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EXAMPLES OF FOUR-DIMENSIONAL GEOMETRIC TRANSITION Joint with S. - - PowerPoint PPT Presentation
EXAMPLES OF FOUR-DIMENSIONAL GEOMETRIC TRANSITION Joint with S. - - PowerPoint PPT Presentation
EXAMPLES OF FOUR-DIMENSIONAL GEOMETRIC TRANSITION Joint with S. Riolo Fribourg, 8th May 2019 W HAT IS GEOMETRIC TRANSITION ? Roughly speaking, a geometric transition is a deformation of geometric structures on a manifold, which at some point
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A simple example of this transition is obtained by considering a hyperbolic sphere with three cone points, i.e. the double of a (regular, say) hyperbolic triangle.
AN EXAMPLE
HYPERBOLIC-EUCLIDEAN-SPHERICAL TRANSITION
One can then “zoom" around this limit point, and obtain a transition
- f the geometric structure to a Euclidean sphere with cone points.
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Similarly, starting from a hyperbolic quadrilateral, we can construct a hyperbolic torus with one cone point.
ANOTHER EXAMPLE
HYPERBOLIC-EUCLIDEAN-SPHERICAL TRANSITION
When you let the cone angle go to 2π, the torus collapses to a
- point. Again, by “zooming", one finds a Euclidean torus as a limit.
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In this talk, we will be interested in hyperbolic structures which “collapse" to a co-dimension one totally geodesic hyperplane. An example (from Cooper-Hodgson-Kerckhoff's book): By glueing opposite faces, and then doubling, one constructs a hyperbolic structure on T3, singular along a link, which collapses to a hyperbolic torus with one cone point. We want to study "rescaled limits” for this type of degenerations.
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In dimension 3, Danciger in 2011 introduced the “right” limit geometry (half-pipe) and understood a fairly general condition which ensures the existence of transition on a compact three-manifold M, with a very simple (conical) singularities along a knot K. He proves that collapsing hyperbolic structures on M, with cone singularities along a knot K, admits a geometric transition to half- pipe and Anti-de Sitter structures under the following condition on the character variety: for a representation
HALF-PIPE GEOMETRY
WHY DO WE CARE?
H1
Adρ(π1(M∖K), 𝔱𝔭(2,1)) ≅ ℝ .
ρ : π1(M∖K) → SO(2,1) .
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In general, it is a difficult problem to determine which manifolds admit a transition of geometric structures. In this talk, we are interested in higher-dimensional examples. It is already pretty hard to produce higher-dimensional examples of hyperbolic manifolds (or other geometric structures) and study their deformations. We will produce here a class of examples of geometric transition from a hyperbolic structure to half-pipe and Anti-de Sitter structures
- n a closed finite-volume four-manifold with a simple not too
complicated singular locus.
HALF-PIPE GEOMETRY
WHY DO WE CARE?
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In 2010, Kerckhoff and Storm described an interesting deformation
- f hyperbolic polytopes in dimension 4, depending on a real
parameter. Starting from the 24-cell and removing two walls, they construct a polytope which can now be deformed to a family Pt, collapsing to a three-dimensional polytope (cuboctahedron) when t=0. This family of polytopes has been used by several authors (Martelli- Riolo, Saratchandran, Kolpakov-Riolo, Riolo-Slavich). In their paper, Kerckhoff and Storm mention that when t=0 the family of polytopes Pt is expected to have interesting geometric
- limits. It turns out that half-pipe geometry is well-suited to describe
this infinitesimal behaviour.
THE KERCKHOFF-STORM POLYTOPES
WHY DO WE CARE?
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A REMINDER
REAL PROJECTIVE STRUCTURES
A real projective structure on a manifold M is given by an atlas with values in n-dimensional projective space, where transition functions are the restrictions of projective transformations.
Ui Uj 'i(Ui)⊆X 'j(Uj)⊆X 'i :Ui !X 'j :Uj !X 'j◦'−1
i
2G
This is the definition of (G,X)-structure, for:
X = ℝPn
G = PGL(n + 1,ℝ)
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SPECIAL CASES
REAL PROJECTIVE STRUCTURES
Here we are interested in three special types of real projective structures:
- Hyperbolic structures
- Anti-de Sitter structures
- Half-pipe structures
X1 = ℍn = P {−x2
0 + x2 1 + … + x2 n−1 + x2 n < 0},
G1 = PO(n,1) X−1 = 𝔹d𝕋n = P {−x2
0 + x2 1 + … + x2 n−1 − x2 n < 0},
G−1 = PO(n − 1,2) X0 = ℍℙn = P {−x2
0 + x2 1 + … + x2 n−1+0 ⋅ x2 n < 0},
G0 = {[ A vT ±1] : A ∈ O(n − 1,1), v ∈ ℝn } < PGL(n + 1,ℝ)
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- 1. LIMIT OF HYP&ADS GEOMETRY
MOTIVATION FOR HALF-PIPE GEOMETRY
Consider the projective transformations Then define Namely, Xt is defined by the inequality Then Xt converges to:
rt = [diag (1,…,1,1 t )] .
