Cone-manifolds and hyperbolic surgeries

(1)

Universit`

a degli Studi di Pisa

Corso di Dottorato in Matematica

XXIX Ciclo

Cone-manifolds and hyperbolic

surgeries

Tesi di Dottorato

Candidato

Stefano Riolo

Relatore

Prof. Bruno Martelli

Universit`

a di Pisa

(2)

(3)

Abstract

We first introduce hyperbolic, Euclidean, and spherical cone-manifolds of arbitrary di-mension. After that, we carefully describe a deforming hyperbolic 4-polytope of finite volume. Finally, we glue copies of that polytope to get some interesting deformations of hyperbolic cone-manifolds of dimension four. In particular, we discover some four-dimensional instances of Thurston’s hyperbolic Dehn surgery and degeneration. We also find the smallest known hyperbolic 4-manifold that is not commensurable with the integral lattice of O(4,1).

(4)

(5)

Acknowledgements

I warmly thank my thesis advisor Bruno Martelli for continuous encouragement. His enthusiasm and patience have been greatly appreciated. I also thank him for having drawn 14 of the 24 figures of this thesis. In particular, Figure 3-15 is taken from his beautiful book [Mar16].

I also express gratitude to all the people who have shared mathematical insight and knowledge with me. Among them: Agnese Barbensi, Federico Buonerba, Alessio Carrega, Daniele Celoria, Carlo Collari, Ga¨el Cousin, Federico Franceschini, Roberto Frigerio, Sasha Kolpakov, Marco Moraschini, Leone Slavich – and, specially, the ex-traordinary Michael McQuillan.

Finally, I choose not to thank the people I love, since, I think, love goes much beyond this stuff.

(6)

(7)

Introduction

The main purpose of this thesis is to introduce hyperbolic cone-manifolds and to describe some applications in low-dimensional topology. More specifically, we provide some new results about complete, finite-volume, hyperbolic 4-manifolds.

The main result consists in exhibiting a concrete example of a phenomenon which may be interpreted as a four-dimensional analogue of Thurston’s hyperbolic Dehn filling.

In this introduction, we say that a manifold M is hyperbolic if it admits a complete, finite-volume, hyperbolic structure (equivalently, a Riemannian metric of constant negative sectional curvature for which M is complete and of finite volume).

Hyperbolic four-manifolds First of all, we remark that the world of hyperbolic manifolds in dimension greater than three appears still rather mysterious. In fact, so little is known that there is yet need of classes of examples.

Moreover, in absence of analogues of the Geometrisation for three-manifolds, the role of hyperbolic geometry in higher dimensional topology is not clear.

The first natural step is to rise one dimension up. We refer the reader to [Mar15] for a recent survey about hyperbolic 4-manifolds, focusing mainly on concrete exam-ples.

Thurston’s ideas Before going into detail about our results, we now give some motivations about hyperbolic cone-manifolds and their deformations.

Before stating his celebrated Geometrisation Conjecture [Thu82] (today a theorem of Perelman [Per02, Per03b, Per03a]), William Thurston revolutionised the field of three-dimensional topology being aware that

• “a lot of” 3-manifolds are hyperbolic;

• the geometric invariants of a hyperbolic manifold are actually topological in-variants of the manifold.

(10)

The last assertion follows by Mostow-Prasad rigidity [Mos73, Pra73], stating that a complete finite-volume hyperbolic structure is essentially unique. Thus, hyperbolic geometry became a very important tool for the study of 3-manifolds.

The most striking results in this direction are doubtless Thurston’s celebrated the-orems known as Hyperbolisation (for Haken manifolds) and Hyperbolic Dehn Filling. Roughly speaking, the former implies that the interior M of every compact 3-manifold N with non-empty boundary satisfying some obvious necessary conditions is hyperbolic. If instead ∂N = ∅ (so that M = N is closed), the theorem guarantees the hyperbolicity of M with the additional hypothesis that M is “Haken”. The last condition is removed with Perelman’s proof of the Geometrisation conjecture; however Haken manifolds form a wide class of 3-manifolds.

Hyperbolic Dehn filling On the other hand, every non-compact orientable hyper-bolic 3-manifold M is diffeomorphic to the interior of a compact N whose boundary consists of tori. Each end of M is an unbounded cusp with Euclidean torus section, exponentially shrinking towards infinity.

A natural way to get a closed manifold Mfill from M is to glue solid tori to ∂N . The possible fillings Mfill are countably-many up to diffeomorphism. Thurston’s hyperbolic Dehn filling theorem states that almost all the possible Mfillare hyperbolic. Thus, countably-many closed hyperbolic manifold are obtained from a single cusped one.

This theorem is proved through a deformation theory of hyperbolic structures on 3-manifolds. The complete structure on M is deformed (necessarily through incomplete structures by Mostow-Prasad rigidity) so that the metric completion is a metric space homeomorphic to Mfill_.

Such deformation can be seen as a path of singular (complete, finite-volume) hyperbolic structures on the filling Mfill, converging to the desired non-singular one. Topologically, the singular locus is the link L ⊂ Mfill _{such that M ≈ M}fill _{− L.}

Metrically, it is a collection of simple closed geodesics. The singularities are of conical type: by measuring the angle around them one gets some cone angle θ different from 2π.

While approaching the original complete structure of M , the cone angles and the length of the singular locus tend to zero: at the limit, L goes to infinity and is drilled away, giving rise to the cusps of M . In the opposite direction, the cusps of M are filled by some small circles with cone angles increasing from 0 to 2π, so to that eventually the hyperbolic structure on Mfill _{is non-singular.}

(11)

carries a Riemannian metric with cone-like singularities.

Cone-manifolds Conical singularities are widely used in differential geometry, com-plex analysis, etcetera. Usually, the idea is to deform singular metrics in order to get results about the non-singular ones. In this thesis, we are interested in some very special cases.

In the literature, (G, X)-cone-manifolds where first introduced by Thurston in the beautiful paper “Shapes of polyhedra” [Thu98], when he discovered that some moduli spaces are naturally complex hyperbolic cone-manifolds.

In any dimension, by gluing in pairs the facets of some hyperbolic polytopes, one gets a hyperbolic cone-manifold. Following [BLP05, McM], we will give an induc-tive definition independent on such a polyhedral decomposition: a hyperbolic (resp. Euclidean, spherical) cone-manifold is a space locally modeled on hyperbolic (resp. Euclidean, spherical) cones over spherical cone-manifolds of one dimension less.

Hyperbolic cone-manifolds are naturally stratified into totally geodesic (often in-complete) hyperbolic manifolds: some pieces of codimension at least two, which alto-gether form the singular locus, and the top-dimensional regular locus. In dimension n = 3, the singular locus is a graph, plus maybe some isolated points.

Each connected k-dimensional stratum has a well defined link, which is a spherical cone-manifold of dimension n − k − 1. We call regions the codimension-two strata. The link of a region is just a circle, whose length is the cone angle of the region.

Hyperbolic surgeries Roughly speaking, a cone deformation is a path t 7→ Xt of

hyperbolic cone-manifolds whose stratification is topologically constant. What vary are the hyperbolic structures on the strata and the isometry classes of the links (in particular, the cone angles).

As a simple example, consider a family Pt⊂ Hn of regular simplices with dihedral

angles varying smoothly. By doubling Pt, one gets a cone deformation with Xt

home-omorphic to the sphere Sn_{. Each stratum carries a deforming hyperbolic structure}

and the cone angles vary smoothly.

We vaguely talk about hyperbolic surgery when, as t → ∞, a cone deforma-tion converges to a cone-manifold X∞ such that the path describes some topological

surgery to pass from Xt to X∞.

