Geometric structures on manifolds: transitions from hyperbolic to anti-de Sitter geometry

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Corso di Laurea Magistrale in

Matematica

Geometric structures on manifolds:

transitions from hyperbolic to

anti-de Sitter geometry

Autore:

Andrea Parma

Relatore: Prof. Bruno Martelli Controrelatore: Prof. Roberto Frigerio

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Introduction

In this work we expose some results about geometric transitions, which have been achieved by Jeffrey Danciger in his thesis ”Geometric transitions: from hyperbolic to AdS geometry”. More precisely, we will talk about geometric structures, and in particular we will focus on deformations between two kinds of such structures (namely hyperbolic and anti-de Sitter) on closed manifolds, mainly in dimension 3 and possibly with a set of singular points. This is done by passing through an intermediate geometry, called half-pipe: we will prove a converse of this phenomenon, by giving a sufficient condition (which involves the representations of a fundamental group in a Lie group of matrices) for recovering a path of hyperbolic and anti-de Sitter structures starting from a half-pipe one.

In Chapter 1 we will give some general definitions about geometric struc-tures: given a smooth n-manifold X and an analytic group G of diffeomor-phisms of X, we define a (X, G)−structure on a n-manifold M as an atlas with values in X whose changes of coordinates locally coincide with elements of G, and we observe that such a structure is fully described by a developing map D : fM → X and a holonomy representation ρ : π1(M ) → G (we will

actually work on these two objects in most of the proofs). We will be mainly interested in the cases where X is either the hyperbolic space Hnor the anti-de Sitter space AdSn, and G is its isometry group: a deformation from one space to the other can be constructed by considering a continuous family of pairs (Xs, Gs), with Xs ⊂ PnR and Gs < PGL(n + 1, R), which is a model

for either Hn or AdSn, depending on the sign of s; the case s = 0 gives the so called half-pipe geometry (HPn, GHP) that will be described in Chapter

2. We can then consider geometric transitions on a generic n-manifold M , i.e. a path of (Xt, Gt)−structures with t varying continuously from positive

to negative values.

Since Mostow’s rigidity theorem does not allow deformations of hyper-bolic structures in dimension n ≥ 3, we introduce the notion of cone singular-ity, so that we will be working on geometric structures with a singular locus of codimension 2, where a sufficiently small neighbourhood of any singular point is modeled on a cone-like space. The main goal is to prove the follow-ing regeneration theorem: if a closed 3−manifold N has a half-pipe structure (DHP, ρHP) with singular locus Σ, and (ρt) is a suitable path of holonomy

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representations, then it is possible to construct a path of hyperbolic/anti-deSitter structures (Dt, ρt) limiting to (DHP, ρHP). We will first prove this

result separately on a tubular neighbourhood of the singular locus and on its complementary in N , and then we will construct a geometric transition on the whole N .

Chapter 3 will give an alternative model for the spaces H3, AdS3 and HP3 by defining P3R as a projectivized space of hermitian 2 × 2 matrices with coefficients in a 2−dimensional R−algebra, so that it will be possible to restate the previous theorem. Furthermore, we will see that the existence of a path of holonomies is implied by a smoothness condition about the representation variety R(π1(M \ Σ), PSL(2, R)). We will finally apply this

regeneration theorem to a half-pipe structure that will be constructed in Chapter 4.

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Chapter 1

Preliminaries

1.1 (X,G)-structures

We start by recalling the well known definition of a smooth (C∞) structure on a n−manifold as a maximal atlas whose charts take values in Rn and whose transition maps are smooth functions: we may replace Rn with any smooth manifold X, and ask the transition maps to be diffeomorphisms that preserve an additional structure of X. We will now expose some general results that can be achieved when the set of all allowed transition maps satisfies an analiticity condition, as in the following definition.

Definition 1. Let X be a smooth manifold, and G an analytic group of diffeomorphisms of X (i.e. such that if g1, g2 ∈ G agree on an open set of

X, then g1 = g2). A (X, G)−structure on a manifold M is a maximal atlas

{(Ui, ϕi)}i∈I with respect to the following properties:

• {Ui}i∈I is an open cover of M , and every ϕimaps Ui diffeomorphically

to an open set of X;

• if i and j are indexes such that Ui ∩ Uj 6= ∅ and U is a connected

component of the latter, then ϕi◦ ϕ−1_j |ϕj(U ) is (the restriction of ) an

element of G, which is uniquely determined since G is analytic. Remark. 1. Some authors use a slightly more general definition in which

G is a pseudogroup, i.e. an analytic family of diffeomorphisms between open sets of X, which is closed under restriction, inversion and (when it makes sense) composition; however, it won’t be necessary in this work.

2. The maximality condition easily implies that if the atlas contains a chart (U, ϕ), it also contains all its restrictions to open subsets of U and all its compositions with elements of G. In particular, we can always find a subatlas made of charts whose domains are trivializing open sets with respect to the universal cover π : fM → M .

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3. It is possible to define an induced structure on fM : given a subatlas of M as above, for every (Ui, ϕi), take the restrictions of ϕi ◦ π to the

connected components of π−1(Ui).

The analiticity of G allows us to define analytic continuations: let M be a (X, G)−manifold, and fix a chart (U0, ϕ0) and a point p0 ∈ U0. For every

path α : [a, b] → M such that α(a) = p0, we define the analytic continuation

of ϕ0 along α (which will be a pathα : [a, b] → X) in the following way: byb the Lebesgue lemma, we can find a partition a = t0 < t1 < ... < tn= b and

charts {(Ui, ϕi)}0≤i<n such that α([ti, ti+1]) ⊆ Ui ∀i. Now, for every i > 0,

we consider the restrictions of ϕi−1and ϕito the connected component Wi ⊆

Ui−1∩ Ui containing α(ti), and find gi ∈ G (which is uniquely determined)

such that gi ◦ ϕi and ϕi−1 agree on Wi. We can finally define α as theb concatenation ϕ0◦ α [t0,t1] g1◦ ϕ1◦ α [t1,t2] ... g1...gn−1◦ ϕn−1◦ α [tn−1,tn]

Notice that, up to replacing the charts with ϕ0, g1ϕ1, ..., g1...gn−1ϕn−1,

we can assume that ϕi−1 and ϕi are already compatible around α(ti), i.e.

they agree on a connected neighbourhood of α(ti): we will say that such a

sequence of charts is coherent. In the following, we will be using the same notation as above: in particular, M will always be a (X, G)−manifold with a base chart (U0, ϕ0) and a basepoint p0 ∈ U0.

Proposition 1. The analytic continuation of any path α starting from p0

is well defined.

Proof. We just give a sketch: one has to check that:

i) if we fix a partition of [a, b], then α does not depend on the choice of_b the charts;

ii) α does not depend on the partition._b

For (i), if {(Vi, ψi)}0≤i<n is another coherent sequence of charts, with ψ0 =

ϕ0, it is enough to prove that ϕi and ψi are compatible around α(ti) ∀i.

This can be done by induction on i: the case i = 0 is trivial, and if i > 0 we have that ϕi and ψi are compatible (respectively) with ϕi−1 and ψi−1,

which are compatible as well by inductive hypothesis and by the analiticity of G.

Finally, we easily see that the construction ofα with respect to a certain_b sequence of charts does not change if we refine the partition, so that if we start from two different partitions we can merge them together and use (i).

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Proposition 2. If q ∈ M is another point, and α is a path from p0 to q,

then the endpoint of α only depends on the homotopy class of α.b

Proof. The idea is to consider a homotopy H : [a, b]×[0, 1] → M between two paths, divide the domain into smaller rectangles (each of which is contained in the domain of a chart) and prove that if two paths in [a, b] × [0, 1] coincide outside a single rectangle, the analytic continuations of their images have the same endpoint (since they can be constructed from the same sequence of covering charts).