{ Xt = rt(ℍn) t > 0 Xt = r|t|(𝔹d𝕋n) t < 0
Xt = P {−x2
0 + x2 1 + … + x2 n−1 + t|t|x2 n < 0} .
X0 = ℍℙn = P {−x2
0 + x2 1 + … + x2 n−1 < 0} .
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- 1. LIMIT OF HYP&ADS GEOMETRY
MOTIVATION FOR HALF-PIPE GEOMETRY
Moreover, Gt converges to: which is what we define as the “isometry" group of half-pipe space. Half-pipe space X0 is topologically the product However, its geometry is substantially
- different. For instance, the second
factor represents a degenerate direction.
G0 = {[ A vT ±1] : A ∈ O(n − 1,1), v ∈ ℝn }
X0 ≅ ℍn−1 × ℝ .
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- 2. DUALITY WITH MINKOWSKI SPACE
MOTIVATION FOR HALF-PIPE GEOMETRY
There is another reason why the "isometry group" G0 of half-pipe geometry is naturally defined as before. In fact, n-dimensional half-pipe space naturally parameterizes spacelike hyperplanes in Minkowski space Given (x0,…,xn), we associate the hyperplane defined by This is a spacelike hyperplane as a consequence of the condition and, by homogeneity, the correspondence is well-defined on
ℝn−1,1 . {(y0, …yn−1) ∈ ℝn−1,1 : − x0y0 + x1y1 + … + xn−1yn−1 = xn} .
−x2
0 + x2 1 + … + x2 n−1 < 0 ,
ℍℙn .
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- 2. DUALITY WITH MINKOWSKI SPACE
MOTIVATION FOR HALF-PIPE GEOMETRY
The "isometry group" G0 of half-pipe space is then the group induced by the action of In fact, it is an exercise to check that the action on spacelike hyperplanes defines an isomorphism Isom(ℝn−1,1) ≅ O(n − 1,1) ⋉ ℝn−1,1 . Isom(ℝn−1,1) ≅ G0 = {[ A vT ±1] : A ∈ O(n − 1,1), v ∈ ℝn } .
There is more additional structure in half-pipe geometry determined by the structure of the group G0. In fact, it makes sense to talk about a (non-degenerate) totally geodesic hyperplane, which corresponds to the hyperplanes in Minkowski space passing through a given point.
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HOW TO PRODUCE EXAMPLES OF TRANSITION
Here’s my favourite recipe:
- Take a continuous family of finite-volume hyperbolic polytopes Pt,
for t in [0,ε), such that P0 collapses to a polytope in a co- dimension one totally geodesic subspace.
- Glue a finite number of copies of Pt, so as to obtain a family of
hyperbolic structures on a fixed manifold (with some singularities!)
- Apply the “rescaling” transformations rt=[diag(1,…,1,1/t)].
- Show that the rescaled polytopes glue to a half-pipe structure
when t=0, and even continue towards Anti-de Sitter structures for t in (-ε,0].
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EXAMPLES OF TRANSITION
A FIRST EXAMPLE IN 3D
Let us now go back to
- ur first 3d example of
geometric transition. For simplicity, we send some of the vertices at infinity. Here is an affine picture of the hyperbolic polyhedron we
- btain (in the projective model).
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EXAMPLES OF TRANSITION
A FIRST EXAMPLE IN 3D
We can now deform this hyperbolic (ideal) polyhedron, by “flattening” towards a hyperbolic quadrilateral. By glueing faces/doubling, one obtains a deformation of geometric structures (with cone singularities!).
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EXAMPLES OF TRANSITION
A FIRST EXAMPLE IN 3D
Let us now apply the projective rescaling rt=[diag(1,…,1,1/t)]. One shows that the polytope limits to a polytope in half-pipe space, and continues to Anti-de Sitter space!
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THE STRUCTURE OF THE SINGULAR LOCUS
When we glue opposite faces, and double, we get finite-volume “cusped” structures, with singularities along a two-component link.
- When t>0, this is a cone singularity, for
which the cone angle approaches 2π.
- When t<0, this is a tachyon singularity.
- When t=0, the singularity is the
infinitesimal version of cone angles and tachyons.