Hyperbolic Dehn filling is such an example, with X∞ = M . Another example,

valid in any dimension, is obtained from the family Ptof regular simplices previously

mentioned. Letting the dihedral angles decrease until the simplex P∞ becomes ideal,

(12)

drilled away. Correspondingly, as t → ∞, the path t 7→ Xt describes a hyperbolic

surgery: the 0-strata of Xt are drilled away, giving rise to the cusps of the

cone-manifold X∞. In the opposite direction, the cusps of X∞ (with section a Euclidean

cone-manifold homeomorphic to Sn−1_{) are “hyperbolically filled” by points.}

Hyperbolic cone 3-manifolds Coming back to hyperbolic cone 3-manifolds, their importance is not limited to Thurston’s hyperbolic Dehn filling theorem.

The celebrated Orbifold theorem – based on Thurton’s ideas and fully proved inde-pendently by Cooper, Hodgson and Kerckhoff [CHK00], and Boileau, Leeb and Porti [BLP05] – was a considerable advance towards the Geometrisation of 3-manifolds. That theorem implies the Geometrisation for 3-orbifolds with non-empty singular locus – so, via coverings, for a big class of manifolds – and its proof relies on cone-manifold deformations.

Further important results through the theory of hyperbolic cone deformations were obtained by Brock and Bromberg [Bro07, BB04], towards a proof of the famous Bers-Sullivan-Thurston’s Density conjecture about Kleinian groups.

A fundamental contribution to the theory of hyperbolic cone 3-manifolds was done by Hodgson and Kerckhoff. In [HK98], they developed a theory of cone deformations through a study of harmonic L2 forms, obtaining an important local rigidity/flexibility theorem.

Higher dimension Higher dimensional hyperbolic cone deformations are still un-explored. First of all, with the previous example of doubling a deforming regular simplex, we already noted that cone deformations exist.

But whether or not they may have a role for the study of higher-dimensional hyperbolic manifolds is unknown. A first negative signal which jumps up in mind may be the following.

Thinking of cone-manifolds as a tool to understand hyperbolic manifolds, one would like to consider cone-manifolds homeomorphic to n-manifolds and with not too complicated singular locus. The simplest desired case would be when the singular locus is a codimension-two submanifold. But probably such example would be useless if n ≥ 5: the singular locus would be a hyperbolic manifold of dimension ≥ 3, hence rigid by Mostow-Prasad (however, for the flexibility of the regular locus, which is incomplete, there are not similar immediate obstructions).

Even if n = 4, with in mind wide moduli spaces of closed hyperbolic surfaces, we do not know if such a hyperbolic cone deformation may exist. In any dimension n ≥ 4, to obtain existence and uniqueness of infinitesimal cone deformations of a closed

(13)

hyperbolic cone-manifold with singular locus a closed codimension-two submanifold, Montcouquiol [Mon05, Mon06] works in the enlarged context of Einstein singular metrics. In this context, one may have the feeling that this Einstein metric should be hyperbolic only in some very lucky cases.

Obstructions for hyperbolic Dehn filling Continuing with some disheartening facts about higher-dimensional hyperbolic cone deformations, we note two remarkable obstructions to extend hyperbolic Dehn filling in higher dimension.

The first is Garlan-Raghunathan rigidity [GR70], which forbids to deform any hy-perbolic lattice in dimension greater than three. In other words, hyhy-perbolic Dehn filling is a three-dimensional phenomenon, because in higher dimension complete finite-volume hyperbolic structures cannot be deformed, not even through incomplete structures (recall that we are dealing only with the finite-volume case, otherwise there is much more flexibility).

As a second obstruction, let us consider a hyperbolic n-manifold M and the com-plement M = M − L of a closed codimension-two totally geodesic submanifold L. If n = 3, then M is hyperbolic (see for instance [Koj98, Theorem 1.2.1] for a proof without using Geometrisation). For n = 4, the manifold M cannot be hyperbolic, since no circle bundle over the hyperbolic surface L can carry a Euclidean metric. In general, for n > 3, by a result of Belegradek [Bel12], M does not even carry any complete negatively-curved pinched Riemannian metric.

The major contribution of this thesis is to show that, despite the previous negative premises, some interesting cone deformations exist in dimension four.

The major result Our main result consists in exhibiting a concrete example of hyperbolic filling in dimension four:

Main Theorem. There exists a compact smooth 4-manifold N with ∂N diffeo-morphic to a 3-torus, which contains a smooth 2-torus and a smooth Klein bottle T, K ⊂ M = int(N ) that intersect transversely in two points (see Figure 0-1), such that the following holds.

There is an analytic path {Mt}t∈(0,1) of complete finite-volume hyperbolic

cone-manifold structures on M with singular locus the immersed cone-surface Σ = T ∪ K. The two cone-surfaces T and K have cone angles 0 < α < 2π and 0 < β < 2π respectively. We have

(14)

T K M

Figure 0-1: A picture of N and the immersed surface Σ = T ∪ K ⊂ M = int(N ).

θ θ θ θ θ θ θ θ θ α β

Figure 0-2: Local models for the cone 4-manifolds Mtand Wtof the main theorems. Each region is

labeled with a cone angle. In the third picture, the singularity is like the cone over the 1-skeleton of a tetrahedron (a sixth region is back). The last picture represents two discs intersecting transversely in a point. The cone-manifold Mthas singularities only of the first and last kind, while for Wtall

kinds occur, depending on the value of t.

When t varies from 0 to 1 the angle α goes from 0 to 2π and β goes from 2π to 0. As t → 0 and t → 1, the path converges to two complete, finite-volume hyperbolic 4-manifolds M0 = M − T and M1 = M − K.

(By analytic path, we mean that the holonomy of the regular locus varies analyt-ically.)

The local models for the singular points of Mt are particularly simple (the first

and the last of Figure 0-2).

A cusp with 3-torus section of the hyperbolic manifold M0 is filled in Mt by the

cone-torus T . The manifold M0 contains a totally geodesic twice-punctured Klein

bottle K×. The two cusps of the hyperbolic surface K× are filled by two cone points of increasing cone angle α (note that K has cone angle β).

The cone-surfaces T and K intersect orthogonally, precisely at these two points. A neighbourhood of each of these two points is represented in Figure 0-2-right. These two points are also the cone points of T . In T (which has cone angle α), they are of cone angle β.

The cone deformation goes on until the cone-surface K is drilled away, giving rise to a cusp of M1, with Euclidean section a circle bundle over K. Moreover, the

cone-torus T becomes in M1 a totally geodesic twice-punctured torus T× (its cone

points are drilled and become cusps), and M1 is non-singular because the “cone angle”

around T× is now α = 2π.

(15)

inter-Figure 0-3: The function Vol(Mt).

changed with that of M1 and T×.

A cusp with 3-torus section of deforming shape exists in Mt for all t ∈ [0, 1].

Some properties of Mt We note that the cone-manifold Mt is particularly nice,

since, topologically, its singular locus is a generically immersed surface. This ap-pears to be the first example of such a kind of hyperbolic cone 4-manifold (and cone deformation).

An interesting feature is the behaviour of the volume of Mtduring the deformation,

shown in Figure 0-3 and expressed in function of the cone angles as

Vol(Mt) = 8π2 3 2 − α + β 2π + αβ 4π2 .

In dimension three, the volume must decrease under hyperbolic Dehn filling (see the Schl¨afli formula in Section 2.2.6). In our case, instead, it increases from both sides. However, for the moment, we do not see any special reason for that: we do not exclude that in some other cases it may decrease.

Hyperbolic 4-manifolds The operations of filling and drilling two surfaces of zero Euler characteristic to go from M0 to M1 and viceversa do not modify their

Euler characteristic χ. So, by the Gauss-Bonnet formula, they have the same volume V = 4π₃2χ, as confirmed by the previous formula. Moreover, we have χ = 2, so their volume is small (the minimal possible being of course 4π₃2).

The hyperbolic manifolds M0 and M1 are arithmetic. The first is

(16)

Maclachlan [Mac11], we show that they are not commensurable.

In particular, at the moment, the manifold M1 appears to be the smallest known

hyperbolic 4-manifold which is not commensurable with that lattice.

