that H(J (RΓ)) = Λ(GΓ).
Corollary
If Γ is 2-connected and outerplanar, then Λ(GΓ) is conformally removable.
Γ is 2-connected and outerplanar =⇒ J (RΓ) is the boundary of a 1,1 John domain =⇒ J (RΓ) is W -removable. W 1,1-removability is preserved under global David homeomorphisms. Conformal Matings, and Univalent Functions We want to mate the action of an anti-polynomial on its filled Julia set with the action of the Nielsen map of a function reflection group on its filled limit set. 2 Simplest example: z ⊥⊥Γ2.
Π
2 1 Topological mating: glue D z and D NΓ2 along S using the topological conjugacy ? between the boundary maps. A conformal model, if it exists, would be an anti-meromorphic map on Ω := Cb\“Π” that fixes ∂Ω.
Mating Function Groups with Anti-polynomials
In classical polynomial mating, one mates the actions of two polynomials on their filled Julia sets. 2 Simplest example: z ⊥⊥Γ2.
Π
2 1 Topological mating: glue D z and D NΓ2 along S using the topological conjugacy ? between the boundary maps. A conformal model, if it exists, would be an anti-meromorphic map on Ω := Cb\“Π” that fixes ∂Ω.
Mating Function Groups with Anti-polynomials
In classical polynomial mating, one mates the actions of two polynomials on their filled Julia sets. We want to mate the action of an anti-polynomial on its filled Julia set with the action of the Nielsen map of a function reflection group on its filled limit set. Π
2 1 Topological mating: glue D z and D NΓ2 along S using the topological conjugacy ? between the boundary maps. A conformal model, if it exists, would be an anti-meromorphic map on Ω := Cb\“Π” that fixes ∂Ω.
Mating Function Groups with Anti-polynomials
In classical polynomial mating, one mates the actions of two polynomials on their filled Julia sets. We want to mate the action of an anti-polynomial on its filled Julia set with the action of the Nielsen map of a function reflection group on its filled limit set. 2 Simplest example: z ⊥⊥Γ2. 2 1 Topological mating: glue D z and D NΓ2 along S using the topological conjugacy ? between the boundary maps. A conformal model, if it exists, would be an anti-meromorphic map on Ω := Cb\“Π” that fixes ∂Ω.
Mating Function Groups with Anti-polynomials
In classical polynomial mating, one mates the actions of two polynomials on their filled Julia sets. We want to mate the action of an anti-polynomial on its filled Julia set with the action of the Nielsen map of a function reflection group on its filled limit set. 2 Simplest example: z ⊥⊥Γ2.
Π A conformal model, if it exists, would be an anti-meromorphic map on Ω := Cb\“Π” that fixes ∂Ω.
Mating Function Groups with Anti-polynomials
In classical polynomial mating, one mates the actions of two polynomials on their filled Julia sets. We want to mate the action of an anti-polynomial on its filled Julia set with the action of the Nielsen map of a function reflection group on its filled limit set. 2 Simplest example: z ⊥⊥Γ2.
Π
2 1 Topological mating: glue D z and D NΓ2 along S using the topological conjugacy ? between the boundary maps. Mating Function Groups with Anti-polynomials
In classical polynomial mating, one mates the actions of two polynomials on their filled Julia sets. We want to mate the action of an anti-polynomial on its filled Julia set with the action of the Nielsen map of a function reflection group on its filled limit set. 2 Simplest example: z ⊥⊥Γ2.
Π
2 1 Topological mating: glue D z and D NΓ2 along S using the topological conjugacy ? between the boundary maps. A conformal model, if it exists, would be an anti-meromorphic map on Ω := Cb\“Π” that fixes ∂Ω. Theorem (Lee-Lyubich-Makarov-M) 2 The map f (z) := z + 1/(2z ) is univalent on Cb \ D, and the Schwarz 2 reflection map σ is the unique conformal mating of z and NΓ2 . In particular, ? is a welding map.
