introduction to hyperbolic 3-manifold ideal triangulations.casella/talks/mapss_2015.pdf ·...
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Introduction to Hyperbolic 3-Manifold IdealTriangulations.
——————————————————————————Presented by: Alex Casella
——————————————————————————School of Mathematics and Statistics
MaPSS——————————————————————————
27th March 2015
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Talk Structure
Talk Structure
i) Motivationii) What is a hyperbolic 3-manifold ideal triangulationiii) Some notions on the hyperbolic spaceiv) How to construct a hyperbolic structure on a triangulated
3-manifold
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Manifolds
Study of n-Manifolds
What is a n-manifold?? A n-manifold M is a topological space which locally
looks like Rn.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Manifolds
3-Manifolds
Dimension 1 and 2 are completely understood.For dimension n ≥ 4 we can not have a classification.
⇒ 3-Manifolds are the main focus.
![Page 5: Introduction to Hyperbolic 3-Manifold Ideal Triangulations.casella/Talks/MaPSS_2015.pdf · Introduction to Hyperbolic 3-Manifold Ideal Triangulations Construction of the Hyperbolic](https://reader033.vdocuments.us/reader033/viewer/2022043005/5f8b0d2d566e9e4fb31aab94/html5/thumbnails/5.jpg)
Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Manifolds
3-Manifolds
Dimension 1 and 2 are completely understood.For dimension n ≥ 4 we can not have a classification.
⇒ 3-Manifolds are the main focus.
![Page 6: Introduction to Hyperbolic 3-Manifold Ideal Triangulations.casella/Talks/MaPSS_2015.pdf · Introduction to Hyperbolic 3-Manifold Ideal Triangulations Construction of the Hyperbolic](https://reader033.vdocuments.us/reader033/viewer/2022043005/5f8b0d2d566e9e4fb31aab94/html5/thumbnails/6.jpg)
Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Manifolds
3-Manifolds
Dimension 1 and 2 are completely understood.For dimension n ≥ 4 we can not have a classification.
⇒ 3-Manifolds are the main focus.
![Page 7: Introduction to Hyperbolic 3-Manifold Ideal Triangulations.casella/Talks/MaPSS_2015.pdf · Introduction to Hyperbolic 3-Manifold Ideal Triangulations Construction of the Hyperbolic](https://reader033.vdocuments.us/reader033/viewer/2022043005/5f8b0d2d566e9e4fb31aab94/html5/thumbnails/7.jpg)
Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Decomposition Approach
Decompose the space in simpler pieces
Intuition: we cut the 3-manifoldalong closed surfaces trying toget 3-manifolds which areeasier to deal with.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Decomposition Approach
Decomposition scheme: preliminary considerations
We will assume the 3-manifold M to be:Connected: work on the connected components.Orientable: the double cover of a non-orientable manifoldis orientable.Without Boundary: if N has non-empty boundary, we canglue M to itself along the boundary to get a closedmanifold.Compact: makes things easier – this is cheating.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Decomposition Approach
Decomposition scheme: cutting to simplify
Let M be a connected, oriented, closed (i .e. compact and withempty boundary) 3-manifold:1) Prime Decomposition Theorem: We cut M along
convenient spheres and fill in the holes with 3-balls.
2) JSJ-Decomposition: We cut the resulting pieces along“good” tori, ending up with compact, oriented 3-manifoldswith (possibily empty) toric boundary.
![Page 10: Introduction to Hyperbolic 3-Manifold Ideal Triangulations.casella/Talks/MaPSS_2015.pdf · Introduction to Hyperbolic 3-Manifold Ideal Triangulations Construction of the Hyperbolic](https://reader033.vdocuments.us/reader033/viewer/2022043005/5f8b0d2d566e9e4fb31aab94/html5/thumbnails/10.jpg)
Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Decomposition Approach
Decomposition scheme: Geometrization Theorem
Theorem (Geometrization Theorem)
Let M be a compact, orientable, irreducible 3-manifold withempty boundary. There exists a (possibly empty) collection ofdisjointly embedded incompressible tori T1, . . . ,Tn in M suchthat each component of M cut along T1∪·· ·∪Tn is Seifertfibered or hyperbolic. Furthermore any such collection of toriwith a minimal number of components is unique up to isotopy.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Decomposition Approach
Decomposition scheme: Mostow - Prasad RigidityTheorem
Theorem (Mostow - Prasad Rigidity Theorem)
Let M,N be finite-volume hyperbolic 3-manifolds. Everyisomorphism π1(M)→ π1(N) is induced by a unique isometryM → N .
Theorem
A hyperbolic 3-manifold hasfinite volume if and only if it iseither closed or has toricboundary and it is nothomeomorphic to T 2× I
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Motivation
Decomposition Approach
Decomposition scheme: Mostow - Prasad RigidityTheorem
Theorem (Mostow - Prasad Rigidity Theorem)
Let M,N be finite-volume hyperbolic 3-manifolds. Everyisomorphism π1(M)→ π1(N) is induced by a unique isometryM → N .
Theorem
A hyperbolic 3-manifold hasfinite volume if and only if it iseither closed or has toricboundary and it is nothomeomorphic to T 2× I
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Hyperbolic 3-Manifold Ideal Triangulations
Ideal Triangulations
Idea:triangulate the 3-manifold M;endow each tetrahedron with an hyperbolic structure;glue the tetrahedra back together by hyperbolic isometries.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Hyperbolic 3-Manifold Ideal Triangulations
Ideal Triangulations
Theorem
Every 3-manifold M which is the interior of a compact manifoldwith toric boundary admits an ideal triangulation.