These results allow us to define the developing map D : fM → X: after fixing pe0 ∈ π

−1_(p

0), together with the base chart ϕ0 ◦ π : π−1(U0) → X,

it makes sense to define D(x) as the endpoint of α, where α is any path_b fromp_e0 to x. Notice that this map is a local diffeomorphism, since it locally

coincides with charts of fM . If we change the inital data, we get another map D0 such that D0 = g ◦ D, where g is a uniquely determined element of G (which can be found by choosing a path between the two basepoints and a sequence of charts whose domains cover its image). In particular, we can consider the composition D ◦γ, where γ ∈ π1(M ) is a deck transformation of

f

M , and observe that it is another developing map (the base chart is the same as before, while the basepoint is another point in π−1(p0)), therefore we get

ρ(γ) ∈ G such that D ◦ γ = ρ(γ) ◦ D (this relation is called equivariance of D with respect of ρ). We also have

ρ(γ1γ2) ◦ D = D ◦ γ1◦ γ2= ρ(γ1) ◦ D ◦ γ2 = ρ(γ1) ◦ ρ(γ2) ◦ D

and since D is an open map, ρ(γ1γ2) and ρ(γ1) ◦ ρ(γ2) are elements of

G that agree on an open set of X, and thus they must be equal, so that ρ : π1(M ) → G is a homomorphism, called holonomy representation. It

is easily seen that if D is replaced by g ◦ D, then ρ is replaced by gρg−1: this defines an equivalence relation among the pairs (D, ρ).

Remark. The pair (D, ρ) completely describes the (X, G)−structure. In fact, there is a natural bijection between maximal (X, G)−atlases like in Definition 1 and equivalence classes of pairs (D, ρ): we have already seen how to get D and ρ from an atlas; conversely, given D and ρ, for every connected trivializing open set U ⊆ M , take che chart D ◦ σ, where σ : U → fM is a section of π (notice that two different choices of σ differ by composition with a certain γ ∈ π1(M ), so that the corresponding charts differ by composition

with ρ(γ)). Thanks to this result, we will often be working with D and ρ instead of the atlas.

We are interested in deformations of such structures: if M is compact and with boundary, a deformation of a structure (D0, ρ0) is a smooth path

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have an equivalence relation for deformations: if gt is a smooth path in G,

ψt is a smooth family of diffeomorphisms, each of which is defined on the

complement in M of a neighbourhood of ∂M , and eψt is a lift of ψt to fM

(with respect to a fixed basepoint), we say that (Dt, ρt) and (D0t, ρ0t) are

equivalent deformations if

D0_t= gtDtψe_t ρ0_t= gtρtgt−1

Since π1(M ) admits a finite presentation, say with m generators, the set

of homomorphisms ρ : π1(M ) → G can be seen as a subspace of Gmgiven by

the relations, which is stable under the action of G by conjugation, so we can consider the quotient space R(π1(M ), G) (however, this space is usually far

from being a manifold); as well, we can can give a suitable topology to the set D(M ; X, G) of all (X, G)−structures of M up to equivalence (by quotienting the set of all developing maps with the compact-open topology), and finally, we get hol : D → R by mapping every structure to its holonomy and passing to the quotients. This map has the following important property:

Theorem 1. (Ehresmann-Thurston theorem) The map hol : D → R is a local homeomorpsism.

An immediate consequence is that if (ρt)t∈(−,) is a path in R and ρ0 is

the holonomy of a (X, G)−structure, it is possible (up to restricting (−, ) to a smaller neighbourhood of 0) to lift (ρt) to a path of (X, G)−structures

in D. This result is proved in [1], and we will use it in the next chapters.

1.2 Hyperbolic geometry and its models

Now we are going to focus on the geometries we are most interested in, starting from hyperbolic geometry. We recall that the hyperbolic space Hn is defined to be the unique (up to isometry) complete and simply connected riemannian manifold with all sectional curvatures equal to −1, and that a riemannian n−manifold is said to be hyperbolic if it is locally isometric to Hn (this is equivalent to saying that all sectional curvatures are −1 as well). We also recall some useful models for the hyperbolic space

• The hyperboloid model: in the Minkowski space Rn+1 _{with the}

inner product η1(x, y) = −x1y1 + x2y2 + ... + xn+1yn+1, the set H

of vectors x such that η1(x, x) = −1 and x1 > 0 is one of the two

connected components (the ”positive” one) of a hyperboloid. In anal-ogy with spherical geometry, the tangent space at a generic x can be

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canonically identified with the hyperplane orthogonal to x (since for every α : [0, ) → H we can derive the identity η1(α(t), α(t)) = −1 and

obtain η1(α(0), α0(0)) = 0). By equipping each tangent space with the

restriction of η1 we get a riemannian manifold, which turns out to be

a model for Hn. Here we can easily see that every linear isometry of (Rn+1, η1) which preserves H (instead of mapping it to the negative

component) restricts to an isometry of H: these transformation form the Lie group O+(n, 1), which is exactly the isometry group of Hn(its action is already transitive on the orthonormal frames): it has dimen-sion n(n + 1)

2 and two connected components, given by the orientation preserving and the orientation reversing isometries. We finally men-tion that the intersecmen-tions of H with the linear k−subspaces are totally geodesic (k − 1)−submanifolds, all isometric to Hk−1: in particular, the geodesic lines are given by the 2−planes, and any two distinct vec-tors x and x0 determine a unique geodesic arc between them, whose length is cosh−1(η1(−x, x0)).

• The Poincar´e disk model: using the same notation as above, and identifying the euclidean space Rn _{with the hyperplane {x}

1 = 0}, we

can map H into Rn by the stereographic projection centered at −e1:

the image is the open unit ball B, and the inverse map allows to define the pullback metric on B, which is

gy(v, w) =

4hv, wi (1 − |y|2₎2

(where gy is the metric in the tangent space at y ∈ B and h·, ·i is

the euclidean scalar product of Rn_{): this is a conformal change of the}

euclidean metric, so that this model (called Poincar´e disk) preserves the angles; one can also check that the stereographic projection maps any k−subspace of H to the intersection of B with either a k−plane containing the origin or a k−sphere orthogonal to ∂B.

• The upper half-space model, i.e. the open half-space {x_n> 0} ⊂ Rn with the metric

gx(v, w) =

hv, wi x2

n

.

It is a conformal model as well, and it is obtained from the Poincar´e disk by a spherical inversion, so that the k−subspaces are the k−half-spaces and the k−hemispheres orthogonal to the boundary {xn= 0}.

• The projective model: we can also replace the hyperboloid H with the union of all the lines of Rn+1 spanned by the vectors of H. This defines an open subset of PnR, and an alternative version of the isom-etry group is PO(n, 1) = O(n, 1)/{±I}: this is exactly the group of

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projective transformations which preserve the quadric η1(x, x) = 0.

Since any vector in H satisfies x1 6= 0, the corresponding line admits

a representative such that x1 = 1, so that this model of Hn is actually

an open subset of Rn, namely the open unit ball; unlike the Poincar´e disk, this is not a conformal model, but the hyperbolic k−subspace are now represented by the affine k−planes of Rn.

Another important tool in hyperbolic geometry is the definition of points at infinity, or ideal points: a boundary ∂Hn _{is added to H}n _{by considering}

the set of equivalence classes of half-lines, where two half-lines are said to be equivalent if their unit speed parametrizations γ1, γ2 : [0, +∞) → Hnsatisfy

sup{d(γ1(t), γ2(t)), t ≥ 0} < +∞; it is also possible to define a topology on

Hn= Hnt ∂Hn such that:

• ∂Hn_{and H}n _{are homeomorphic, respectively, to a (n − 1)−sphere and}

a closed n−ball;

• every isometry of Hn _{extends uniquely and continuously to H}n_.

We also mention that every isometry has at least a fixed point in Hn_,

and it is called:

• elliptic (or sometimes rotation) if it fixes a point in Hn_;

• parabolic if it fixes no points in Hn _{and exactly one point in ∂H}n_;

• hyperbolic if it fixes no points in Hn _{and exactly two points in ∂H}n_.

Although there is no reasonable way to define a distance between ideal points, it turns out that the isometries act on ∂Hnas conformal transforma-tions: this is well seen in the upper half-space model, where we can define the angles between curves in ∂Hn in the usual way, and every isometry is a composition of reflections and/or spherical inversions, which preserve the angles. This has an interesting consequence in the case n = 3: a conformal and orientation preserving diffeomorphism of ∂H3 ∼= S2 corresponds to a holomorphic automorphism of the Riemann sphere P1C = C ∪ {∞} (where C is canonically identified with the plane x3 = 0), and conversely, it is not

difficult to prove that any such automorphism is the restriction of a unique isometry, so that Isom+(H3) ∼= Aut(P1C) = PSL(2, C).

Since Isom(Hn) is an analytic group of diffeomorphisms, (Hn, Isom(Hn)) is a first example of (X, G)−pair. We can easily check that such a structure on a smooth manifold M corresponds to a hyperbolic metric, i.e. a rieman-nian metric with constant curvature −1: in fact, an atlas like in Definition 1 allows to pull back on M the metric of Hnthrough all charts, and this makes sense because all the changes of coordinates are isometries; conversely, if M has a hyperbolic metric, every point admits an ε−ball which is isometric to an ε−ball in the hyperbolic space, and by mapping all such balls into Hn we get the desired atlas.

