Background: 5. Pared 3-manifolds A careful treatment of geometrically finite hyperbolic structures on 3-manifolds with cusps requires the discussion of pared manifolds . For pared manifold M = ( M 0 , P ):
Background: 5. Pared 3-manifolds A careful treatment of geometrically finite hyperbolic structures on 3-manifolds with cusps requires the discussion of pared manifolds . For pared manifold M = ( M 0 , P ): ◮ M 0 is a compact 3-manifold with boundary
Background: 5. Pared 3-manifolds A careful treatment of geometrically finite hyperbolic structures on 3-manifolds with cusps requires the discussion of pared manifolds . For pared manifold M = ( M 0 , P ): ◮ M 0 is a compact 3-manifold with boundary ◮ P ⊂ ∂ M 0 is a disjoint union of incompressible tori and annuli
Background: 5. Pared 3-manifolds A careful treatment of geometrically finite hyperbolic structures on 3-manifolds with cusps requires the discussion of pared manifolds . For pared manifold M = ( M 0 , P ): ◮ M 0 is a compact 3-manifold with boundary ◮ P ⊂ ∂ M 0 is a disjoint union of incompressible tori and annuli ◮ ∂ M = ⊔ i Σ i denotes ∂ M 0 \ ∂ P
Background: 5. Pared 3-manifolds A careful treatment of geometrically finite hyperbolic structures on 3-manifolds with cusps requires the discussion of pared manifolds . For pared manifold M = ( M 0 , P ): ◮ M 0 is a compact 3-manifold with boundary ◮ P ⊂ ∂ M 0 is a disjoint union of incompressible tori and annuli ◮ ∂ M = ⊔ i Σ i denotes ∂ M 0 \ ∂ P GF ( M ) = { [ ρ ] ∈ GF ( M 0 ) | ρ ( γ ) is parabolic ⇔ γ ∈ π 1 ( P ) }
Background: 5. Pared 3-manifolds A careful treatment of geometrically finite hyperbolic structures on 3-manifolds with cusps requires the discussion of pared manifolds . For pared manifold M = ( M 0 , P ): ◮ M 0 is a compact 3-manifold with boundary ◮ P ⊂ ∂ M 0 is a disjoint union of incompressible tori and annuli ◮ ∂ M = ⊔ i Σ i denotes ∂ M 0 \ ∂ P GF ( M ) = { [ ρ ] ∈ GF ( M 0 ) | ρ ( γ ) is parabolic ⇔ γ ∈ π 1 ( P ) } Theorem (Ahlfors, Bers, Marden, Sullivan) GF ( M ) ∼ = � i T (Σ i )
Background: Review
Background: Review Most important things to keep in mind:
Background: Review Most important things to keep in mind: ◮ The cartoon of a quasi-Fuchsian 3-manifold
Background: Review Most important things to keep in mind: ◮ The cartoon of a quasi-Fuchsian 3-manifold ◮ QF (Σ) ∼ = T (Σ) × T (¯ Σ)
Background: Review Most important things to keep in mind: ◮ The cartoon of a quasi-Fuchsian 3-manifold ◮ QF (Σ) ∼ = T (Σ) × T (¯ Σ) ◮ GF ( M ) ∼ = � i T (Σ i )
Background: Review Most important things to keep in mind: ◮ The cartoon of a quasi-Fuchsian 3-manifold ◮ QF (Σ) ∼ = T (Σ) × T (¯ Σ) ◮ GF ( M ) ∼ = � i T (Σ i ) ◮ The IF: Stab (Ω i ) < ˆ Γ is quasi-Fuchsian
The Definition
The Definition In everything that follows, M = ( M 0 , P ) is a geometrically finite pared 3-manifold with incompressible boundary Σ = ∂ M = ⊔ i Σ i .
The Definition In everything that follows, M = ( M 0 , P ) is a geometrically finite pared 3-manifold with incompressible boundary Σ = ∂ M = ⊔ i Σ i . The inclusion π 1 (Σ i ) ֒ → π 1 ( M ) induces the restriction map r i on (conjugacy classes of) representations.