Recall the need of concrete examples of hyperbolic 4-manifolds to work with. In fact, most of the concrete examples of cusped hyperbolic 4-manifolds that we know from the literature are commensurable with M0. In some sense, we got the “new” M1

from a “known” M0.

We do not exclude that ours is just a lucky case. In fact, nothing is known. However, it seems natural to ask whether (or in which cases, in general) it is possible to vary the cone angles along immersed cone-surfaces in hyperbolic cone 4-manifolds. An affirmative answer may bring to construct new hyperbolic 4-manifolds.

Why hyperbolic filling is not forbidden We note that the 4-dimensional hy-perbolic structures that deform in the theorem are supported on the manifold

M − (T ∪ K) ≈ M0− K× ≈ M1− T×.

In particular, such structures are never complete, so that Garland-Raghunathan rigid-ity is not violated and the previously mentioned obstruction of Belegradek is com-pletely avoided.

Coming back to the results of Montcouquiol about Einstein cone deformations, we note that some differences jump out between his situation and ours. First, our cone-manifolds are non-compact. Second, the singular locus is, topologically, a generically immersed (rather than embedded) surface and, metrically, a union of cone-surfaces (whose cone points are precisely the locus of self-intersection of the immersed surface). We note that, in dimension three, a generically immersed manifold of codimension two is just a submanifold.

Differences with n = 3 In addition to the behaviour of the volume, we note some differences with the three-dimensional hyperbolic Dehn filling.

In dimension three, one starts with M0 = M and deforms its hyperbolic

struc-ture, whose metric completion is a cone-manifold. In our case, to avoid Garland-Raghunathan rigidity, the filling of a cusp forces somehow the appearance of other singularities elsewhere.

Another important difference is that the situation of our example is much more rigid than the 3-dimensional one. First, by construction, the two cone angles are not independent. Moreover, in dimension three, all possible fillings of M0 admit a

(17)

one possible filling of M0.

The main ingredient: Kerckoff and Storm’s deformation As we said, the main theorem is proved by constructing an explicit example. The main ingredient of the proof is a family of infinite-volume deforming polytopes Ft ⊂ H4, discovered in

[KS10] by Kerckhoff and Storm.

Their aim was to deform a discrete group of isometries of H4_{, showing the existence}

of an infinite-volume orbifold version of hyperbolic Dehn filling in dimension four. The necessity of infinite covolume is due to the Garland-Ragunathan rigidity.

The existence of such a deformation shows also that their result about the rigid-ity of convex cocompact hyperbolic manifolds [KS12] is optimal in dimension four. Moreover, they found countably-many orbifolds in the family.

They start with the regular ideal 24-cell P1, and remove from its associated

reflec-tion group two generators corresponding to opposite facets of P1. Then, the associated

Coxeter, rigth-angled, polytope F1 is obtained by attaching two “Fuchsian ends” to

P1, one for each generator removed.

The orbifold F1 has some cusps of virtual rank three, some others of virtual rank

two. Kerckoff and Storm show that it is possible to deform the associated reflection group and describe the deformation geometrically in full detail. As soon as t 6= 1, the rank-two cusps are hyperbolically filled.

Sketch of proof We truncate the infinite-volume ends of the polytope Ftwith two

additional hyperplanes, to get a family Pt of finite-volume polytopes such that, for

t = 1, we have the original 24-cell P1.

What is important for our constructions is the abundance of right dihedral angles in Pt. Moreover, during the first part of the deformation, the 2-faces of Pt with

non-right dihedral angle intersect pairwise only at some vertices.

At the beginning of the deformation, a combinatorial change occurs to the 24-cell P1: some ideal vertices are substituted by new 2-faces. The first part of the

deformation ends with a new combinatorial change: some 2-faces go to infinity and are replaced by ideal vertices. At this point, we get a Coxeter polytope with dihedral angles π₂ and π₃. In particular, this first part of the deformation interpolates between two Coxeter polytopes.

By gluing some copies of Pt and manipulating a bit the construction, we get the

(18)

More hyperbolic surgeries By gluing copies of Pt in a different fashion, we get

another interesting four-dimensional cone deformation. Again, every combinatorial change of the polytope Ptdetermines a hyperbolic surgery to the corresponding

cone-manifold.

Theorem. There exists an analytic path {Wt}t∈(0,1] of complete finite-volume

ori-entable hyperbolic cone 4-manifolds with cone angles < 2π, with some times 1 > t1 >

t2 > ¯t > 0, such that W1 is a manifold and Wt1, W¯t are orbifolds.

When t ∈ (0, t2), the cone-manifold Wt is homeomorphic to C × S1, where C is a

hyperbolic 3-manifold.

At the critical times 1, t1, t2, 0 the topology of Wt changes as follows:

• at t = 1, by hyperbolic Dehn filling twelve 3-torus cusps by adding twelve 2-tori;

• at t = t1, by hyperbolic surgery substituting eight small S2 with eight small S1;

• at t = t2, by hyperbolic surgery substituting four small S3 with four S0;

• as t → 0, the cone angles tend to 2π and Wt degenerates to C.

The singular locus of Wt consists of:

• an immersed surface made of 12 cone-tori and 8 cone-spheres intersecting or-thogonally at their cone points, when t ∈ (t1, 1);

• 12 embedded twice-punctured tori, when t = t1;

• a 2-complex with generic singularities (whose topology changes as t = t2), when

t ∈ (0, t1).

The local models for the singular points of Wt are represented in Figure 0-2.

The cone-manifold Wt has some cusps with 3-torus section of deforming shape

that exist for all t ∈ (0, 1]. At the critical times 1, t1 and t2, there are some additional

cusps with section respectively homeomorphic to

S1× S1_{× S}1_, _S2_{× S}1_, _S3_.

Metrically, each Sk _{factor is the double of a Euclidean regular k-simplex.}

To pass from from W1 to W1−ε we have a hyperbolic Dehn filling similar to that of

the Main Theorem to pass from M0 to Mε: an S1 factor is shrunk and a core S1× S1

is inserted. To pass from Wti to Wti±ε, each additional cusp is hyperbolically filled by shrinking an Sk factor, and thus by inserting a core S3−k. To pass from Wti−ε

(19)

to Wti+ε, an S

k _{with trivial normal bundle is substituted with an S}3−k _{with trivial}

normal bundle.

We will see also another kind of hyperbolic filling in the cone-manifold context: a cusp with 3-torus section that is filled with a small circle. In other words, instead of shrinking an S1 _{factor of S}1 _{× S}1 _{× S}1_{, an S}1 _{× S}1 _{factor is collapsed.} _The

resulting space, not homeomorphic to a manifold, will be however a hyperbolic cone-manifold. From a topological point of view, this kind of filling was already considered by Fujiwara and Manning in [FM10, FM11].

Degeneration As t → 0, we see in the previous theorem a phenomenon very famil-iar in dimension three: a degeneration. Degenerations were first noted by Thurston [Thu79], for instance in the famous figure-eight knot complement example, and are fundamental in the proof of the Orbifold theorem.

This phenomenon may be interpreted as follows. One starts with the hyperbolic orbifold Wt¯ (with all cone angles π) supported on the product C × S1. Then, as

t → 0, the family {Wt}t∈(0,¯t ] of hyperbolic cone-manifolds collapses to the hyperbolic

manifold C and their holonomy tends algebraically to that of C.

In some sense, the fibred geometry H3× R of the manifold C × S1 _{is approximated}

by hyperbolic geometry H4 through singular structures. As the cone angles tend to 2π, so that the singularities tend to disappear, hyperbolic geometry forces the family to collapse to the hyperbolic base C of the “Seifert” fibration C × S1.

This phenomenon was already noted in the orbifold context by Kerckhoff and Storm in [KS10]. Indeed, for t ≤ t2 we have Pt = Ft and the family degenerates as

t → 0 to an ideal right-angled cuboctahedron P0, which can be seen as a hyperbolic

3-orbifold. In fact, we have P_t¯= W¯_t

(Z/2Z)3 and P0 = C

(Z/2Z)2.