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1.3 The transition to AdS geometry in the

projec-tive space

We have seen that Hn _{and Isom(H}n_{) can be realized, respectively, as an}

open subset of PnR and a subgroup of projective transformations, and as a consequence every hyperbolic structure also has a naturally induced (Pn

R, PGL(n + 1))−structure. In this section, we will always use the pro-jective model, and Hn will always be the open ball

X1 := {[x1, ..., xn+1] ∈ PnR | − x21+ x22+ ... + x2n+1 < 0}.

The same construction can be done from any nonzero inner product on Rn+1, by taking the vectors whose square-norm is negative and the group of linear transformations (up to ±I) which preserve the product. We will consider the family of inner products

ηs:=   −1 0 0 0 In−1 0 0 0 sgn(s)s2  

where s is an arbitrary real number (coherently with the notation we had used in the previous section). For a generic s 6= 0, we define the space Xsand the group Gs from ηs. We now consider che case s = −1: in analogy

with the hyperboloid model of Hn, we can identify the tangent space at a line with its orthogonal hyperplane, and notice that the restriction of η−1

there has signature (n − 1, 1): this means that X−1 is a lorentzian manifold,

called anti-de Sitter space and denoted by AdSn, with isometry group PO(n − 1, 2); furthermore, one can define the curvature tensor as well as it is done in riemannian geometry, and check that all sectional curvatures are −1, so that AdSn _{is the lorentzian analogue of H}n_.

Remark. We recall that in a vector space with a Minkowski product g, a vector x 6= 0 is called space-like, light-like or time-like if g(x, x) is (respec-tively) positive, null or negative. In particular, the set of time-like vectors has two connected components, which are commonly called past and future light cones, and a Lorentz transformation is called time preserving or time reversing, respectively, if it preserves the two cones or if it exchanges them. A lorentzian manifold is said to be time-orientable if it is possible to choose, with continuity, a time orientation on every tangent space.

We will always use the projective model of AdSn: since this space is not contained in any affine chart of PnR, it will be sometimes convenient to use the conformal model of the projective space, which is obtained from

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the north hemisphere of Sn ⊂ Rn+1 _{by the stereographic projection on the}

hyperplane xn= 0 centered at the south pole (the result is the closed unit

ball of Rn, where the antipodal points on the boundary are identified, and the projective lines are represented by circular arcs which join two antipodal points). We can observe that:

• AdSn _{is topologically a R}n−1_{−bundle over S}1_{, and it is orientable if}

and only if n is odd;

• it is always time-orientable, so that the isometry group has two con-nected components (time preserving and time reversing) if n is even, and four otherwise;

• every time-like vector in any tangent space is the speed of a periodic geodesic, and its orthogonal hyperplane (i.e. the totally geodesic (n − 1)−submanifold ”spanned” by all its orthogonal vectors in the tangent space) is isometric to Hn−1; similarly, every space-like vector generates an infinite euclidean line, and its orthogonal hyperplane is isometric to AdSn−1;

• PO(n − 1, 2) acts transitively on AdSn, and the stabilizer of a point is the full Lorentz group O(n − 1, 1).

We finally observe that the hyperplane xn+1= 0 has the same

intersec-tion with all the spaces Xs: not surprisingly, this intersection is a subspace

of Xs isometric to Hn−1, and we will call it the distinguished hyperplane of

Xs, whose stabilizer in Gs is the subgroup consisting of blockwise diagonal

matrices (with an element of O(n − 1, 1) in the top left block and a ±1 in the bottom right one).

It turns out that Xsis actually another model for Hnor AdSn, depending

on the sign of s: in fact, the rescaling map

rs:=

In 0

0 |s|−1

maps Xsgn(s)to Xs, and transforms Gsgn(s)into Gsby conjugation;

how-ever, our goal is to describe deformations from hyperbolic to anti-de Sitter geometry, so we will need a path of (Xs, Gs)−structures with s varying

con-tinuously from 1 to −1. Of course, this implies that we will have to define a new geometry for s = 0.

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Chapter 2

Transitions on manifolds

with cone singularities

Now that we have described a transition from Hn to AdSn in P3R, we will start studying analogous transitions between hyperbolic and anti-de Sitter structures on a generic manifold (by passing through a new type of structure that still has to be defined). However, since Mostow’s rigidity theorem does not allow deformations of hyperbolic structures in dimension at least 3, we must work with slightly different objects, which will be manifolds with geometric structures outside a set of singular points, called singular locus. On the other hand, neighbourhoods of the singular points will be modeled on cone-like singularities: as well as a neighbourhood of the vertex of a cone can be seen as an angle in H2 whose sides are identified, in dimension 3 we can consider a dihedral angle in H3 whose faces are identified, and the singular locus will be a line. We call Cα the space obtained from a dihedral

angle of width α (notice that C2π is the trivial case, i.e. a manifold which

doesn’t actually have any singularities); such a construction can be done also for α > 2π, by gluing more angles along their faces ”consecutively”. These are models for hyperbolic manifolds with a cone singularity.

Definition 2. Let N be a closed orientable 3−manifold, Σ ⊂ N a link and M = N \ Σ. A hyperbolic cone-manifold structure on N with singular locus Σ is a hyperbolic structure on M which extends to N in the following sense: for every singular point p ∈ Σ there exist a neighbourhood U 3 p and a chart ϕ : U → Cα for some α ∈ R such that U ∩ Σ is mapped into the singular

line of Cα, and such charts, together with those of M , give an atlas of N

whose transition maps are isometries in the sense of metric spaces.

Remark. We recall that a distance-preserving bijection between riemannian manifolds is automatically a smooth isometry, so there is nothing new about

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the transition maps which do not involve singular points; as well, a generic transition map is a riemannian isometry outside the singular locus.

Using the same notation as above, let T be a tubular neighbourhood of a connected component Σ1 ⊆ Σ, and µ ⊂ T \ Σ1 a meridian, i.e. a simple

loop that is trivial in T and winds around Σ1 once. After fixing a base chart

whose domain contains the basepoint of µ, we easily see that: • N is the metric completion of M ;

• the developing map D of T \ Σ₁ extends to the completion of ^T \ Σ1,

which is the union of ^T \ Σ1 and a line fΣ1 which covers Σ1;

• D(fΣ1) is a line ` ⊂ H3 and, if ρ is the holonomy representation

asso-ciated to D, ρ(µ) is a rotation by α mod 2π around `; in particular, we have a uniquely determined [α] ∈ R/2πZ for each connected com-ponent of Σ;

• If λ ⊂ T \ Σ1 is a loop which is parallel to Σ1 in T , ρ(λ) is a hyperbolic

isometry acting on ` as a translation, whose size defines the length of Σ1.

We can also consider the lifted developing map eD on ^T \ Σ1, which takes

values in ^H3\ `: the completion of this space can be seen as a dihedral angle of infinite width, whose rotation group is isomorphic to R, and in this case e

ρ(µ) gives exactly α and completely describes a neighbourhood of a singular point.

An analogous construction can be done in anti-de Sitter geometry: we consider a tubular neighbourhood T of a space-like line `, so that the orthog-onal plane at any point of ` inherits a Minkowski product. As in the previous case, a neighbourhood of a singular point is described by the lifted holonomy of a loop encircling Σ1, which corresponds to an orientation-preserving

isom-etry of_R1,1^_{\ {0}, where R}1,1 _{denotes the 2−dimensional Minkowski space:}

the universal cover has infinitely many time-like components, and the group of orientation-preserving isometries is isomorphic to Z × R, where the first component indicates the action on the the time-like components (which is basically a translation). Now, if we require the singular points to have a past and a future, both non-empty and connected (as well as the smooth points), we must have two time-like components, and therefore the discrete component of the lifted holonomy must be 2 (or −2, but the sign only de-pends on the orientation); on the other hand, the second component φ ∈ R gives the cone angle. Like before, we call Cφthe space having a singular line

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Definition 3. Let N , Σ and M be as before. An anti-de Sitter cone-manifold structure on N with singular locus Σ is an anti-de Sitter structure on M such that every p ∈ Σ has a neighbourhood modeled on Cφ for some

φ ∈ R (i.e. which is mapped into Cφby a chart which preserves the singular

points), and the transition maps are isometries.