Theorem (Epstein–Penner)
Every finite volume hyperbolic 3-manifold M can be realized asa union of (possibly flat) hyperbolic ideal tetrahedra with gluedfaces.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Hyperbolic 3-Manifold Ideal Triangulations
Ideal Triangulations
Theorem
Every 3-manifold M which is the interior of a compact manifoldwith toric boundary admits an ideal triangulation.
Theorem (Epstein–Penner)
Every finite volume hyperbolic 3-manifold M can be realized asa union of (possibly flat) hyperbolic ideal tetrahedra with gluedfaces.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
The Hyperbolic Space H3
Upper half hyperbolic space
The 3-dimensional hyperbolic space is the metric spaceconsisting of the upper half–space
H3 = {(x ,y ,z) ∈ R3;z > 0}
endowed with the hyperbolic metric
ds2 =dx2 +dy2 +dz2
z2
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
The Hyperbolic Space H3
Hyperbolic Tetrahedra
We will call hyperbolic tetrahedron the convex hull of four noncollinear distinct points in H3.
In particular we will call idealhyperbolic tetrahedra thosewhose vertices lie on theboundary of H3.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
The Hyperbolic Space H3
Hyperbolic Tetrahedra
We will call hyperbolic tetrahedron the convex hull of four noncollinear distinct points in H3.
In particular we will call idealhyperbolic tetrahedra thosewhose vertices lie on theboundary of H3.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
The Hyperbolic Space H3
Isometries of H3
Orientation preserving hyperbolic isometries acts on ∂H3 asthe group PSL(2,C), hence
Isom+(H3)∼= PSL(2,C) = {(
a bc d
)∈ C4 ; ad −bc = 1}
Theorem
Let z1,z2,z3 ∈ ∂H3 be three distinct points, then
f (w) =(w −z1)(z3−z2)
(z2−z1)(z3−w)
is the only orientation preserving isometry sending thehyperbolic ideal triangle (z1z2z3) to (01∞).
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
The Hyperbolic Space H3
Isometries of H3
Orientation preserving hyperbolic isometries acts on ∂H3 asthe group PSL(2,C), hence
Isom+(H3)∼= PSL(2,C) = {(
a bc d
)∈ C4 ; ad −bc = 1}
Theorem
Let z1,z2,z3 ∈ ∂H3 be three distinct points, then
f (w) =(w −z1)(z3−z2)
(z2−z1)(z3−w)
is the only orientation preserving isometry sending thehyperbolic ideal triangle (z1z2z3) to (01∞).
![Page 21: Introduction to Hyperbolic 3-Manifold Ideal Triangulations.casella/Talks/MaPSS_2015.pdf · Introduction to Hyperbolic 3-Manifold Ideal Triangulations Construction of the Hyperbolic](https://reader033.vdocuments.us/reader033/viewer/2022043005/5f8b0d2d566e9e4fb31aab94/html5/thumbnails/21.jpg)
Introduction to Hyperbolic 3-Manifold Ideal Triangulations
The Hyperbolic Space H3
Isometries of H3
Corollary
Given any two ideal hyperbolic triangle, there exists a uniqueorientation preserving isometry between them.
Faces of ideal hyperbolic tetrahedra are ideal hyperbolictriangles, therefore we can always glue two faces of two (notnecessarily distinct) ideal tetrahedra by a unique orientationpreserving isometry.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
The Hyperbolic Space H3
Hyperbolic Ideal Tetrahedron Labels
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
The Hyperbolic Space H3
Hyperbolic Ideal Tetrahedron Labels
Every complex number z uniquely determines a hyperbolicshape of an ideal tetrahedron.
Theorem
Two hyperbolic structures z and w of an hyperbolic idealtetrahedron are the same structure, i .e. are isometric, if andonly if
w ∈ {z,z ′,z ′′}
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Construction of the Hyperbolic Structure from an Ideal Triangulation
Construction of the Hyperbolic Structure from an IdealTriangulation
Let M be a finite volume hyperbolic 3-manifold and let T anEpstein–Penner ideal triangulation of M.
Let T1, . . . ,Tn be the set of tetrahedra, with shape parametersz1, . . . ,zn.
The structure on M induced by T depends on the parametersz1, . . . ,zn. How is this structure?
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Construction of the Hyperbolic Structure from an Ideal Triangulation
Construction of the Hyperbolic Structure from an IdealTriangulation
Points in the interior of tetrahedra are good.Points in the interior of faces are good.Points in the interior of edges are good if and only if:
loops around each edge have angle 2π
the product of the modules of the shape parameters aroundeach edge is 1.
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Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Construction of the Hyperbolic Structure from an Ideal Triangulation
Construction of the Hyperbolic Structure from an IdealTriangulation
Points in the interior of tetrahedra are good.Points in the interior of faces are good.Points in the interior of edges are good if and only if:
loops around each edge have angle 2π
the product of the modules of the shape parameters aroundeach edge is 1.
![Page 27: Introduction to Hyperbolic 3-Manifold Ideal Triangulations.casella/Talks/MaPSS_2015.pdf · Introduction to Hyperbolic 3-Manifold Ideal Triangulations Construction of the Hyperbolic](https://reader033.vdocuments.us/reader033/viewer/2022043005/5f8b0d2d566e9e4fb31aab94/html5/thumbnails/27.jpg)
Introduction to Hyperbolic 3-Manifold Ideal Triangulations
Construction of the Hyperbolic Structure from an Ideal Triangulation
Construction of the Hyperbolic Structure from an IdealTriangulation
Points in the interior of tetrahedra are good.Points in the interior of faces are good.Points in the interior of edges are good if and only if:
loops around each edge have angle 2π
the product of the modules of the shape parameters aroundeach edge is 1.