Simultaneous Uniformization
The corresponding Riemann map f : Cb \ D → Ω must be a rational map, and the desired anti-meromorphic map would be given by σ := f ◦ (1/z) ◦ (f | )−1. Cˆ\D f Cb \ D −−−−→ Ω σ y1/z y f D −−−−→ Cb Simultaneous Uniformization
The corresponding Riemann map f : Cb \ D → Ω must be a rational map, and the desired anti-meromorphic map would be given by σ := f ◦ (1/z) ◦ (f | )−1. Cˆ\D f Cb \ D −−−−→ Ω σ y1/z y f D −−−−→ Cb Theorem (Lee-Lyubich-Makarov-M) 2 The map f (z) := z + 1/(2z ) is univalent on Cb \ D, and the Schwarz 2 reflection map σ is the unique conformal mating of z and NΓ2 . In particular, ? is a welding map. The Deltoid Reflection Map The Schwarz reflection map is piecewise defined on Theorem (Lee, Lyubich, Makarov, M) Conformal matings of geometrically finite quadratic anti-polynomials with
the Nielsen map NΓ2 are realized in the C&C family.
+ =
0 7→ ∞ 7→ 0; the “Basilica".
More on Quadratic Schwarz Reflection Maps
The Circle and Cardioid (C&C) Family: Let ♥ be a cardioid. For c a ∈ C \ (−∞, −1/12), set Ωa := ♥ ∪ B(a, ra) . Theorem (Lee, Lyubich, Makarov, M) Conformal matings of geometrically finite quadratic anti-polynomials with
the Nielsen map NΓ2 are realized in the C&C family.
+ =
0 7→ ∞ 7→ 0; the “Basilica".
More on Quadratic Schwarz Reflection Maps
The Circle and Cardioid (C&C) Family: Let ♥ be a cardioid. For c a ∈ C \ (−∞, −1/12), set Ωa := ♥ ∪ B(a, ra) .
The Schwarz reflection map is piecewise defined on + =
0 7→ ∞ 7→ 0; the “Basilica".
More on Quadratic Schwarz Reflection Maps
The Circle and Cardioid (C&C) Family: Let ♥ be a cardioid. For c a ∈ C \ (−∞, −1/12), set Ωa := ♥ ∪ B(a, ra) .
The Schwarz reflection map is piecewise defined on Theorem (Lee, Lyubich, Makarov, M) Conformal matings of geometrically finite quadratic anti-polynomials with
the Nielsen map NΓ2 are realized in the C&C family. More on Quadratic Schwarz Reflection Maps
The Circle and Cardioid (C&C) Family: Let ♥ be a cardioid. For c a ∈ C \ (−∞, −1/12), set Ωa := ♥ ∪ B(a, ra) .
The Schwarz reflection map is piecewise defined on Theorem (Lee, Lyubich, Makarov, M) Conformal matings of geometrically finite quadratic anti-polynomials with
the Nielsen map NΓ2 are realized in the C&C family.
+ =
0 7→ ∞ 7→ 0; the “Basilica". C&C Family and the Tricorn
Theorem (Lee, Lyubich, Makarov, M) The connectedness locus of the C&C family is combinatorially equivalent to the basilica limb of the Tricorn (connectedness locus of {z2 + c}). C&C Family and the Tricorn
Theorem (Lee, Lyubich, Makarov, M) The connectedness locus of the C&C family is combinatorially equivalent to the basilica limb of the Tricorn (connectedness locus of {z2 + c}). The anti-polynomials P and RΓ (which is critically fixed) are mateable ⇐⇒ they are Moore-unobstructed. Perform David surgery to replace the zk -dynamics by Nielsen maps of ideal polygon groups.
+ =
A General Mating Theorem
Theorem (Lyubich-Merenkov-Ntalampekos-M) Let P be a post-critically finite, hyperbolic anti-polynomial of degree d, and GΓ be a function kissing reflection group generated by reflections in (d + 1) circles. Then, P and GΓ are conformally mateable ⇐⇒ they are Moore-unobstructed. Perform David surgery to replace the zk -dynamics by Nielsen maps of ideal polygon groups.
+ =
A General Mating Theorem
Theorem (Lyubich-Merenkov-Ntalampekos-M) Let P be a post-critically finite, hyperbolic anti-polynomial of degree d, and GΓ be a function kissing reflection group generated by reflections in (d + 1) circles. Then, P and GΓ are conformally mateable ⇐⇒ they are Moore-unobstructed.
The anti-polynomials P and RΓ (which is critically fixed) are mateable ⇐⇒ they are Moore-unobstructed. + =
A General Mating Theorem
Theorem (Lyubich-Merenkov-Ntalampekos-M) Let P be a post-critically finite, hyperbolic anti-polynomial of degree d, and GΓ be a function kissing reflection group generated by reflections in (d + 1) circles. Then, P and GΓ are conformally mateable ⇐⇒ they are Moore-unobstructed.