Remark. Since lorentzian manifolds are not metric spaces, we need an al-ternative definition of isometry for the open sets containing singular points: we ask the transition maps to be smooth isometries which can be extended to the singular locus.

Although we will mainly work in dimension 3, we mention that these definitions of cone singularities can be generalized in any dimension: the singular locus is assumed to be a totally geodesic submanidfold of codimen-sion 2 (space-like in the anti-de Sitter case), and the spaces Cα and Cφ are

defined analogously. In the following, we will also give some examples in dimension 2, where the singular locus is a discrete set.

In the anti-de Sitter case, these singularities also arise from an alternative construction: in dimension 2, we can cut AdS2 along a light-like half-line originating from a point p, and glue the two resulting half-lines in a different way. More precisely, any isometry that fixes p and acts as a Lorentz boost on its tangent plane gives a gluing map, and the result is a surface with a cone point. Similarly, we can construct the so called tangent cone at the singular point by repeating the same procedure on R1,1: given a Lorentz boost with angle φ, the gluing map between the two light-like half-lines is a scalar multiplication by eφ (see Figure 2.1). These constructions also describe the orthogonal sections of a cone singularity in dimension 3.

2.1 Half-pipe geometry

A deforming hyperbolic structure on a manifold M may collapse to a lower dimensional object. As an example, we may construct a family of hyperbolic structures with a cone point on a torus: if we start with a quadrilateral Q in H2 whose opposite sides have the same length, we can define isometric gluing maps on the boundary, and the resulting torus T has a smooth hyperbolic structure outside the point p given by the four vertices; not surprisingly, the cone angle at such point is the sum of the interior angles of Q (which is always strictly less than 2π, and its difference from 2π gives the area of T ). A developing map for the hyperbolic structure on T0 = T \ p can be defined so that Q minus its vertices is the image of a fundamental domain of eT0, and the holonomies of the generators of π1(T0) are two orientation

preserving isometries, each of which maps a side of Q to its opposite. If we try to deform Q by making the sum of its angles approach 2π (i.e. if we try

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Figure 2.1: The alternative construction of cone singularities for AdS geometry: instead of directly gluing the space-like half-lines s and s0, we can make a cut along the light-like half-line ` and transform the region between s0and ` by the Lorentz boost which maps s0 to s; now, we have to glue the two copies of ` (the shaded lines in the second image show how the half-lines are matched).

to remove the singularity), its area approaches 0, and if we choose such a deformation so that the distance between two opposite sides is fixed, then Q collapses to a segment, and the developing map of the punctured torus collapses to a submersion on the corresponding line.

In the following, we assume to have a family of developing maps whose limit is a submersion and gives a smooth foliation of M ; however, we shall consider an additional structure.

Definition 4. Let X and G be as in Definition 1, with dim X = n − k. A (X, G)−foliation on a n−manifold M is a smooth foliation of M by k− dimensional leaves, with trivializing charts ϕi : Ui → X × Rk such that

the transition maps are of the form ϕj ◦ ϕ−1i (x, y) = (gij(x), fij(x, y)), with

gij ∈ G.

In other words, we have a foliation such that the space of leaves has, locally, a (X, G)−structure, and we can still define a pseudo-developing map D : fM → X (which will be a submersion instead of a local diffeomorphism) and the corresponding holonomy ρ : π1(M ) → G, in analogy with what we

did in Section 1.1. In the case k = 1, X = Hn−1 and G = Isom(X) (the one we are interested in), we talk about hyperbolic foliations.

We now recall the space X0 ⊂ PnR, which can equivalently be defined as the quotient of the quadric {x ∈ Rn+1| − x2

1+ x22+ ... + x2n= −1} by the

antipodal map (or just the connected component consisting of points such that x1> 0, with no quotient): this equation is the same as the hyperboloid

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Hn−1 × R, and the projection on the distinguished hyperplane xn+1 = 0

gives a hyperbolic foliation. We will see how this structure is intermediate between hyperbolic and anti-de Sitter geometry.

Let (Dt, ρt) be a smooth family of hyperbolic structures on M (defined

for 0 < t < ) limiting, with respect to the compact-open topology, to a submersion D0 on a hyperplane P ⊂ Hn, equivariant with respect to

the limit holonomy ρ0 (we can assume wlog that P is the distinguished

hyperplane in the projective model of Hn); as a consequence, ρ0 takes values

in the subgroup of isometries that preserve P , which consists of matrices of the form

±A 0 0 ±1

(where A ∈ O(n − 1, 1)), so the pair (D0, ρ0) gives a hyperbolic

fo-liation of M in the following way: in analogy with the case of ordinary (X, G)−structures, we can take all the charts from open sets of M to Hn−1 × R such that the first component is the composition of D0 with a

section of π : fM → M .

Now, since we actually don’t want the developing maps to collapse, we can compose them with the rescaling maps, and obtain rt◦ Dt : fM → Xt

(for t > 0): notice that the path of hyperbolic structures on M doesn’t change, but the limit for t → 0, which must be seen as a limit of projective structures, may be different. In particular, we assume that rt◦ Dtconverges

to a local diffeomorphism D : fM → X0, which will be the developing map

of a new projective structure; on the other hand, the holonomy of rt◦ Dt is

rt◦ ρt◦ r−1t : if we fix γ ∈ π1(M ), we have a smooth path

ρt(γ) =

A(t) w(t) v(t)T a(t)

.

This is actually a lift in GL(n + 1, R), wlog with constant determinant ±1: by assumption, this path converges to a blockwise diagonal matrix as t → 0, and the lift is uniquely determined if we require that a(t) → 1. Since the rescaling map is represented by the diagonal matrixI 0

0 t−1 , it follows that rt◦ ρt(γ) ◦ rt−1=   A(t) tw(t) v(t)T t a(t)  

Since, by assumption, the limits of v(t), w(t) and a(t) are respectively 0, 0 and 1, we find lim t→0rtρt(γ)r −1 t = A(0) 0 v0(0)T 1 (2.1)

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Letting ρ(γ) be the limit, we say that the ρt are compatible to first

order with ρ.

Definition 5. Let HPn be the space X0, and let GHP be the group

A 0 vT ±1 | _{A ∈ O(n − 1, 1), v ∈ R}n /{±I} < PGL(n + 1, R) Any (HPn, GHP)−structure is called a half-pipe structure.

This is a slight modification of the pair (X0, G0): the space is the same,

but we have restricted the group of isometries: if we only required the isometries to preserve the degenerate metric η0, the bottom right coefficient

could be any nonzero real number, but here we want it to be ±1.

As already mentioned, a model for HPn is the connected component of the quadric Q ⊂ Rn+1 given by x2₁− x2

2− ... − x2n = 1 and x1 > 0, and in

analogy with the hyperboloid model of Hn, we can define the k−subspaces as the intersections of Q with the (k + 1)−linear subspaces of Rn+1_{; in}

particular, if p is a generic point of HPnand T is its tangent space, there is a sorrespondence between the k−linear subspaces of T and the k−subspaces of HPn containing p. Since we have the diffeomorphism HPn ∼= Hn−1× R (where Hn−1is identified with the subspace given by xn+1= 0) by separating

the first n coordinates from the last one, T turns out to be the direct sum of he_n+1i and T_π(p)_Hn−1_{, where π is the projection on the hyperplane x}

n+1 = 0:

this is also the hyperplane of Rn+1orthogonal to p (as a vector) with respect to the metric η0, and η0|T is still a degenerate metric which vanishes on

he_n+1i: for this reason, he_n+1i ⊂ T will be called degenerate direction, as well as any vertical line in HPnwill be called degenerate line. On the other hand, η0 is positive-definite on every hyperplane H ⊂ T not containing

en+1; furthermore, since T ∼= hen+1iL H ∼= hen+1iL Tπ(p)Hn−1, we have

an isomorphism between H and Tπ(p)Hn−1(given by the projections), which

turns out to be an isometry, and as a consequence π gives an isometry between the subspace spanned by H and Hn−1.

Now we recall the definition of GHP: every element is the equivalence

class of two matrices which differ by a change of sign, so that exactly one of them preserves the connected component of Q (the one whose top left coefficient is positive). If we only consider the ”positive” component as a model, like before, it may be more convenient to replace every equivalence class with the representative which preserves the components:

GHP = A 0 vT ±1 | A ∈ O+(n − 1, 1), v ∈ Rn (2.2)

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Figure 2.2: The hyperboloid model of HP2, and an example of a degenerate line (the vertical one) and a nondegenerate line.