� � � The Definition In everything that follows, M = ( M 0 , P ) is a geometrically finite pared 3-manifold with incompressible boundary Σ = ∂ M = ⊔ i Σ i . The inclusion π 1 (Σ i ) ֒ → π 1 ( M ) induces the restriction map r i on (conjugacy classes of) representations. Definition The skinning map σ M is given by ∼ r i ∼ = � GF ( M ) � QF (Σ i ) � = � T (Σ i ) × T ( ¯ � i T (Σ i ) Σ i ) � � � � � � � � � � � � � � p 2 � � � � � � � σ i � � � � � � M � � � � � � � T ( ¯ � � � Σ i ) i T ( ¯ i σ i σ M = � M : � i T (Σ i ) → � Σ i )
� � � The Definition In everything that follows, M = ( M 0 , P ) is a geometrically finite pared 3-manifold with incompressible boundary Σ = ∂ M = ⊔ i Σ i . The inclusion π 1 (Σ i ) ֒ → π 1 ( M ) induces the restriction map r i on (conjugacy classes of) representations. Definition The skinning map σ M is given by ∼ r i ∼ = � GF ( M ) � QF (Σ i ) � = � T (Σ i ) × T ( ¯ � i T (Σ i ) Σ i ) � � � � � � � � � � � � � � p 2 � � � � � � � σ i � � � � � � M � � � � � � � T ( ¯ � � � Σ i ) i T ( ¯ i σ i σ M = � M : � i T (Σ i ) → � Σ i ) r i lands in QF (Σ i ) because of the IF.
Background: The Definition, Again
Background: The Definition, Again Assume for simplicity that M has only one boundary component.
Background: The Definition, Again Assume for simplicity that M has only one boundary component. The cover of M corresponding to π 1 (Σ) < π 1 ( M ):
Background: The Definition, Again Assume for simplicity that M has only one boundary component. The cover of M corresponding to π 1 (Σ) < π 1 ( M ): X X M σ M (X )
Background: The Definition, Again Assume for simplicity that M has only one boundary component. The cover of M corresponding to π 1 (Σ) < π 1 ( M ): X X IF - The cover is quasi-Fuchsian M σ M (X )
Background: The Definition, Again Assume for simplicity that M has only one boundary component. The cover of M corresponding to π 1 (Σ) < π 1 ( M ): X X IF - The cover is quasi-Fuchsian Note that σ M depends only on the topology of ( M , Σ). M σ M (X )
Background: A Simple Example
Background: A Simple Example Suppose M is quasi-Fuchsian. That is, M ∼ = H 3 / Γ , for Γ < PSL 2 ( C ) quasi-Fuchsian. Topologically, M ∼ = Σ × [0 , 1].
Background: A Simple Example Suppose M is quasi-Fuchsian. That is, M ∼ = H 3 / Γ , for Γ < PSL 2 ( C ) quasi-Fuchsian. Topologically, M ∼ = Σ × [0 , 1]. Then ∂ M ∼ = Σ ⊔ ¯ Σ, so T ( ∂ M ) ∼ = T (Σ) × T (¯ Σ)
Background: A Simple Example Suppose M is quasi-Fuchsian. That is, M ∼ = H 3 / Γ , for Γ < PSL 2 ( C ) quasi-Fuchsian. Topologically, M ∼ = Σ × [0 , 1]. Then ∂ M ∼ = Σ ⊔ ¯ Σ, so T ( ∂ M ) ∼ = T (Σ) × T (¯ Σ) In this case, σ M ( X , ¯ Y ) = ( ¯ Y , X )
A Symmetry Lemma
A Symmetry Lemma Let M = ( M 0 , P ) is a geometrically finite pared 3-manifold with incompressible boundary Σ.
A Symmetry Lemma Let M = ( M 0 , P ) is a geometrically finite pared 3-manifold with incompressible boundary Σ. Suppose φ ∈ Diff ( M 0 ) satisfies φ ( P ) = P . Then φ induces Φ ∈ MCG ∗ ( M ) ⊂ MCG ∗ (Σ). In this case,
A Symmetry Lemma Let M = ( M 0 , P ) is a geometrically finite pared 3-manifold with incompressible boundary Σ. Suppose φ ∈ Diff ( M 0 ) satisfies φ ( P ) = P . Then φ induces Φ ∈ MCG ∗ ( M ) ⊂ MCG ∗ (Σ). In this case, Symmetry Lemma σ M ( Fix Φ) ⊂ Fix Φ
A Symmetry Lemma Let M = ( M 0 , P ) is a geometrically finite pared 3-manifold with incompressible boundary Σ. Suppose φ ∈ Diff ( M 0 ) satisfies φ ( P ) = P . Then φ induces Φ ∈ MCG ∗ ( M ) ⊂ MCG ∗ (Σ). In this case, Symmetry Lemma σ M ( Fix Φ) ⊂ Fix Φ This lemma is an immediate consequence of the observation that σ M is MCG ∗ ( M )-equivariant.