More examples The cone-manifolds Wtand Mtare not special in any sense: there

are many ways one can modify their construction to produce different deforming cone-manifolds from Pt with different types of behaviour. By taking finite covers one can

also get infinitely many examples of various kinds.

Outline In Chapter 1, we fix some notations and briefly introduce some preliminary notions.

In Chapter 2, we introduce hyperbolic, Euclidean and spherical cone-manifolds and prove some basic facts about these spaces. The main sources for this chapter are [CHK00, BLP05, McM]. However, the exposition and most of the proofs are original (when not quoted).

(20)

Finally, in Chapter 3, we prove the previously mentioned results about hyperbolic surgeries in dimension four. These results are the outcome of a work, [MR16], joint with Bruno Martelli.

(21)

Chapter 1 Preliminary notions

In this chapter, we fix some notations and introduce some preliminary notions. Some of them are actual prerequisites for the thesis, some others are inserted just to con-textualise somehow the subject of cone-manifolds.

There is no claim of completeness about these (well-known) topics; we refer the reader to the bibliography. However, the more the language will be informal, the less the content will be needed in the following.

1.1 Convex polytopes

Let Xn _{= X}n κ be Xn =         

the hyperbolic space Hn _{if κ = −1,}

the Euclidean space En _{if κ = 0,}

the unit sphere Sn if κ = 1.

In this section, we briefly introduce convex finite polytopes in Xn, with particular emphasis on finite-volume, acute-angled ones. We refer the reader mainly to [Vin85, AVS93].

1.1.1 Generalities

For simplicity of exposition, we concentrate on the hyperbolic case. The Euclidean and spherical cases are treated similarly with the obvious modifications. Some care is needed in the spherical case, but we will omit the details.

(22)

Half-spaces in Hn _{We represent the hyperbolic space H}n_{as the upper sheet of the}

hyperboloid hv, vi = −1 in Rn+1 _{with respect to the Lorentzian product}

hv, wi = −v0w0+ v1w1+ . . . + vnwn.

Every space-like vector v determines a half-space in Hn_{, that consists of all w ∈}

Hn with hv, wi ≤ 0. We are interested in the case where two space-like vectors v and v0 determine two half-spaces whose intersection is non-empty and is a proper subset of both half-spaces. There are three possible configurations to consider, easily determined by the number

α = −hv, v

0_i

phv, vihv0_{, v}0_i (1.1)

as follows:

• if −1 < α < 1, the boundary hyperplanes of the two half-spaces intersect with a dihedral angle θ such that cos θ = α;

• if α = 1, the boundary hyperplanes are asymptotically parallel;

• if α > 1, the boundary hyperplanes are ultra-parallel, and their distance d is such that cosh d = α.

Convex polytopes We define as usual a (finite convex ) polytope to be the in-tersection P of finitely many half-spaces in Xn_{, with the additional hypothesis that}

int(P ) 6= ∅. The boundary ∂P is naturally stratified into a finite number of k-dimensional faces, with k = 0, . . . , n − 1. For k = 0, 1 or n − 1, a k-face is called respectively vertex, edge or facet.

In the hyperbolic case, if the closure P of P in the compactification Hn _intersects

∂∞Hn in finitely many points (possibly none), the volume of P is finite; otherwise it is infinite. These points in ∂∞Hn are called ideal vertices.

Let us come back to the general case Xn _{= H}n_{, E}n _{or S}n_{. In the spherical case,}

every polytope is compact. A spherical polytope is said to be degenerate if it contains two antipodes. In the Euclidean and hyperbolic cases, there are of course unbounded polytopes. In any case, a non-degenerate finite-volume polytope is the convex envelop of its vertices (including ideal vertices in the hyperbolic case).

Faces _{A k-dimensional face F of P lies in a unique subspace X}k _{⊂ X}n and is a k-polytope in Xk. The interior of the face F is the interior of F in Xk. Given a point p ∈ P , there exists a unique face Fp of P such that p belongs to the interior of F .

(23)

Geometric links _{Given a polytope P ⊂ X}n_{, the link of a point p ∈ P is the set}

SpP of unit vectors at p pointing into P . More precisely, denoting by SpXn the unit tangent sphere to Xn _{at p and by exp}

p: TpXn → Xn the exponential map, SpP = SpXn∩ exp−1p (P ) ⊂ TpXn.

Being a subset of the unit tangent sphere, SpP is identified with a subspace of

Sn−1 (the distance in SpXn is the angle between unit vectors). Up to isometry, SpP

depends only on the face Fp whose interior contains p.

The point p lies in the interior of P if and only if SpP ' Sn−1 (in this thesis, the

symbol ' means “isometric”). Otherwise, SpP is an (n − 1)-dimensional spherical

convex polytope. Such polytope is non-degenerate if and only if p is a vertex of P . The link of the face F = Fp is the subspace LF ⊂ SpP of unit vectors orthogonal

to F . Specifically, we take the normal space NpF ⊂ TpXn to F at p and define

LF = SpP ∩ NpF ⊂ TpXn.

If the face F has dimension k ∈ {0, . . . , n−1}, then its link LF is a non-degenerate

spherical polytope of dimension n − k − 1.

If Xn = Hn and p ∈ ∂∞Hn is an ideal vertex of P , we still can define its link Lp as

the intersection of P with a “sufficiently small” horosphere centred at p (that is, an horosphere intersecting all faces F of P with p ∈ ∂∞F ). In that case, Lp is a compact

(n − 1)-dimensional Euclidean polytope, well-defined up to homothety.

1.1.2 Acute-angled polytopes

The theory of acute-angled hyperbolic polytopes is beautifully introduced in a paper of Vinberg [Vin85] and we briefly recall some of the facts described in that paper.

Gram matrix _{Let P ⊂ H}n _{be a polytope, defined as the intersection of the}

half-spaces dual to some unit space-like vectors v1, . . . , vm. We calculate αij from vi, vj

using (1.1) for any i, j. The m × m matrix −αij is the Gram matrix of P , see [Vin85].

We say that P is acute-angled if αij ≥ 0 for all i 6= j. Acute-angled polytopes have

many nice properties. In this section, we will always suppose that P is acute-angled.

Remark 1.1.1. By a theorem of Andreev [And70] a polytope P in Xnis acute-angled if and only if all its faces have dihedral angles ≤ π₂, and this explains the terminology.

(24)

Generalised Coxeter diagrams The Gram matrix of an acute-angled polytope P is nicely encoded via the generalised Coxeter diagram D of P , which is constructed as follows: every vertex of D represents a vector vi, and every edge between two distinct

vertices vi and vj has a label that depends on αij ≥ 0 as follows:

• if αij > 1 the edge is dashed (and sometimes labeled with the number d > 0

such that cosh d = αij, but we will not do that);

• if αij = 1 the edge is thickened;

• if 0 ≤ αij < 1 the edge is labeled with the angle π₂ ≥ θ > 0 such that cos θ = αij.

To simplify the picture, the edges labeled with an angle π₂ are not drawn, and in those with π₃ the label is omitted.

Faces The following facts are proved in [Vin85, Section 3]. Every acute-angled polytope P in Xn _{is simple, that is each face F of P of codimension k is contained in}

exactly k facets.

Every non-degenerate acute-angled spherical polytope is a simplex, while every bounded Euclidean acute-angled polytope is a product of simplices. In particular, the link of each face of an acute-angled polytope is a spherical simplex, while the link of each ideal vertex is a product of Euclidean simplices.

All the faces of P may be easily determined from D as follows:

• the vertices vi represent the facets of P ;

• the pairs of vertices connected by an edge labeled with some angle θ represent the codimension-two faces of P ; the angle θ is the dihedral angle of that face;

• more generally, the faces F of codimension k correspond to the k-uples of ver-tices of D whose subdiagram represents a (k − 1)-dimensional spherical simplex LF, which is the link of the face F .