Here we can see that GHP has dimension

n(n + 1)

2 like Isom(H

n_{) and}

Isom(AdSn), and with an abuse of notation we will call GHP the isometry

group of HPn.

Clearly, every isometry preserves the degenerate lines, so it descends to an isometry of the leaf-space Hn−1 given by A. The group has four connected components, one for each choice for the signs of det A and of the bottom right coefficient; more precisely, the signs of det A, the bottom right coefficient and the determinant of the whole matrix tell, respectively, if the isometry preserves the orientation of Hn−1, the degenerate lines and HPn.

If two points lie on the same degenerate line, the corresponding vec-tors differ by a multiple of en+1, say ken+1; the quantity |k| is invariant

under isometries, so we can call it the length of the vertical path which joins the two points. Of course, an analogous formula for this length can be obtained for the projective model: two different points in X0 ⊂ PnR lie on the same degenerate line if and only if they can be expressed as [x1, x2, ..., xn, y] and [x1, x2, ..., xn, z], with −x21+ x22+ ... + x2n < 0, so that

the corresponding points in Q are given by the same (n − 1)−uples rescaled

by 1 x1 r 1 − x2 x1 2 − ... −xn x1 2

, and the signed distance along the

degen-erate line turns out to be

z − y x1 r 1 −x2 x1 2 − ... −xn x1 2 (2.3)

If, instead of what we did in (2.2), we choose the representatives whose bottom right coefficient is 1, we see that GHP∼= Rno O(n − 1, 1), where the second factor is the stabilizer of the distinguished hyperplane, while the first one shall be thought as a group of infinitesimal isometries: in fact, recalling

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(2.1), such a transformation arises from a path of isometries limiting to the identity. More precisely, we can consider the matrices

Ev=

0 −v v∗ 0

∈ so(n − 1, 1)

(where v ∈ Rn_{and v}∗_{is obtained from v}T _{by changing the sign of the first}

component), and observe that every infinitesimal isometry can be expressed as lim t→0 rtexp(Etv)r −1 t = I 0 v∗ 1

This transformation acts on any degenerate line π−1(x) ⊂ HPn (where x ∈ Hn−1) as a translation, whose size is exactly the vertical speed at 0 of the path exp(Etv) · x in Hn (these two quantities are both equal to

η1(v, x), as one could easily check). We will talk about infinitesimal

el-liptic/parabolic/hyperbolic isometries if v is (respectively) space-like/light-like/time-like, since the path exp(Etv) consists of isometries of the

corre-sponding type.

We can now see, as an example, how a family of collapsing singular hyperbolic structures (Dt, ρt) on a torus like at the beginning of this section

limits to a singular half-pipe structure: for every t > 0 we choose, as a fundamental domain, a quadrilateral Qt such that the distances between

opposite sides are a and b(t), where b(t) → 0, and we arrange these domains in the hyperboloid model of H2 so that, if x and y are the generators of the fundamental group of the punctured torus, we have

ρt(x) =   cosh a sinh a 0 sinh a cosh a 0 0 0 1   and ρ_t(y) =   cosh b(t) 0 sinh b(t) 0 1 0 sinh b(t) 0 cosh b(t)  

These quadrilaterals collapse to a segment and ρt(y) limits to the

iden-tity. When we rescale, every Qtis contained in Xt, and the holonomy changes

by conjugation: in particular, rtρt(y)rt−1 now limits to an infinitesimal

hy-perbolic isometry. Similarly, the commutator [x, y] is represented by a loop that winds around the singular point, and as already observed, its holonomy is a rotation that gives the cone angle α(t). Like before, we can compute the limit of rtρt([x, y])r−1t , and not surprisingly, the result is an infinitesimal

rotation that gives the so called infinitesimal cone angle ω, which turns out to be α0(0).

This construction is summarized in Figure 2.3, where every Xt ⊂ P3R is represented by its projection on the plane {x1 = 0} (when t = 1, this is

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Figure 2.3: The figures in the first row show a fundamental domain for a hyperbolic structure collapses to a segment in the Poincar´e disk model (the two figures in the middle are intermediate steps), while those in the second one show how the same fundamental domains, when rescaled, limit to a polygon in HP2_.

2.2 The regeneration theorem in the smooth case

We have seen how HP geometry arises as a limit of collapsing hyperbolic or anti-de Sitter structures; conversely, we can construct such a family of collapsing structures from a HP structure (and from a family of holonomy representations like in (2.1)).

Theorem 2. Let M0a compact manifold with boundary, and M a thickening

of M0 (i.e. M0 ⊂ M and M \ M0 is a collar neighbourhood of ∂M0).

Let (DHP, σHP) be a HP-structure on M , and ρt : π1(M0) → Isom(X)

a continuous path of holonomy representations, where X is either Hn _or

AdSn and t varies, respectively, in a right/left neighbourhood of 0. Finally, assume that the holonomies ρt are compatible to first order with σHP: then,

for sufficiently small values of t, there exists a family of (X, G)−structures on M0 (where G = Isom(X)) with holonomy ρt.

Proof. The holonomy representation σt:= rtρtrt−1takes values in the group

Gt⊂ PGL(n+1, R), and σtconverges to σHPby assumption. Now we can use

the Ehresmann-Thurston theorem, and find, for small values of t, a family of projective structures on M0 with holonomy σt: we call Ft the corresponding

developing maps, which converge to DHP with respect to the compact-open

topology. We have to prove that these are actually (X, G)−structures: since σttakes values in Gt, it is sufficient to prove that the image of Ftis contained

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give, as we had already observed, a family of developing maps with images in X±1 and equivariant with the holonomy representations ρt given by the

hypothesis).

Let K ⊆ fM0 be a compact fundamental domain, so that DHP(K) is a

compact subset of X0; it is easy to see that for a generic p ∈ X0 we have

p ∈ Xs ⇔ s < µ(p), where µ : X0 → (0, +∞] is a continuous function, so

that µ ◦ DHP has a minumum value > 0 on K, and DHP(K) ⊂ X

2. Now,

since the maps Ft converge to DHP, we also have Ft(K) ⊂ X

2 when t is

sufficiently close to 0, and if t <

2 this implies that Ft(K) ⊂ Xt (because, in general, s < s0 ⇒ Xs0 ⊂ X_s). Finally, for every x ∈ fM₀, we can find

γ ∈ π1(M0) and y ∈ K such that x = γ(y), and by equivariance it follows

that

Ft(x) = Ft(γ(y)) = (σt(γ))(Ft(y)) ∈ (σt(γ))(Xt) = Xt

where the third step is allowed for small values of t, and the last equality holds because σt(γ) ∈ Gt is an isometry of Xt.

Our next goal is to prove an analogous result for hyperbolic/anti-de Sitter structures with cone singularities: the theorem we have just proved will apply to the smooth part, and in the following we will see what to do in a neighbourhood of the singular locus.

2.3 Deformations and cone singularities in

projec-tive geometry

As suggested by the previous proof, we shall work with the induced projec-tive structures.

Definition 6. A geometric transition from hyperbolic to anti-de Sitter ge-ometry on a n−manifold M is a smooth path of projective structures (Dt, ρt)

on M , where t varies in a neighbourhood of 0, and:

• ∀t > 0, (Dt, ρt) is projectively conjugate to a hyperbolic structure

(which means that there exists gt∈ PGL(n + 1, R) such that gt◦ Dt⊆

X1 = Hn and the image of gtρtgt−1 is contained in G1 = Isom(Hn));

• ∀t < 0, (Dt, ρt) is projectively conjugate to an anti-de Sitter structure

(in the same sense as above).

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Definition 7. Let N be an orientable 3−manifold, Σ ⊂ N a link, and M = N \ Σ. A projective structure on N with cone singularity at Σ is a projective structure on M given by an atlas {Ui, ϕi}i∈I such that:

• every ϕ_i extends to a continuous map ϕi : Ui → P3R; moreover, if Ui∩ Σ is not empty, it is diffeomorphic to an open interval, and it is

mapped by ϕi into a line of P3R;

• for every p ∈ Σ there exist a neighbourhood B of p and finitely many charts (U1, ϕ1), ..., (Uk, ϕk) such that B ⊆ U1∪ ... ∪ Uk and B ∩ Σ ⊆

Uj∩ Σ ∀j.