The anti-polynomials P and RΓ (which is critically fixed) are mateable ⇐⇒ they are Moore-unobstructed. Perform David surgery to replace the zk -dynamics by Nielsen maps of ideal polygon groups. A General Mating Theorem
Theorem (Lyubich-Merenkov-Ntalampekos-M) Let P be a post-critically finite, hyperbolic anti-polynomial of degree d, and GΓ be a function kissing reflection group generated by reflections in (d + 1) circles. Then, P and GΓ are conformally mateable ⇐⇒ they are Moore-unobstructed.
The anti-polynomials P and RΓ (which is critically fixed) are mateable ⇐⇒ they are Moore-unobstructed. Perform David surgery to replace the zk -dynamics by Nielsen maps of ideal polygon groups.
+ = The coefficient problem (analogue of De Brange’s Theorem in class S) is open for class Σ. Truncated families: Σ∗ := n d o f (z) = z + a1 + ··· + ad : a = − 1 and f | is univalent. z zd d d Cˆ\D S ∗ Since Σ = d≥1 Σd , it is natural to study the coefficient problem for ∗ Σd . ∗ The coefficients are maximized at the extremal points of Σd . ∗ The extremal points of Σd are functions f for which f (T) has (d + 1) cusp points and (d − 2) double points (these are the maximal number of singularities).
An Extremal Problem for Univalent Functions
n o a1 ad Σ := f (z) = z + z + ··· + zd + ··· : Cb \ D → Cb is univalent. Truncated families: Σ∗ := n d o f (z) = z + a1 + ··· + ad : a = − 1 and f | is univalent. z zd d d Cˆ\D S ∗ Since Σ = d≥1 Σd , it is natural to study the coefficient problem for ∗ Σd . ∗ The coefficients are maximized at the extremal points of Σd . ∗ The extremal points of Σd are functions f for which f (T) has (d + 1) cusp points and (d − 2) double points (these are the maximal number of singularities).
An Extremal Problem for Univalent Functions
n o a1 ad Σ := f (z) = z + z + ··· + zd + ··· : Cb \ D → Cb is univalent. The coefficient problem (analogue of De Brange’s Theorem in class S) is open for class Σ. S ∗ Since Σ = d≥1 Σd , it is natural to study the coefficient problem for ∗ Σd . ∗ The coefficients are maximized at the extremal points of Σd . ∗ The extremal points of Σd are functions f for which f (T) has (d + 1) cusp points and (d − 2) double points (these are the maximal number of singularities).
An Extremal Problem for Univalent Functions
n o a1 ad Σ := f (z) = z + z + ··· + zd + ··· : Cb \ D → Cb is univalent. The coefficient problem (analogue of De Brange’s Theorem in class S) is open for class Σ. Truncated families: Σ∗ := n d o f (z) = z + a1 + ··· + ad : a = − 1 and f | is univalent. z zd d d Cˆ\D ∗ The coefficients are maximized at the extremal points of Σd . ∗ The extremal points of Σd are functions f for which f (T) has (d + 1) cusp points and (d − 2) double points (these are the maximal number of singularities).
An Extremal Problem for Univalent Functions
n o a1 ad Σ := f (z) = z + z + ··· + zd + ··· : Cb \ D → Cb is univalent. The coefficient problem (analogue of De Brange’s Theorem in class S) is open for class Σ. Truncated families: Σ∗ := n d o f (z) = z + a1 + ··· + ad : a = − 1 and f | is univalent. z zd d d Cˆ\D S ∗ Since Σ = d≥1 Σd , it is natural to study the coefficient problem for ∗ Σd . ∗ The extremal points of Σd are functions f for which f (T) has (d + 1) cusp points and (d − 2) double points (these are the maximal number of singularities).