In particular, the set of vectors v1, . . . , vm defining P is minimal (no proper subset

defines P ), and k facets in P intersect if and only if the hyperplanes containing them do. These nice facts are not true in general for non acute-angled polytopes.

Diagrams of the faces Every face F of an angled polytope P is also acute-angled, and one can deduce a Coxeter diagram DF for F from that, D, of P . We

explain how this works in the easier case when F is a facet; the procedure can then be applied iteratively.

(25)

The diagram DF is formed by all the vertices of D that represent facets that

are incident to F ; that is, DF is constructed from D by removing the vertex vi

corresponding to F and all the vertices vj that are connected to vi by either a dashed

or a thickened edge.

The resulting diagram DF is not yet a generalised Coxeter diagram for F , because

the value of α from formula (1.1) needs to be recomputed for every edge. To do so, in the hyperbolic case we must substitute each space-like vector vj with its projection

P (vj) in the time-like hyperplane vi⊥ containing F , using the formula

P (vj) = vj−

hvj, vii

hvi, vii

vi.

The new α ≥ 0 is computed using the projections P (vj) and is equal or bigger than

the original one (in particular S is still acute-angled).

Ideal vertices The ideal vertices v of P are also detected in a similar fashion: they correspond to the subdiagrams of D that represent some compact (n − 1)-dimensional Euclidean acute-angled polytope Lv, which is in fact the link of v. The polyhedron

Lv must be a product of simplexes, so the subdiagram is a disjoint union of diagrams

representing Euclidean simplexes.

There is a combinatorial criterion that one can use to check from D whether P is compact and/or has finite volume, see [Vin85, Proposition 4.2]. We suppose that P contains at least one (finite or ideal) vertex.

Theorem 1.1.2. The polytope P is compact (resp. has finite volume) if and only if each of its edges joins exactly two finite (resp. finite or ideal) vertices.

This condition is designed to exclude the presence of hyper-ideal vertices, see [Vin85]. In this thesis, we will only deal with finite-volume polytopes.

Coxeter polytopes If all the dihedral angles of P are of type π_n for some n ≥ 2, then P is a Coxeter polytope. In this case the group Γ < Isom(Xn) generated by the reflections along its facets is discrete and has P as a fundamental domain, so that P = Xn_/

Γ may be interpreted as an orbifold. We refer to Section 1.3.2 for a brief

introduction to geometric orbifolds.

1.1.3 The regular 24-cell

We now introduce a quite special hyperbolic 4-polytope: the ideal regular 24-cell. This section may be useful for Chapter 3.

(26)

In what follows, we call a k-face of a polytope a vertex if k = 0, an edge if k = 1, a face if k = 2, and wall if k = 3.

Combinatorics The 24-cell is a curious regular convex polytope – see for instance [Cox73, 8.2] for an introduction.

In this paragraph, we briefly describe some of its combinatorial properties (that is, roughly speaking, that properties which do not depend on the fact that we are talking about a hyperbolic, spherical or Euclidean 24-cell).

The 24-cell is the only self-dual convex regular polytope which is not a polygon nor a simplex. It has 24 vertices, 96 edges, 96 triangular faces and 24 octahedral walls. The vertex figure is a cube.

Like the 3-dimensional cube, the 24-cell has a standard three-colouring: an as-signment of one among three colours to each of the walls, such that any two walls sharing a face have different colours. For each colour, there are eight walls with that colour. The 24 walls are thus divided into three octets (of course, independently on the chosen colours!).

The octet to which each wall W belongs consists in the six walls sharing a vertex with W and the opposite of W , defined as follows. As every regular polytope, the 24-cell has a center : the only point fixed by all the symmetries. The opposite of W is obtained from W by the inversion through the center of P .

The symmetry group of the 24-cell has order 24 · 8 · 3 · 2 = 1152.

Ideal hyperbolic realisation The regular hyperbolic ideal 24-cell can be defined up to isometries of H4 _{as follows.}

Let {e0, e1, e2, e3, e4} be the standard basis of R1,4. We define P to be the convex

hull in H4 _{of the 24 points of ∂}

∞H4 defined by the light-like vectors n√

2e0± ei± ej

o

1≤i6=j≤4.

Following [KS10, Section 3], it can be shown that P is the intersection of the 24 half-spaces defined by the space-like vectors

n√ 2e0± e1± e2± e3± e4 o ∪ne0± √ 2ei o 1≤i≤4 .

By applying Formula (1.1) to this list of vectors, one concludes that if two hy-perplanes (boundary of the corresponding half-spaces) intersect in H4, they do it orthogonally. Thus, P is a right-angled polytope and, in particular, a Coxeter poly-tope.

(27)

The opposite of each wall is obtained by changing the signs of the spatial coordi-nates of the corresponding space-like vector.

The center of P is (1, 0, 0, 0, 0).

Colouring the walls Recall the standard 3-colouring of P : its walls are divided into three octets. Kerckhoff and Storm call each of these octets respectively letter walls, positive walls and negative walls as follows (see also Table 3.1 of Section 3.1.1 with t = 1):

• the letter walls are associated to the vectors e0±

√

2ei with 1 ≤ i ≤ 4;

• the positive (resp. negative) walls are associated to the vectors √2e0± e1± e2±

e3± e4 with an even (resp. odd ) number of negative coordinates;

It is not difficult to prove, by applying Formula (1.1), that this corresponds to a standard 3-colouring of the walls of P . Thus, a wall is tangent at infinity to six walls of its octet, the seventh being its opposite.

Symmetries The symmetry group of P is computed explicitly in [KS10, Section 4]

Sym(P ) ∼= O 4; Z1₂ < O(4),

where the orthogonal group O(4) acts naturally on the space-like coordinates of R1,4_.

The symmetry group acts on the set of octets, inducing a short exact sequence

1 → SO (4; Z) → Sym(P ) → S3 → 1,

where S3 is the symmetric group of order 6.

The ideal right-angled cuboctahedron _{Consider now the hyperplane H}3 _{⊂ H}4

defined by the space-like vector e4. The closure H3∪ ∂∞H3is disjoint from the closure of the opposite letter walls defined by e0 ±

√

2e4 (following [KS10], these walls are

called G and H in Table 3.1).

All the other 12 = 24 − 6 · 2 ideal vertices of P (which are not ideal vertices of G or H) lie in ∂∞H3. The intersection P ∩ H3 is thus the convex envelope of these 12 ideal points and results an ideal right-angled cuboctahedron.

The cuboctahedron is a uniform polyhedron (see [Cox73]) with 12 vertices, 24 edges, 8 (regular) triangular faces and 6 (regular) quadrilateral faces.

(28)

Figure 1-1: A cuboctahedron.

1.2 Cones and joins

In this section, we recall some constructions of metric spaces of interest for us. We follow almost entirely [BH11]: proofs of unjustified assertions made in this section can be found there. Henceforth, the symbol ' stands for “isometric”.

1.2.1 Geodesics and length spaces

Let (X, d) be a metric space.

Geodesics A geodesic segment is the image of an isometric embedding of a closed interval in X. A geodesic is a curve in X which is locally an isometric embedding.

The space (X, d) is said to be geodesic (resp. uniquely geodesic) if every two points of X are joined by a geodesic segment (resp. a unique geodesic segment).

The length of a curve c : [a, b] → X is defined as

l(c) = sup a=t0≤t1≤...≤tn=b n−1 X i=0 d(c(ti), c(ti+1)),

where the supremum is taken over all partitions of the interval [a, b] with arbitrary n. A curve is called rectifiable if its length is finite.

Length spaces To any metric d on X we can associate another metric dintr on

X, called the intrinsic metric. For any two points x, y ∈ X, their intrinsic distance dintr(x, y) is defined as the the infimum of the lengths (measured with the metric d)

(29)

The space (X, d) is called a length space if d = dintr. The completion of a length

space is a length space.