Strategy Strategy Use the Symmetry Lemma to cut down dimensions and complexity, making σ M accessible.
Strategy Strategy Use the Symmetry Lemma to cut down dimensions and complexity, making σ M accessible. In the example that follows, T (Σ) ∼ = H , so σ M is ’only’ a holomorphic map H → H .
Strategy Strategy Use the Symmetry Lemma to cut down dimensions and complexity, making σ M accessible. In the example that follows, T (Σ) ∼ = H , so σ M is ’only’ a holomorphic map H → H . Non-monotonicity restricted to a real one-dimensional submanifold guarantees the existence of a critical point.
The Example: Glueing an Octahedron
The Example: Glueing an Octahedron Glue the green faces of the octahedron, in pairs, with twists:
The Example: Glueing an Octahedron Glue the green faces of the octahedron, in pairs, with twists:
The Example: Glueing an Octahedron Glue the green faces of the octahedron, in pairs, with twists:
The Example: Glueing an Octahedron Glue the green faces of the octahedron, in pairs, with twists:
The Pared Manifold
The Pared Manifold In the resulting pared 3-manifold M = ( M 0 , P ), M ◦ is a genus 2 handlebody, and P consists of (annuli neighborhoods of) 2 essential curves in ∂ M 0 .
The Pared Manifold In the resulting pared 3-manifold M = ( M 0 , P ), M ◦ is a genus 2 handlebody, and P consists of (annuli neighborhoods of) 2 essential curves in ∂ M 0 . The curves in P are disk-busting , so by a Lemma of Otal, M is acylindrical and Σ = ∂ M is incompressible.
The Pared Manifold In the resulting pared 3-manifold M = ( M 0 , P ), M ◦ is a genus 2 handlebody, and P consists of (annuli neighborhoods of) 2 essential curves in ∂ M 0 . The curves in P are disk-busting , so by a Lemma of Otal, M is acylindrical and Σ = ∂ M is incompressible. The boundary Σ is a four-holed sphere.
A Path in GF ( M )
A Path in GF ( M ) Consider the regular ideal octahedron in H 3 , with vertices { 0 , ± 1 , ± i , ∞} , and perform the indicated face identifications with M¨ obius transformations. This determines a representation ρ 1 : π 1 ( M ) → PSL 2 ( C ).
A Path in GF ( M ) Consider the regular ideal octahedron in H 3 , with vertices { 0 , ± 1 , ± i , ∞} , and perform the indicated face identifications with M¨ obius transformations. This determines a representation ρ 1 : π 1 ( M ) → PSL 2 ( C ). One may check: ◮ ρ 1 ( π 1 ( P )) is purely parabolic
A Path in GF ( M ) Consider the regular ideal octahedron in H 3 , with vertices { 0 , ± 1 , ± i , ∞} , and perform the indicated face identifications with M¨ obius transformations. This determines a representation ρ 1 : π 1 ( M ) → PSL 2 ( C ). One may check: ◮ ρ 1 ( π 1 ( P )) is purely parabolic ◮ ρ 1 ( π 1 (Σ)) is Fuchsian
A Path in GF ( M ) Consider the regular ideal octahedron in H 3 , with vertices { 0 , ± 1 , ± i , ∞} , and perform the indicated face identifications with M¨ obius transformations. This determines a representation ρ 1 : π 1 ( M ) → PSL 2 ( C ). One may check: ◮ ρ 1 ( π 1 ( P )) is purely parabolic ◮ ρ 1 ( π 1 (Σ)) is Fuchsian ◮ ρ 1 ∈ GF ( M )
A Path in GF ( M ) Consider the regular ideal octahedron in H 3 , with vertices { 0 , ± 1 , ± i , ∞} , and perform the indicated face identifications with M¨ obius transformations. This determines a representation ρ 1 : π 1 ( M ) → PSL 2 ( C ). One may check: ◮ ρ 1 ( π 1 ( P )) is purely parabolic ◮ ρ 1 ( π 1 (Σ)) is Fuchsian ◮ ρ 1 ∈ GF ( M ) Since geometric finiteness is an open condition, we can deform the representation ρ 1 in GF ( M ). Let ρ t indicate the same face identifications, for the octahedron with vertices { 0 , ± 1 , ± it , ∞} .