We easily see that, for any hyperbolic or anti-de Sitter manifold with cone singularities, the underlying projective structure satisfies the requests of the previous definition: in fact, any singular point p is mapped to a point q in the singular line of a certain Cα by a chart, and a small neighbourhood

of q is covered by its intersections with a collection of dihedral angles, whose interiors will be the open sets U1, ..., Uk. In particular, we can still prove

that:

• if T is a tubular neighbourhood of a component Σ1of Σ, the developing

map of ^T \ Σ1 extends to the universal branched cover ( ^T \ Σ1∪ fΣ1),

and fΣ1 is mapped to a line in ` ⊂ P3R;

• the holonomy of a meridian µ encircling Σ1is an orientation preserving

projective transformation which fixes ` pointwise (the transformations with this property form a group G`).

We now focus on the last point: after choosing an appropriate system of homogeneous coordinates, we have

G`= I B 0 A | A, B ∈ M2×2(R), det A > 0

We are only interested in the conjugacy class of ρ(µ) (two conjugate matrices may be seen as the same holonomy with respect to two different coordinate systems). Of course, we get an invariant by considering the eigenvalues of A: if they are complex, a straightforward calculation shows that we can assume, up to conjugation, that

B = 0 and A = kcos(θ) − sin(θ) sin(θ) cos(θ)

,

where k is a positive real number and θ ∈ R

2πZ; furthermore θ is uniquely determined, because θ and −θ are the arguments of the eigenvalues of A,

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and there is no ambiguity since I 0 0 A and I 0 0 AT

are not conjugate in G`. As well, if the eigenvalues are real, they have the

same sign, so we can take θ = ±π. In both cases, we call eiθ the rotational part of ρ(µ), so that we have a map R : G` → S1, which turns out to be

a homotopy equivalence. In particular, π1(G`) ∼= Z, and by lifting R to

e

R : fG` → R, we can define the total rotational part, which is given by the

lifted holonomy; if the eigenvalues of A are real, it is also called discrete rotational part and it is a multiple of π. As an example, we may recall the construction of anti-de Sitter cone singularities in Figure 2.1: ρ(µ) has two real positive eigenvalues (with light-like eigenvectors), so that the discrete rotational part is an integer multiple of 2π; furthermore, if we consider µ as a curve defined on [0, 1] and choose a path (gt)t∈[0,1] in PGL(2, R) such that

g0=Id, g1 = ρ(µ) and gt◦ D(µ(0) = D(µ(t)), the corresponding path of the

rotational parts is a loop in S1 with winding number 1; as a consequence, the discrete rotational part is always 2π, but unlike the hyperbolic case, this does not imply that the cone singularity is trivial. More generally, this implication does not hold for projective cone singularities: we may also have the same situation as in Figure 2.1, where a neighbourhood of a singular point is obtained by cutting and gluing along a half-plane which is stabilized by ρ(µ), and the latter may fix that half-plane pointwise without being the identity (we will see how this is exactly what happens in half-pipe geometry).

Definition 8. Let N , Σ and M be as usual. A HP-structure with cone sin-gularity is a smooth HP-structure on M whose underlying projective struc-ture has a cone singularity at Σ, and such that for every p ∈ Σ there are exactly two rays originating from p and transverse to Σ which correspond to degenerate lines.

Remark. The last request is coherent with with the definition of cone singu-larities for AdS-structures, where we asked the singular points to have a past and a future, both nonempty and connected: the tangent space of a point in HP3 may be considered as a Minkowski space where the two cones consisting of all time-like vectors have shrunk to a line (the degenerate one).

If we restrict to a small neighbourhood B of a singular point p, we have a developing map from ^B \ Σ to HP3_{⊂ P}3

R, which extends to the singular locus: if q ∈ P3R is the image of p and ` 3 q is the line which contains the image of B ∩ Σ, we may consider a degenerate ray r in B (with respect to the HP-structure) originating from p, whose image is contained in a half-line

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Figure 2.4: The two degenerate rays r and r0 originating from a singular point p must be mapped by the developing map into the two degenerate half-lines s and s0originating from q ∈ HP3, which determine two degenerate half-planes, both bounded by the line ` containing the image of B ∩ Σ (in red), and consequently two degenerate half-planes originating from Σ in M .

s ⊂ P3R originating from q. The plane spanned by ` and s gives a half-plane in HP3 containing a degenerate half-line, so it is a union of degenerate half lines, and those near s correspond to degenerate rays originating from the singular points near p. By the previous definition, we can find two degenerate half-planes in B, and if µ is a small nontrivial loop in B \ Σ with basepoint in one of them, its image is a loop in HP3 _which:

• starts from one of the two degenerate half-planes of HP3_{with boundary}

`;

• winds around ` by crossing the opposite half-plane;

• ends on the starting half-plane.

This implies that the holonomy of µ fixes ` pointwise and has discrete rotational part 2π. Similarly, if λ is a longitude, i.e. a loop in M such that hλ, µi = π₁(T ), where T is a tubular neighbourhood of the component of Σ containing p, ρ(λ) is an element of GHP which preserves `.

Since ` is a nondegenerate line of HP3 (because the degenerate rays intersect it transversely by definition), we can assume up to conjugation that ` is the line given by x3= x4 = 0, so that

ρ(µ) =I ∗ 0 ∗

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(we are using the description of GHP given by (1.2)), and since the

dis-crete rotational part is 2π, all its eigenvalues must be positive, so that ρ(µ) must be an infinitesimal rotation:

ρ(µ) =     1 0 0 0 0 1 0 0 0 0 1 0 0 0 ω 1    

An analogous result holds for λ:

ρ(λ) =     cosh d sinh d 0 0 sinh d cosh d 0 0 0 0 ±1 0 0 0 ψ ±1    

As elements of GHP, these two matrices can be expressed as limits of

rescaled isometries of either H3 or AdS3, like in (2.1): the most natural choices, for the hyperbolic case, are

ρt(µ) =     1 0 0 0 0 1 0 0 0 0 cos ωt − sin ωt 0 0 sin ωt cos ωt     and ρt(λ) =     cosh d sinh d 0 0 sinh d cosh d 0 0 0 0 ± cos ψt − sin ψt 0 0 sin ψt ± cos ψt     .

In particular, ρt(µ) is a rotation whose angle tends to 0 as t → 0, but the

cone angle actually limits to 2π, and as already observed, ω represents the variation of the cone angle 2π + ωt (notice that, since in most cases the cone angle increases when a path of hyperbolic structures limit to a half-pipe one, ω will usually be negative): of course, a completely analogous path of holonomies can be constructed in Isom(AdS3) by using the hyperbolic trigonometric functions in the bottom right block).

The geometry of a neighbourhood of a singular point in a HP-structure is described by ρ(µ), and such a neighbourhood can be constructed in analogy

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Figure 2.5: A construction of cone singularities in HP geometry: in two dimensions, the holonomy always gives an isometry of HP2 which fixes a degenerate line, so we can make a cut along a degenerate half-line h and glue the two copies of the latter in a different way (like we did in Figure 2.1): in this model, the nondegenerate lines which cross h appear with a change of slope (which is given by the infinitesimal angle ω). Of course, the same construction can be done in dimension 3 by gluing two copies of a degenerate half-plane.

with what we had done for AdS geometry: if we choose, in B, a nondegen-erate half-plane α bounded by `, we have a developing map from B \ α to a dihedral angle in HP3, whose faces α1 and α2 both correspond to α and

must be glued in the only possible way. We can also repeat the procedure shown in the description of Figure 2.1, i.e. make a cut along a degenerate half-plane β and transform one of the two pieces by the isometry ρ(µ), which maps α1 to α2: we are left with a dihedral angle whose faces are two copies

of β, and the gluing map between them is always the identity (since ρ(µ) fixes the degenerate plane containing `), but now some straight segments in B \ Σ are represented by distorted segments with a ”corner point” on β (the figure above shows a 2−dimensional section).

2.4 The regeneration theorem for singular

struc-tures

Now that we have described geometric transitions from hyperbolic to anti-de Sitter structures around a cone singularity, we shall prove a new version of Theorem 2: more precisely, our goal is to give a sufficient condition to

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construct a path of hyperbolic/anti-de Sitter structures, starting from a singular half-pipe manifold and a path of holonomies which limit, in a certain sense that will be clarified later, to the half-pipe holonomy. We will start by constructing such strucures around the singular locus, and we will finally ”glue” them with those on the smooth locus given by Theorem 2.