Hopf-Rinow It is well-known that Riemannian manifolds with the distance func-tion induced by the Riemannian length of curves are length spaces. Moreover, a “Hopf-Rinow” theorem for length spaces holds: every complete, locally compact length space is a geodesic space. In particular, complete connected Riemannian man-ifolds are geodesic spaces.

1.2.2 Cones

We now define a metric construction of fundamental importance for us: the hyperbolic (resp. Euclidean or spherical ) cone C_HX (resp. C_EX or C_SX) over a metric space (X, dX). As sets, we put

• C_HX = C_EX = [0, +∞) × X .

∼ and

• C_SX =0,π₂ × X .

∼,

where the relation ∼ is generated by (0, x1) ∼ (0, x2) for all x1, x2 ∈ X. The

equiva-lence class [(r, x)] is denoted by rx ∈ C_XX. The special point 0x, denoted by 0 ∈ C_XX, is called the vertex of the cone.

The metric To define the distance d(rx, r0x0), we take a triangle OP P0 _{in X}2 _such

that d_X2(O, P ) = r, d

X2(O, P 0_{) = r}0

and the inner angle ∠(OP, OP0) = ¯dX(x, x0) :=

min{dX(x, x0), π} (such a triangle always exists and is unique up to isometry). Then,

we put (see Figure 1-2)

d(rx, r0x0) := d_X2(P, P0).

In formulas, recalling the cosine law in X2 _{[AVS93], we have}

cosh d(rx, r0x0) = cosh r cosh r0− sinh r sinh r0cos ¯dX(x, x0),

d(rx, r0x0)2 = r2+ r02− 2rr0cos ¯dX(x, x0),

cos d(rx, r0x0) = cos r cos r0+ sin r sin r0cos ¯dX(x, x0),

respectively for the hyperbolic, Euclidean and spherical cone (in the last case, we require d(rx, r0x0) ≤ π).

The metric d on the cone C_XX is defined so that, for every x, x0 ∈ X, the sets {rx : 0 ≤ r ≤ r0} are geodesic segments and the Aleksandrov angle (see [BH11]) at

(30)

¯ dX(x, x0) r

X

2 r0 d(rx, r0x0) O P P0

Figure 1-2: Defining the distance between points rx, r0x0 of the cone C_XX through the model space X2_.

the vertex 0 between the geodesic segments {rx : 0 ≤ r ≤ r0} and {rx0 : 0 ≤ r ≤ r00}

is the distance dX(x, x0).

We note that C_H_Sn−1 _{' H}n and C_E_Sn−1 _{' E}n, while the spherical cone C_S_Sn−1 is isometric to a hemisphere of Sn.

Geodesics through the vertex For r, r0 > 0, we have d(rx, r0x0) = r + r0 if and only if dX(x, x0) ≥ π. This implies that a path c : [−ε, ε] → CXX with c(0) = 0

defines a geodesic segment if and only if c(t) = tx and c(−t) = tx0 for t > 0 and some x, x0 ∈ X with dX(x, x0) ≥ π.

In particular, if the diameter of X is less than π, the vertex of the cone can never be an interior point of a geodesic segment.

1.2.3 Spherical joins

Another metric notion of importance for us is the spherical join X1 ∗ X2 of two

non-empty metric spaces (X, dX) and (Y, dY). As a set,

X ∗ Y =0,π₂ × X × Y .

∼,

where the equivalence relation ∼ is generated by (0, x, y1) ∼ (0, x, y2) and (π₂, x1, y) ∼

(π₂, x2, y) for all x, x1, x2 ∈ X and all y, y1, y2 ∈ Y . By convention, we put

X ∗ ∅ = X.

We will denote the point z = [(θ, x, y)] ∈ X ∗ Y by

(31)

With the conventions z = x if θ = 0 and z = y if θ = π

2, we sometimes consider X

and Y as subsets of X ∗ Y .

The metric For technical reasons, as in the previous section, given a metric d, we call ¯d the metric ¯d = min{π, d}.

For every two points z = cos θ x + sin θ y and z0 = cos θ0 x0+ sin θ0 y0, the metric d on the spherical join X ∗ Y is defined requiring that d(z, z0) ≤ π and

cos d(z, z0) = cos θ cos θ0 cos ¯dX(x, x0) + sin θ sin θ0 cos ¯dY(y, y0).

Any two points x ∈ X and y ∈ Y are joined in the spherical join X ∗ Y by the geodesic segment {cos θ x + sin θ y : 0 ≤ θ ≤ π₂} of length π

2.

The operation ∗ is commutative and associative up to isometry. Moreover, the natural inclusion X ⊂ X ∗ Y is an isometric embedding of (X, ¯dX).

Joins and spheres _{In this thesis, by S}0 _{we mean the set {1, −1} with metric d}

such that d(1, −1) = π. Moreover, we put S−n = ∅ for every integer n > 0. With these conventions, there are natural isometries Sn_{∗ S}m _{' S}n+m+1 _{for all integers n, m.}

The spherical join S0∗ X is called the spherical suspension of the metric space X. Each of the special points 1, −1 ∈ S0∗ X is called vertex of the suspension.

Cones and spherical joins To relate spherical joins to spherical or Euclidean cones, we note that:

1. the spherical cone is the spherical join with a point, that is

C_SX ' {0} ∗ X;

2. by (1), there are two copies of the spherical cone C_SX isometrically embedded in the spherical suspension S0 ∗ X;

3. the associativity of the spherical join and (1) imply

C_S(X ∗ Y ) ' C_SX ∗ Y ' X ∗ C_SY ;

in particular, the spherical cone and the spherical suspension commute up to isometry;

(32)

Figure 1-3: The spherical tetrahedron ∆ = Iα∗ Iβ. The black, red and green edges have length

respectively π₂, α and β dihedral angles respectively π₂, β and α.

4. the following natural map is an isometry

C_E(X ∗ Y ) → C_EX × C_EY

r(cos θ x + sin θ y) 7→ (r cos θ x, r sin θ y). (1.2)

For the observation (4) recall that, given two connected metric spaces A and B, the product metric on A × B is defined by

d((a, b), (a0, b0))2 = d(a, a0)2 + d(b, b0)2.

Note that given two topological spaces A and B, the isometry (1.2) gives homeomor-phisms

C(A ∗ B) ≈ CA × CB,

where C and ∗ are the usual topological cone and join.

Polytopes and spherical joins _{We note that if P ⊂ S}n _{and Q ⊂ S}m _{are spherical}

convex polytopes (or P = Sn _{or Q = S}m_{), then their spherical join P ∗ Q is isometric}

to a spherical convex polytope of dimension n + m + 1 (allowing P ∗ Q ' Sn+m+1_).

In particular, the spherical cone C_S_{P and the spherical join S}m_{∗ P are (isometric to)}

spherical polytopes.

For instance, the join Iα ∗ Iβ of two arcs of length 0 < α, β < π is a spherical

tetrahedron with two opposite edges of dihedral angles β and α and the other edges right-angled (see Figure 1-3).

Recall from Section 1.1.1 that a spherical polytope is said to be degenerate if it contains antipodal points. A spherical polytope P is degenerate if and only if there exists a spherical polytope Q such that P ' S0∗ Q.

(33)

and a point p in the interior of F (we have p = F if k = 0). Recall from Section 1.1.1 the definition of the link SpP of the point p and of the link LF of the face F . These

two spherical polytopes are related by

SpP ' Sk−1∗ LF.

1.3 Geometric structures

Geometric structures were introduced by Charles Ehresmann [Ehr36] and exploited in a revolutionary way forty years later by William Thurston [Thu79]. For a general picture with rigorous definitions and proofs, the reader may consult [Gol88, Rat06, Kap09].