An Extremal Problem for Univalent Functions
n o a1 ad Σ := f (z) = z + z + ··· + zd + ··· : Cb \ D → Cb is univalent. The coefficient problem (analogue of De Brange’s Theorem in class S) is open for class Σ. Truncated families: Σ∗ := n d o f (z) = z + a1 + ··· + ad : a = − 1 and f | is univalent. z zd d d Cˆ\D S ∗ Since Σ = d≥1 Σd , it is natural to study the coefficient problem for ∗ Σd . ∗ The coefficients are maximized at the extremal points of Σd . An Extremal Problem for Univalent Functions
n o a1 ad Σ := f (z) = z + z + ··· + zd + ··· : Cb \ D → Cb is univalent. The coefficient problem (analogue of De Brange’s Theorem in class S) is open for class Σ. Truncated families: Σ∗ := n d o f (z) = z + a1 + ··· + ad : a = − 1 and f | is univalent. z zd d d Cˆ\D S ∗ Since Σ = d≥1 Σd , it is natural to study the coefficient problem for ∗ Σd . ∗ The coefficients are maximized at the extremal points of Σd . ∗ The extremal points of Σd are functions f for which f (T) has (d + 1) cusp points and (d − 2) double points (these are the maximal number of singularities). Associating Bi-angled Trees to Extremal Functions Main idea: QC deformation of Schwarz reflection maps and ∗ compactness of Σd (surjectivity) + rigidity of Schwarz reflection maps (injectivity). Alternative proof of surjectivity: mate zd with ‘maximal cusp’ kissing reflection groups associated to outerplanar graphs (Lyubich-Merenkov-Ntalampekos-M).
∗ Describing Extremal Functions in Σd
Theorem (Lazebnik-Makarov-M) Let d ≥ 2. There is a canonical bijection between:
∗ {f ∈ Σd : f (T) has (d + 1) cusps and (d − 2) double points} , and {Bi-angled trees with (d − 1) vertices} . Alternative proof of surjectivity: mate zd with ‘maximal cusp’ kissing reflection groups associated to outerplanar graphs (Lyubich-Merenkov-Ntalampekos-M).
∗ Describing Extremal Functions in Σd
Theorem (Lazebnik-Makarov-M) Let d ≥ 2. There is a canonical bijection between:
∗ {f ∈ Σd : f (T) has (d + 1) cusps and (d − 2) double points} , and {Bi-angled trees with (d − 1) vertices} .
Main idea: QC deformation of Schwarz reflection maps and ∗ compactness of Σd (surjectivity) + rigidity of Schwarz reflection maps (injectivity). ∗ Describing Extremal Functions in Σd
Theorem (Lazebnik-Makarov-M) Let d ≥ 2. There is a canonical bijection between:
∗ {f ∈ Σd : f (T) has (d + 1) cusps and (d − 2) double points} , and {Bi-angled trees with (d − 1) vertices} .
Main idea: QC deformation of Schwarz reflection maps and ∗ compactness of Σd (surjectivity) + rigidity of Schwarz reflection maps (injectivity). Alternative proof of surjectivity: mate zd with ‘maximal cusp’ kissing reflection groups associated to outerplanar graphs (Lyubich-Merenkov-Ntalampekos-M). ∗ Detailed analysis of Schwarz reflection maps arising from Σd .
Σ as a Universal Teichmüller Space
Theorem (Lazebnik, Makarov, M)
Bers slice of regular ideal (d + 1) − gon group ⊥⊥{zd } ∼ ∗ = Σd ,→ Σ. Σ as a Universal Teichmüller Space
Theorem (Lazebnik, Makarov, M)
Bers slice of regular ideal (d + 1) − gon group ⊥⊥{zd } ∼ ∗ = Σd ,→ Σ.
∗ Detailed analysis of Schwarz reflection maps arising from Σd . References
R. Lodge, M. Lyubich, S. Merenkov, and S. Mukherjee, On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections, https://arxiv.org/abs/1912.13438, 2019. R. Lodge, Y. Luo, and S. Mukherjee, Circle packings, kissing reflection groups, and critically fixed anti-rational maps, https://arxiv.org/abs/2007.03558, 2020. S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee, Dynamics of Schwarz reflections: the mating phenomena, https://arxiv.org/abs/1811.04979, 2018. S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee. Schwarz reflections and the Tricorn, https://arxiv.org/abs/1812.01573, 2018. K. Lazebnik, N. G. Makarov, and S. Mukherjee, Univalent polynomials and Hubbard trees, https://arxiv.org/abs/1908.05813, 2019. K. Lazebnik, N. G. Makarov, and S. Mukherjee, Bers slices in families of univalent maps, https://arxiv.org/abs/2007.02429, 2020.