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THE STRUCTURE OF THE SINGULAR LOCUS
Why does there exist such an element of G0, the analogue of a rotation along a line, in half-pipe geometry? Recalling that two totally geodesic hyperplanes P1 and P2 in half-pipe space correspond to two points q1 and q2 in Minkowski space, P1 and P2 intersect if and only if q1 and q2 are separated by a spacelike segment. In this case, the element of G0 that we are looking for corresponds to the translation sending q1 to q2. P2 P1 q1 q2
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EXAMPLES OF TRANSITION
A SECOND EXAMPLE IN 3D
We shall now see another simple example which will be useful later on. All the angles of this polytopes are π/2, except at the three top edges. Let us now send these three dihedral angles to π.
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EXAMPLES OF TRANSITION
A SECOND EXAMPLE IN 3D
Again, by applying the projective rescaling rt=[diag(1,…,1,1/t)],
- ne obtains a limit polytope in half-pipe space. As before, the
three lateral faces are degenerate. We c a n a g a i n p r o d u c e a transition of finite-volume structures on a manifold, by doubling along lateral faces, and then along top and bottom faces. The singular locus is more complicated here (a θ-graph).
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FOUR-DIMENSIONAL EXAMPLES
DEFINITION: A cuboctahedral manifold is a hyperbolic finite-volume 3-manifold obtained by glueing a finite number of copies of an ideal right-angled cuboctahedron. It is good if the glueings preserve a checkerboard coloring of the triangular faces. THEOREM (Riolo-S.): Let M be a good cuboctahedral
- manifold. Then MxS1 admits a
g e o m e t r i c t r a n s i t i o n f r o m a hyperbolic structure to an Anti-de Sitter structure, singular along a 2- dimensional simplicial complex.
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FOUR-DIMENSIONAL EXAMPLES
The starting point to construct this transition is the family of 4- dimensional polytopes Pt introduced by Kerckhoff and Storm, by starting from the hyperbolic 24-cells and removing two walls. For all t, the polytope Pt intersects a totally geodesic plane in a
- cuboctahedron. When t→0, Pt collapses to the cuboctahedron itself.
We show that, after rescaling, one can continue the polytopes rt(Pt) in projective space to a path of polytopes. For t=0, one obtains a half-pipe polytope, for which some of the walls are degenerate, and this continues to Anti-de Sitter space. Moreover, all isometries of the cuboctahedron extend to isometries
- f Pt. This permits to “glue" copies and obtain geometric structures.
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SOME WORDS ABOUT THE COMBINATORICS
To get a flavour of how this looks like, here is the list of hyperplanes defining the polytope Pt.
0+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 0− = h
- p
2 : 1 : 1 : 1 : t i , 1+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 1− = h
- p
2 : 1 : 1 : 1 : t i , 2+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 2− = h
- p
2 : 1 : 1 : 1 : t i , 3+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 3− = h
- p
2 : 1 : 1 : 1 : t i , 4+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 4− = h
- p
2 : 1 : 1 : 1 : t i , 5+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 5− = h
- p
2 : 1 : 1 : 1 : t i , 6+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 6− = h
- p
2 : 1 : 1 : 1 : t i , 7+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 7− = h
- p
2 : 1 : 1 : 1 : t i , A = h 1 : p 2 : 0 : 0 : 0 i , B = h 1 : 0 : p 2 : 0 : 0 i , C = h 1 : 0 : 0 : p 2 : 0 i , D = h 1 : 0 : 0 : p 2 : 0 i , E = h 1 : 0 : p 2 : 0 : 0 i , F = h 1 : p 2 : 0 : 0 : 0 i .
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SOME WORDS ABOUT THE COMBINATORICS
A technical point: we work in the double cover of real projective
- space. For instance, the vector A=[-1:√2:0:0] defines the half-space
The polytope Pt is actually contained in a single affine chart {x0>0}.
−x0 + 2x1 ≤ 0
0+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 0− = h
- p
2 : 1 : 1 : 1 : t i , 1+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 1− = h
- p
2 : 1 : 1 : 1 : t i , 2+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 2− = h
- p
2 : 1 : 1 : 1 : t i , 3+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 3− = h
- p
2 : 1 : 1 : 1 : t i , 4+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 4− = h
- p
2 : 1 : 1 : 1 : t i , 5+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 5− = h
- p
2 : 1 : 1 : 1 : t i , 6+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 6− = h
- p
2 : 1 : 1 : 1 : t i , 7+ = h
- p
2 : 1 : 1 : 1 : 1
- |t|
i , 7− = h
- p
2 : 1 : 1 : 1 : t i , A = h 1 : p 2 : 0 : 0 : 0 i , B = h 1 : 0 : p 2 : 0 : 0 i , C = h 1 : 0 : 0 : p 2 : 0 i , D = h 1 : 0 : 0 : p 2 : 0 i , E = h 1 : 0 : p 2 : 0 : 0 i , F = h 1 : p 2 : 0 : 0 : 0 i .