A Path in GF ( M ) Consider the regular ideal octahedron in H 3 , with vertices { 0 , ± 1 , ± i , ∞} , and perform the indicated face identifications with M¨ obius transformations. This determines a representation ρ 1 : π 1 ( M ) → PSL 2 ( C ). One may check: ◮ ρ 1 ( π 1 ( P )) is purely parabolic ◮ ρ 1 ( π 1 (Σ)) is Fuchsian ◮ ρ 1 ∈ GF ( M ) Since geometric finiteness is an open condition, we can deform the representation ρ 1 in GF ( M ). Let ρ t indicate the same face identifications, for the octahedron with vertices { 0 , ± 1 , ± it , ∞} . Let Γ t = ρ t ( π 1 (Σ)).
’Rhombic’ Symmetry
’Rhombic’ Symmetry There is an order 4 orientation-reversing diffeomorphism of the genus 2 handlebody, that preserves P , and thus descends to a mapping class Φ ∈ MCG ∗ (Σ).
’Rhombic’ Symmetry There is an order 4 orientation-reversing diffeomorphism of the genus 2 handlebody, that preserves P , and thus descends to a mapping class Φ ∈ MCG ∗ (Σ). In fact, there are two curves ξ, η ∈ π 1 (Σ) preserved by Φ.
’Rhombic’ Symmetry There is an order 4 orientation-reversing diffeomorphism of the genus 2 handlebody, that preserves P , and thus descends to a mapping class Φ ∈ MCG ∗ (Σ). In fact, there are two curves ξ, η ∈ π 1 (Σ) preserved by Φ. By the Symmetry Lemma, the subset Fix Φ is preserved by σ M .
’Rhombic’ Symmetry There is an order 4 orientation-reversing diffeomorphism of the genus 2 handlebody, that preserves P , and thus descends to a mapping class Φ ∈ MCG ∗ (Σ). In fact, there are two curves ξ, η ∈ π 1 (Σ) preserved by Φ. By the Symmetry Lemma, the subset Fix Φ is preserved by σ M . One may check that Φ has a realization as a hyperbolic isometry normalizing Γ t , i.e. [ ρ t ] ∈ Fix Φ.
’Rhombic’ Symmetry There is an order 4 orientation-reversing diffeomorphism of the genus 2 handlebody, that preserves P , and thus descends to a mapping class Φ ∈ MCG ∗ (Σ). In fact, there are two curves ξ, η ∈ π 1 (Σ) preserved by Φ. By the Symmetry Lemma, the subset Fix Φ is preserved by σ M . One may check that Φ has a realization as a hyperbolic isometry normalizing Γ t , i.e. [ ρ t ] ∈ Fix Φ. Question What is Fix Φ in T (Σ)?
A Detour: 4-Punctued Spheres
A Detour: 4-Punctued Spheres Definition Φ ∈ MCG ∗ (Σ) is { ξ, η } -rhombic if it is order-4, orientation- reversing, and preserves simple closed curves ξ and η .
A Detour: 4-Punctued Spheres Definition Φ ∈ MCG ∗ (Σ) is { ξ, η } -rhombic if it is order-4, orientation- reversing, and preserves simple closed curves ξ and η . X ∈ T (Σ) is { ξ, η } -rhombic if X ∈ Fix Φ, for { ξ, η } -rhombic Φ.
A Detour: 4-Punctued Spheres Definition Φ ∈ MCG ∗ (Σ) is { ξ, η } -rhombic if it is order-4, orientation- reversing, and preserves simple closed curves ξ and η . X ∈ T (Σ) is { ξ, η } -rhombic if X ∈ Fix Φ, for { ξ, η } -rhombic Φ. Important facts about rhombic 4-punctured spheres:
A Detour: 4-Punctued Spheres Definition Φ ∈ MCG ∗ (Σ) is { ξ, η } -rhombic if it is order-4, orientation- reversing, and preserves simple closed curves ξ and η . X ∈ T (Σ) is { ξ, η } -rhombic if X ∈ Fix Φ, for { ξ, η } -rhombic Φ. Important facts about rhombic 4-punctured spheres: ◮ X can be formed by gluing isometric Euclidean rhombi
A Detour: 4-Punctued Spheres Definition Φ ∈ MCG ∗ (Σ) is { ξ, η } -rhombic if it is order-4, orientation- reversing, and preserves simple closed curves ξ and η . X ∈ T (Σ) is { ξ, η } -rhombic if X ∈ Fix Φ, for { ξ, η } -rhombic Φ. Important facts about rhombic 4-punctured spheres: ◮ X can be formed by gluing isometric Euclidean rhombi ◮ Fix Φ = { ξ, η } ⊂ ML (Σ)
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