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

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

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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.

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.

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.

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

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.

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.

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∞).

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∞).

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.

Introduction to Hyperbolic 3-Manifold Ideal Triangulations

The Hyperbolic Space H3

Hyperbolic Ideal Tetrahedron Labels

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 ′′}

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?

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.

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.

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.