- When t→0, the “plus”
walls converge to a totally geodesic plane.
- The “minus" & “letter"
walls are timelike when t < 0 , a n d b e c o m e degenerate in the half- pipe limit.
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SOME WORDS ABOUT THE COMBINATORICS
We already know some of the walls! The “letter” walls look like today's first 3d example… …whereas the “minus" walls are copies of our second example.
We can compare them with the cuboctahedron, which is the “horizontal” slice of t h e f o u r- d i m e n s i o n a l polytope.
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THE STRUCTURE OF THE SINGULAR LOCUS
The singular locus of these geometric structures on MxS1 is a compact foam, namely a 2-complex locally modelled on the cone
- ver the 1-skeleton of the tetrahedron.
To visualize the singularities, let us picture the link of a singular
- point. It is endowed with:
- a spherical structure for t>0,
- a HS-structure for t<0, namely, is locally modelled on the space of
rays from a point in ,
- an intermediate geometric structure for t=0, which is locally
modelled on the space of rays in the tangent space of a point in half-pipe space.
ℝ3,1
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THE STRUCTURE OF THE SINGULAR LOCUS
On the 2-strata of the singular locus, we have a conical singularity. For t>0, the (spherical) link of a point looks like: The analogue for t<0 is a HS-structure on the 3-sphere:
θ θ
The (interior of) the red balls in S3 represent the timelike rays. There is a transition of the geometric structure in the link of this singular point, from spherical to HS. In the transition, the two red spheres collapse to points, representing the degenerate direction in half-pipe geometry.
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THE STRUCTURE OF THE SINGULAR LOCUS
The other points on the singular locus have a more complicated structure: At these points, the singular 2-complex is of the form: F i n a l l y, t h e m o s t complicated singularity is of the form:
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THE STRUCTURE OF THE CUSPS
Finally, let us visualize the structure of the cusps along the transition. In a suitable coordinate system, we can write hyperbolic space (t>0), Anti-de Sitter space (t<0) and half-pipe space (t=0) as the upper half- plane with the bilinear form: Hence the link of an ideal point is endowed with:
- a Euclidean structure for t>0,
- a Minkowski structure for t<0,
- an intermediate structure for t=0, which is the flat analogue of
half-pipe geometry.
dx2
0 + dx2 1 + dx2 2 + t|t|dx2 3
x2
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THE STRUCTURE OF THE CUSPS
A section of the cusp for the hyperbolic structures when t>0 looks like (glue opposite faces by translations): After rescaling, the geometric structure on a section of the cusp looks like:
t = 0 t < 0 t = 0 t > 0
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SOME RESULTS ON THE CHARACTER VARIETY
Kerckhoff and Storm also proved that their path of polytopes, interpreted as a path of hyperbolic orbifolds, forms a smooth 1- dimensional component in the character variety: Here Γ22 is the orbifold fundamental group, which is generated by 22 reflections and has 80 orthogonality conditions. Roughly speaking, this means that at every point of this path, the constructed deformation is “the only possible one”. Their proof relies on the fundamental property that, in dimension 4, “cusp groups stay cusp groups”. This property is applied to each “peripheral" cube group.
X(Γ22, Isom(ℍ4)) := Hom(Γ22, Isom(ℍ4))/ /Isom(ℍ4)
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SOME RESULTS ON THE CHARACTER VARIETY
We prove that the same results hold (surprisingly?) for the Anti-de Sitter version of the path of orbifolds. Moreover, here is a picture of the character variety at the collapse: The "vertical" component is a smooth curve which is the Kerckhoff-Storm polytope (or its Anti-de Sitter brother). The “horizontal” one corresponds to deformations which arise from deforming the cuboctahedron in .
ℍ3
This is a little lie: the picture represents Hom(Γ22, Isom(ℍ4))/
/Isom+(ℍ4)
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SOME RESULTS ON THE CHARACTER VARIETY
To see the actual character variety, one has to take a 2:1 quotient:
[ρ0] [ρt]
The 12-dimensional horizontal component is now composed of “mirror” points. At the level of Zariski tangent space, this reflects the decomposition The 1-dimensionality of the last factor can be proved directly and is a higher-dimensional analogue of Danciger’s sufficient condition for the regeneration.
H1
Adρ0(Γ22, 𝔱𝔭(1,4)) = H1 Adρ0(Γ22, 𝔱𝔭(1,3)) ⊕ H1 ρ0(Γ22, ℝ(1,3))
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