1.3.1 (G, X)-structures

Geometries In agreement with Felix Klein’s Erlangen Program (1871), we will call geometry a couple (G, X) where G is a Lie group and X = G/H is the associated homogeneous space. We require that G acts on X analytically.

More generally, we could require just that X is a manifold and that G is a Lie group acting (not necessarily transitively) on X by diffeomorphisms, provided that the following holds: for every non-empty open set U ⊂ X and any g, g0 ∈ G such that g|U ≡ g0|U, then g = g0.

For instance, we can consider any Riemannian manifold X with a group G of isometries of X. In particular, X could be one of the three model spaces Sn_{, E}n _or

Hnwith G = Isom(X) the respective group of isometries, to get respectively spherical, Euclidean or hyperbolic geometry.

However, the notion of geometry is much more general: think for instance of projec-tive geometry (G, X) = (PGLn+1(R), RPn), affine geometry (G, X) = (Aff(Rn), Rn)

and so on, where there is no notion of distance.

Geometric manifolds A connected manifold M locally modelled on a geometry (G, X) will be called geometric – more specifically a (G, X)-manifold. By locally modelled, we mean that there exists an atlas of M with charts onto open sets of X and transition functions which are restrictions of elements of G. A (G, X)-structure will be an equivalence class of such atlases.

The morphisms between geometric manifolds, called (G, X)-maps, are local dif-feomorphisms compatible with the (G, X)-structures.

(34)

Holonomy and developing It is useful to globalise the local datum of a geometric atlas on M passing to a universal covering fM → M . The space fM inherits the (G, X)-structure by making tautologically the covering fM → M a (G, X)-map.

With a choice of base points, by analytic continuation one gets a (G, X)-map D : fM → X called developing map, well defined up to post-composition with elements of G, and a group homomorphism ρ : π1M → G from the fundamental group of M

called holonomy representation, well defined up to inner automorphisms of G. The developing map D is ρ-equivariant, meaning that

D(γp) = ρ(γ)(D(p))

for all p ∈ fM and all γ ∈ π1M , where as usual the fundamental group π1M acts on

the universal covering by deck transformations.

Viceversa, a representation ρ ∈ Hom(π1M, G) and a ρ-equivariant local

diffeomor-phism D : fM → X are respectively the holonomy representation and the developing map of a (G, X)-structure on M .

Roughly speaking, the developing map builds M piece by piece inside the model space X and the holonomy represents the loops of π1M as geometric automorphisms

of X. The local datum of a geometric atlas is thus globalised by the couple (D, ρ). The image Γ = ρ(π1M ) ⊂ G of the holonomy representation will be called the

holonomy group of the geometric structure – of course it is well defined only up to conjugation.

1.3.2 Geometric orbifolds

Complete structures A (G, X)-structure on M is said complete if the developing map D is a covering.

This notion of completeness coincides with the metric one in the Riemannian case, that is when X is a Riemannian manifold and G is a group of isometries of M . In that case, if moreover X is simply connected, the developing map D will be a diffeomorphism, the holonomy representation ρ will be faithful and the holonomy group Γ will be discrete, thus acting properly discontinuously on X.

If the manifold M carries a complete (G, X)-structure, with X simply connected and Γ acting freely and properly discontinuously, it is possible to identify the “topolog-ical” action π1M y fM with the “geometric” one Γ y X, so that M ≡ fM /π1M ≡ X/Γ. Geometric orbifolds For any geometry (G, X) and any subgroup Γ < G acting on X properly discontinuously with finite stabilisers, the quotient O = X/Γ has a

(35)

natural structure of complete (G, X)-orbifold1_{. The orbifold O is a (G, X)-manifold}

if, and only if, the action of Γ is free.

The image through the quotient map X → O of the points of X with non-trivial stabilizer is called the singular locus of the geometric orbifold O. The complement of the singular locus, called regular locus, is an open dense set.

Orbifold coverings Given a subgroup Γ0 < Γ, the natural map X/Γ0 → X/_Γ is an orbifold covering (and the index of the covering is the index [Γ : Γ0]). As a map between the underlying topological spaces – that is, forgetting the orbifold structure – only its restriction to the preimage of the regular locus is a covering.

The group of deck transformations of the orbifold universal covering X → O is the orbifold fundamental group πorb

1 (O). As for complete geometric manifolds, if X

is simply connected there is a faithful holonomy ρ : πorb

1 (O) → Γ.

Two orbifolds O0and O00are said commensurable if there exist finite-index orbifold coverings O → O0 and O → O00.

By the Bieberbach theorem, every Euclidean n-orbifold is finitely covered by an n-torus. By the Selberg lemma, every hyperbolic orbifold is finitely covered by a complete hyperbolic manifold. Every spherical orbifold is of course finitely covered by Sn.

The orbifold Euler characteristic is the rational number χorb_{(O) = χ(π}orb 1 (O)),

where the latter is the Euler characteristic of the group πorb

1 (O) (defined for instance

in [Bro94]). This invariant generalises the usual Euler characteristic for manifolds: it is multiplicative under finite orbifold coverings and χ(O) = χorb_{(O) if O is a manifold.}

Coxeter polytopes Recall the end of Section 1.1.2 about Coxeter polytopes. By a theorem of Poincar´e [dLH91], Hyperbolic (resp. Euclidean, spherical) Cox-eter polytopes have naturally the structure of hyperbolic (resp. Euclidean, spherical) orbifolds, whose singular locus is the union of the faces.

The orbifold fundamental group is a Coxeter group with presentation

π₁orb(P ) = hg1, . . . , gN | (gigj)niji,

where there is a generator gi for each facet Fi, two facets Fi and Fj intersect with

1_{As for manifolds, the notion of orbifold is independent of that of geometry. Roughly speaking, an}

orbifold is locally Rn_/

Γfor some finite groups Γ. Such quotients should be understood as stacky, not

just topological. Moreover, it is possible to define appropriately the notion of geometric structures on orbifolds. We refer to [Thu79] for definitions.

In this thesis, we will be interested only in geometric orbifolds. A theorem of Thurston [MMA91] states that every (G, X)-orbifold is a global quotient X/Γ.

(36)

dihedral angle π

nij (there is not relation if Fi∩ Fj = ∅) and nii = 1. The holonomy group Γ ≡ πorb

1 (P ) is discrete in G = Isom(Xn) and is a hyperbolic (resp. Euclidean,

spherical) reflection group: the isometry ρ(gi) ∈ Γ is the reflection through the unique

hyperplane Xn−1 _{⊂ X}n _{containing the facet F} i.

The orbifold Euler characteristic of a Coxeter polytope P is given by the formula [Ser71]

χorb(P ) =X

F

(−1)dim(F )

|Stab(F )|, (1.3)

where the sum is over all the faces F of the polytope (ideal vertices are excluded) and Stab(F ) is the stabilizer of a stratum inside the Coxeter reflection group Γ.

The ideal right-angled 24-cell Recall the hyperbolic ideal regular 24-cell P , in-troduced in Section 1.1.3. As orbifold, this Coxeter polytope is commensurable with the integral lattice in G = O(4, 1) ∼_{= Isom(H}4_{) [RT00]. A computation shows that}

χorb_{(P ) = 1.}

1.3.3 Deformations

The following exposition will be quite informal. We refer the reader mainly to [Gol88, Kap09, Apa00].

A space of structures In the following sections, we fix a compact connected man-ifold N with (possibly empty) boundary and call M = int(N ) its interior. The manifold M is considered up to diffeomorphism. We are briefly going to introduce a space of equivalence classes of (G, X)-structures over M .

We give the set of (G, X)-structures over M the topology induced by the C∞ -topology on the compact subsets of fM for developing maps, considered up to post-composition with elements of G.

By defining two (G, structures over M equivalent if there exists a (G, X)-diffeomorphism isotopic to the identity between them outside of a collar of ∂N , one gets the space Def(G,X)(M ) (just denoted by Def(M ) if there is no risk of ambiguity)

of equivalence classes of (G, X)-structures over M .