Proposition 3. Consider a projective structure on a 3−manifold, with sin-gular locus a knot Σ: there exist a tubular neighbourhood T of Σ and a system of coordinates (r, x, y) on ^T \ Σ such that:

• letting µ and λ be as before, the corresponding deck transformations of ^

T \ Σ are, respectively, the maps (r, x, y) 7→ (r, x + 1, y) and (r, x, y) 7→ (r, x, y + 1)

• if D is the developing map of T , the map (r, x, y) 7→ lim

s→0D(s, x, y) is

a submersion f (y) independent of x (whose image is the line in P3R corresponding to the singular locus)

Remark. There is a strong analogy with the usual cylindrical coordinates of R3, so we will say that (r, x, y) are cylindrical coordinates as well. Theorem 3. Assuming that N is orientable and that Σ ⊂ N is a knot, let ρt: π1(M ) → PGL(4, R) be a path of representations such that:

• ρ0 is the holonomy of a projective structure with cone singularity at Σ;

• if µ is a meridian encircling Σ, there exists a line ` ⊂ P3

R such that each ρt(µ) fixes ` pointwise.

Then, for sufficiently small values of t, ρt is the holonomy of a projective

structure with cone singularity at Σ.

Proof. Let T0 b T be two tubular neighbourhoods of Σ (they are both solid tori, since N is orientable): if we restrict to M0 := M \ T0, we can

use the Ehresmann-Thurston theorem, and find a path of developing maps Dt: fM0 → P3R, which give a family of projective structures on M0. On the

other hand, we can define a family of developing maps for T as follows: let D0 be the developing map of the projective structure with holonomy ρ0, and

define a system of cylindrical coordinates (r, x, y) on ^T \ Σ like in Proposition 3; now, since both ρ0(µ) and ρ0(λ) are represented by matrices in GL+(4, R),

they both admit real valued logarithms log(ρ0(µ)) and log(ρ0(λ)), and by

analytic continuation we can define log(ρt(µ)) and log(ρt(λ)) for sufficiently

small values of t, so that the power function ρz_t := exp(z · log ρt) is a well

defined smooth function. Now we can define the developing maps

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Recalling that the deck transformations of T correspond to translations in the last two coordinates, we check the equivariance condition:

∀m, n ∈ Z, D_t0(r, x + m, y + n) = = ρt(µ)x+mρt(λ)y+nρ0(µ)−x−mρ0(λ)−y−nD0(r, x + m, y + n) = = ρt(µ)x+mρt(λ)y+nρ0(µ)−x−mρ0(λ)−y−nρ0(λ)nρ0(µ)mD0(r, x, y) = = ρt(µ)x+mρt(λ)y+nρ0(µ)−xρ0(λ)−yD0(r, x, y) = = ρt(µ)mρt(λ)nρt(µ)xρt(λ)yρ0(µ)−xρ0(λ)−yD0(r, x, y) = = ρt(µ)mρt(λ)nD0t(r, x, y)

(we used the fact that ρt(µ) and ρt(λ) commute, and so all their powers).

We constructed the developing maps separately in eT and outside eT0_{. The two}

maps may not agree on the intersection ^T \ T0_{; however, they do for t = 0,}

so that for small values of t they can be glued by a bump function.

Remark. We can actually assume that ρt(µ) pointwise fixes a line `t and

that `t → `0: in fact, in a sufficiently small neighbourhood U of ` (in the

space of lines of P3R), we can define a smooth function Φ : U → PGL(4, R) such that Φ(`) = Id and ∀`0 ∈ U , Φ(`0) maps `0 to `; if we conjugate every ρt by Φ(`t), we get a smooth path of holonomies with the same fixed line,

and we can prove the theorem in the same way as before.

Now, as promised, we prove an analogous result for transition from hy-perbolic to Anti-de Sitter geometry.

Theorem 4. Let N be a closed orientable 3− manifold, Σ ⊂ N a knot, M = N \ Σ and µ a small loop encircling Σ. Let ρt: π1(M ) → Isom(X) be

a path of representations defined for t > 0, where X is either H3 or AdS3. Furthermore, suppose that:

• the rescaled representations r_tρtrt−1 converge, for t → 0, to a

repre-sentation ρHP;

• ρ_HP is the holonomy of a HP-structure with cone singularity at Σ; • ∀t, ρt(µ) is either a rotation of H3 or a Lorentz boost of AdS3;

then, for sufficiently small values of t, ρtis the holonomy of a

hyperbolic/anti-de Sitter structure with cone singularity at Σ.

Proof. We can apply Theorem 3 to the representations σt := rtρtr−1t , and

find a family of projective structures with cone singularity at Σ, which realize the σt. Now, as in the proof of the regeneration theorem in the smooth case,

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we can consider the image of a compact fundamental domain K by the developing maps Dt: since D0(K) ⊂ X0, we will have Dt(K) ⊂ X±t (where

the sign depends on X) for small values of t, so that the structures (Dt, ρt)

are conjugage by rt to hyperbolic/anti-de Sitter structures with the same

singular locus.

Remark. Recall that the Dtare defined on the universal cover of N branched

over Σ: if T is a tubular neighbourhood of Σ, we can take the universal cover of N \ Σ and ”complete” it with a copy of eΣ for every connected component of π−1(T \ Σ); the compact fundamental domain must be taken in this object (and it must include singular points).

Of course, the last two theorems can be proven, with slight modifications, in the general case when Σ is a link.

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Chapter 3

The PGL models in

dimension 3

3.1 _{The spaces X}

s

as spaces of hermitian matrices

Since we are mainly working in dimension 3, we shall give an alternative description of H3_{, AdS}3 _{and HP}3_{: more precisely, we will introduce a new}

model for the 3−dimensional spaces Xs(where s is an arbitrary real number)

and, consequently, a generalization of the following facts about hyperbolic geometry:

• a compactification of H3 _{can be defined by adding a set ∂H}3 _{of points}

at infinity, and every isometry of H3extends uniquely and continuously to the ideal boundary ∂H3.

• the orientation-preserving isometries act faithfully on ∂H3 _as

projec-tive transformations of P1

C.

We will finally give a description of Isom+(Xs) as a group of projective

transformations on the ideal boundary (with respect to a projective structure that will be defined soon). The main goal of this construction is to state another regeneration theorem, where we only start from a half-pipe structure (D, ρ) on a closed 3−manifold with cone singularities, and see that a family of holonomies like in Theorem 4 (and then a geometric transition) can be constructed if a certain regularity condition about the representation variety R(π1(M ), O(2, 1)) holds in a neighbourhood of ρ0 = π◦ρ, where, as usual, M

is the smooth locus and π : Isom(HP3) → Isom(H2) is the same projection as in the previous chapter.

We start by defining the algebras Bs := R[κs] as the quadratic extensions

of R such that κ2s= −sgn(s)s2: we have Bs∼= C when s > 0 (an isomorphism

is given by κs ↔ _si), and similarly, if κ−1 is denoted by τ , Bs ∼= B−1= R[τ ]

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σ, so that B0 = R[σ]. In analogy with the standard notations for complex

numbers, we say that a generic a + bκs∈ Bs (with a, b ∈ R) has real part a

and imaginary part b; furthermore, we can define:

• the conjugation a + bκs:= a − bκs;

• the square-norm |a + bκs|2 := (a + bκs)(a + bκs) = a2 − b2κ2s, which

is always real (i.e. it has imaginary part 0), but not always positive if s ≤ 0; notice that a + bκsis invertible if and only if its square-norm is

not 0.

In Chapter 1, the spaces Xs (in dimension 3) had been defined by −x21+

x2₂+ x2₃+ sgn(s)s2x2₄< 0 in P3R: if s > 0, we have already given a definition for the ideal boundary ∂Xs, which turns out to be the topological boundary

of Xs as a subspace of P3R, so it looks reasonable to do the same for every value of s. Figure 3.1 shows how these boundaries look in the conformal model of P3R: we can observe that if s < 0 the boundary is homeomorphic to a torus, while if s = 0 (so that X0 = HP3) there is a ”critical point”,

namely [0, 0, 0, 1], which can be thought as the common point at infinity of all degenerate lines, and it is convenient to exclude this point in order to preserve some properties that hold for s 6= 0, like:

• ∂Xs is a manifold;

• the isometries of Xs act transitively on ∂Xs.

Definition 9. The ideal boundary ∂Xs is the topological boundary of Xs in

P3R, with [0, 0, 0, 1] removed if s = 0.

We will actually give these boundaries an additional structure, which is preserved by the isometries like in the hyperbolic case. The inequality that defines Xs can be rewritten as:

−(x1+ x2)(x1− x2) + (x3+ x4κs)(x3− x4κs) < 0 ⇐⇒ ⇐⇒ det x1+ x2 x3− x4κs x3+ x4κs x1− x2 > 0 . (3.1)

This suggests to identify R4 with the space of hermitian 2 × 2 matrices (with respect to the conjugation of Bs), so that

P3R = (Herm(2, Bs) \ {0})/R∗

and then Xs consists of classes of matrices with positive determinant. It is

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Figure 3.1: The ideal boundaries of the spaces X1= H3, X0 = HP3 and X−1 = AdS3,

from left to right, in the conformal model of P3R; notice that if s = 1 this is not the Poincar´e disk model: the lines of H3 are represented the lines of P3R (i.e. the circular arcs between two antipodal points), which are not orthogonal to ∂H3_{. We can observe}

that in HP3 the so called critical point is the only point at infinity of all degenerate lines, while the nondegenerate ones have two distinct points at infinity in ∂HP3. Similarly, a space-like, light-like or time-like line in AdS3_{has 2, 1 or 0 points at infinity (respectively).}

• the subspace of real symmetric matrices intersects Xs at the

distin-guished hyperbolic plane;

• ∂Xs corresponds to the subspace of P3R given by det X = 0 (or equiv-alently by rk(X) = 1), but if s = 0 the point

0 −σ σ 0

∈ P(Herm(2, B0))

is removed.

We can also observe that any hermitian matrix X decomposes uniquely as

X = Z + tJ κs, where Z ∈ Sym(2, R), t ∈ R and J =

0 −1 1 0

(3.2)

whose coordinates in P3R are [Z11+ Z22, Z11− Z22, 2Z12, 2t]: if s = 0,

we have Z 6= 0 ∀X | [X] ∈ HP3 and det Z = det X > 0, and the projection [Z + tJ σ] → [Z] is well defined: this corresponds to the projection on the distinguished hyperbolic plane, whose fibers are the degenerate lines: in particular, [Z + tJ σ] and [Z0+ t0J σ] lie on the same degenerate line if and only if Z = Z0, and recalling (2.3), their signed distance along such line is

2t − 2t0 (Z11+ Z22) r 1 −Z11−Z22 Z11+Z22 2 − 4 Z12 Z11+Z22 2 = (t − t 0_{) · sgn(tr Z)} √ det Z (3.3)

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Now we consider the ideal boundaries: after fixing s ∈ R, we immediatly see that any point in ∂Xsis represented, in this new model, by an equivalence

class of hermitian matrices whose determinant is 0.

It is well known that a rank 1 hermitian matrix X with complex co-efficients uniquely determines [v] ∈ P1C such that X ∼ v · v∗, where v∗ denotes the conjugate transpose of the column vector v and ∼ means ”equal up to a nonzero real scalar”: this gives the already mentioned correspon-dence between ∂H3 and P1C, and the isomorphism Bs ∼= C gives the same

correspondence for all s > 0, since the following diagram commutes:

P1C ∂H3

P1Bs ∂Xs ψ1

ψs

(3.4)

(the vertical maps are naturally induced by the isomorphism Bs ∼= C,

while the horizontal ones map a generic [v] to [v · v∗]).

An analogous result holds for s ≤ 0, but we need to be careful about the definition of P1Bs.

Proposition 4. A vector v ∈ B2_s satisfies v · v∗ = 0 if and only if there exists λ ∈ Bs\ 0 such that λv = 0.

Proof. We only consider the least obvious case s < 0, wlog s = −1. Re-call that the set of non invertible elements of B−1 is the set of elements

whose square-norm is zero, i.e. the union of the two lines spanned (as linear subspaces of R2) by 1 + τ and 1 − τ ; now, if

v = z w and v · v∗ =|z| 2 _zw zw |w|2 ,

then one of the following holds:

1. v · v∗ has a nonzero coefficient in the main diagonal: this implies that at least one component of v is invertible, so that λv = 0 implies λ = 0;

2. v · v∗ has two zeros in the main diagonal, and consequently z and w are both non invertible: we have

v · v∗ = 0 ⇔ zw = 0 ⇔ z and w lie on the same line.

If z and w lie on the same line, any λ 6= 0 in the other one satisfies λv = 0; otherwise, an easy calculation shows that there are no solutions for λ other than 0.

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Now, if we define

P1Bs:= {v ∈ Bs2 | λv = 0 ⇒ λ = 0}/B∗s

the previous proposition implies that the map ψs(see (3.4) above) is well

defined.

Proposition 5. The map ψs is a bijection.

Proof. We are going to construct the inverse function. If X is an arbitrary rank 1 hermitian matrix which represents a point of ∂Xs, there are two

possibilities:

• X has a nonzero element in the main diagonal, wlog 1: then the corresponding component of v also has to be invertible, wlog 1, and the other component is uniquely determined;

• X has two zeros in the main diagonal: this can only happen when s < 0, wlog s = −1, and X can be rescaled to

0 1 − τ 1 + τ 0 or 0 1 + τ 1 − τ 0 .

As well, both components of any vector v such that v · v∗ ∼ X are easily seen to be non invertible and non null, and so v can be rescaled to

1 ∓ τ k(1 ± τ )

, with k ∈ R∗: the choice of the signs is forced by X, while the choice of k is irrelevant, since

1 ∓ τ k(1 ± τ ) ∼ 1 ∓ τ 2 + 1 ± τ 2k · 1 ∓ τ k(1 ± τ ) =1 ∓ τ 1 ± τ .

Remark. One could also check that ψs is a homeomorphism.

It is worth observing that the complement of Bs in P1Bs under the

iden-tification z ↔z 1 is: • the line 1 kσ , k ∈ R if s = 0; • the set 1 + τ + k(1 − τ ) 1 − τ , k ∈ R ∪1 0 ∪1 − τ + k(1 + τ ) 1 + τ , k ∈ R

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Our next goal is to give an alternative description of the isometry groups based on their action on the ideal boundaries. Of course, the group PGL(2, Bs)

acts on P1Bsby [A] ∗ [v] = [Av] (it is well defined since λv = 0 ⇔ λ · Av = 0),

and the correspondence ψsgives an analogous action on Xs by conjugation,

namely

[A] ∗ [v] = [Av · (Av)∗] = [A · vv∗· A].

This is actually the restriction of an action on the whole P3R, since it makes sense to send an arbitrary [X] ∈ P3R to [AXA∗] (the result is still a hermi-tian matrix); furthermore, any A ∈ GL(2, Bs) acts linearly on Herm(2, Bs),

so that [A] ∈ PGL+(2, Bs) acts on P3R as a projective transformation. In order to describe the orientation preserving isometries of Xs, we need

to restrict to the subgroup PGL+(2, Bs) given by the matrices whose

de-terminant has positive square-norm: this is a good definition, since if we multiply all coefficients of a 2 × 2 matrix by λ ∈ B_s∗, the determinant is mul-tiplied by |λ2|2_{, which is always a positive real number. Such a restriction is}

only needed for s < 0: we will see how this excludes the projective transfor-mations that preserve ηs up to a negative constant and, as a consequence,

do not preserve the set Xs.

Also, it is not difficult to prove that:

• PGL+_{(2, B}

s) is a Lie group of dimension 6;

• PGL+(2, Bs) has one connected component if s > 0 and two if s ≤ 0

(in the second case, they are detected by the determinant being or not being a square).

Proposition 6. The map PGL(2, Bs) → PGL(4, R) is injective, and its

image consists exactly of the orientation preserving isometries of Xs.

Remark. Since the conjugation [X] 7→ [X] is a reflection, this implies that every orientation reversing isometry can be uniquely expressed as [X] 7→ [AXA∗].

Proof. If [A] acts trivially on P3R, it also acts trivially on ∂Xs, so that

[Av] = [v] ∀[v] ∈ P1_B

s: by checking this equality for [v] = [e1], [v] = [e2]

and [v] = [e1 + e2] we easily find A ∼ I, which means that the map is

injective. Now, ηs is a nondefinite quadratic form, so its conformal class is

uniquely determined by its zero locus, and since PGL+(2, Bs) preserves this

locus, it acts conformally with respect to ηs: more precisely, the rescaling

factor associated to A ∈ GL+(2, Bs) is

ηs(AA∗)

ηs(I)

= det(AA∗) = | det A|2 > 0, and if s 6= 0 this implies that

i) the action of [A] is an isometry of Xs, and it preserves the orientation.

Geometric structures on manifolds: transitions from hyperbolic to anti-de Sitter geometry

Corso di Laurea Magistrale in

Matematica