As an example, if S is a closed oriented surface of genus g > 1 and G = Isom+_(H2) is the group of orientation-preserving isometries of the hyperbolic plane X = H2, then Def(G,X)(S) is the well-known Teichm¨uller space of S and is homeomorphic to R6g−6.

Representations Consider a finitely generated group Γ and a representation ρ : Γ → G. Every point in a small (path-connected) neighbourhood of ρ in Hom(Γ, G) is a

(37)

deformation of ρ. A deformation ρ0 is trivial if there exists a g ∈ G such that ρ0 = gρg−1.

Let us consider the adjoint representation Ad : G → g into the Lie algebra of G. The composition with ρ gives a representation Adρ : Γ → g into the vector space g. Roughly speaking, a cohomology class in H1_{(Γ; Adρ) (see for instance [Bro94] for an}

introduction to the twisted cohomology of groups) can be seen as an infinitesimal deformation of the representation ρ up to trivial deformations.

Rigidity The representation ρ is said to be locally rigid if it has no non-trivial defor-mations; while it is said infinitesimally rigid if H1_{(Γ; Adρ) = 0 (in other words, it has}

no non-trivial infinitesimal deformations). By a theorem of Weil [Wei64], infinitesimal rigidity implies local rigidity.

Now, given a (G, X)-structure s with holonomy ρ, by the so called Ehresmann-Thurston Principle [Thu79] (see also [BG04] for a proof), every point in a sufficiently small neighbourhood of ρ in Hom(π1M, G) is the holonomy of a (G, X)-structure.

If the holonomy ρ is locally rigid, then the class of s is isolated in Def(M ) (see for instance [Gol88, Deformation Theorem] for details). In other words, the structure s cannot be non-trivially deformed.

There are several important results about infinitesimal rigidity of representations into semisimple Lie groups. In the next section we state some of them in the particular cases of interest for us.

1.3.4 Rigidity of hyperbolic lattices

Through the following, we focus on hyperbolic structures. Thus, we consider

(G, X) = (Isom(Hn), Hn).

The theory of deformations of hyperbolic structures is very rich. We will specialise the discussion only to complete and finite-volume hyperbolic structures in dimension greater than two.

Margulis lemma First of all, we recall a classical fact in hyperbolic geometry. The Margulis lemma [Zas37, KM68] implies that a complete finite-volume hyperbolic manifold M is diffeomorphic to the interior of a compact manifold N .

Each (if any) end of M is a cusp, isometric to a warped product [a, +∞) ×f E,

with f (r) = e−r, for a compact Euclidean manifold E – see [Thu79]. An analogous statement holds in the orbifold context.

(38)

Hyperbolic lattices A (hyperbolic) lattice is a discrete subgroup Γ < G such that the hyperbolic orbifold X/Γ has finite volume. The lattice Γ is said uniform if X/Γ

is compact. The lattice Γ is said to be locally (resp. infinitesimally) rigid if so is its inclusion into G; otherwise it is said flexible.

As we are going to see, “the rigidity of hyperbolic lattices increases with the dimension”. We list some important results in chronological order.

Flexibility In dimension n = 2, lattices are typically flexible: there are almost always non-trivial deformations. There are big spaces of (complete, finite-volume) hyperbolic structures on every closed surface, namely Teichm¨uller and moduli spaces – see for instance [FM12]. The theory is so wide that we do not add more, being mainly interested in higher dimensions.

Local rigidity _{By Calabi-Weil rigidity theorem [Wei62], uniform lattices in Isom(H}n₎

with n ≥ 3 are infinitesimally rigid. In particular, if n ≥ 3 and M is closed, then Def(M ) must be discrete (we will soon see that it is actually either empty or a point). Thus, there are no non-trivial deformations in that case.

By Garland-Raghunathan rigidity theorem [GR70], also non-uniform lattices in Isom(Hn_{) are infinitesimally rigid, provided that n ≥ 4 (see also [BG04] for a more}

elementary proof via geometric structures). In particular, if n ≥ 4, complete finite-volume hyperbolic structures on M have no non-trivial deformations; in other words, they are isolated points in Def(M ).

Assuming Calab-Weil local rigidity, Garland-Raghunathan local rigidity is essen-tially a theorem about the impossibility of deforming parabolic peripheral subgroups of Γ in G; in other words, “cusps stay cusps”.

Global rigidity By Mostow-Prasad rigidity theorem [Mos73, Pra73], two isomor-phic lattices Γ, Γ0 _{< Isom(H}n) with n ≥ 3 are conjugated. In other words, Γ ∼= Γ0 implies Γ0 = gΓg−1 _{for a g ∈ Isom(H}n). Equivalently, every isomorphism π₁orb(O) ∼= π₁orb(O0) between the orbifold fundamental groups of two finite-volume hyperbolic n-orbifolds with n ≥ 3 is induced by a unique isometry O ' O0.

In particular, if n ≥ 3, up to equivalence there exists at most one complete finite-volume hyperbolic structure on M . If moreover M is closed, every structure must be complete, thus Def(M ) is either empty or a point.

Mostow-Prasad rigidity is of fundamental importance in geometric topology: if M has a complete finite-volume hyperbolic structure, then all invariants of that structure are actually topological invariants.

(39)

Hyperbolic Dehn filling Let us now focus on the case n = 3, which is special. By Thurston’s hyperbolic Dehn filling theorem [Thu79] (see also [BP92, Mar16] for a proof), a non-uniform lattice Γ < Isom(H3_{) is flexible. Thus, Garland-Raghunathan}

rigidity cannot be extended to the dimension three. Some details follow.

Suppose a non-compact oriented 3-manifold M has a complete hyperbolic struc-ture s0 ∈ Def(M ) (unique by Mostow-Prasad). We have a holonomy ρ0: π1M → G =

Isom+_(H3) and a lattice Γ = ρ0(π1M ).

The theorem is proved by introducing explicit coordinates, called Dehn surgery coefficients, on a neighbourhood U of s0 in Def(M ). The set U turns to be

homeo-morphic to an open set of R2c, where c is the number of cusps of M .

By Mostow-Prasad rigidity, each structure s 6= s0 is necessarily incomplete; thus

the associated holonomy representation ρ : π1M → G is either non-faithful or has not

discrete image. However, for countably-many s ∈ U the associated holonomy ρ has discrete image and factorises through the holonomy of a closed hyperbolic manifold. More specifically, each such s determines a Dehn filling of M : a manifold Mfill such that M ≈ Mfill− L. The kernel of the projection π1M π1Mfill is generated

the peripheral loops γ1, . . . , γc (meridians of L in Mfill) and ρ(γi) = 1G.

With the Dehn surgery coordinates, we have a straight line t 7→ st∈ U joining s0

to s. For every t, the holonomy ρt(γi) is a rotation in H3 of some angle θi(t) which

increases from 0 to 2π. The metric completion of st is homeomorphic to Mfill (and is

(40)

(41)

Chapter 2 Cone-manifolds

This chapter is an introduction to Hyperbolic, Euclidean, and spherical cone-manifolds. These were first defined and used by Thurston [Thu98]. A systematic study of some aspects of such spaces is done in [CHK00, BLP05, McM].

2.1 Generalities

2.1.1 Definitions

In what follows, for a metric space Y and a point p ∈ Y , we denote by Nε(p) ⊂ Y

the ε-neighbourhood of y. Recall the notion of hyperbolic, Euclidean and spherical cone from Section 1.2.2 and that the symbol ' stands for “isometric”. We use the notation ˚C_SY = {ry ∈ C_SY : r 6= π₂} = C_SY − Y .

We give a definition of cone-manifold by induction on the dimension.

Definition 2.1.1. For a positive integer n, a hyperbolic (respectively Euclidean or spherical ) cone-manifold of dimension n is

• for n = 1, a length space supported on a 1-manifold;

• for n > 1, a length space X such that for each point p ∈ X there exist a compact connected spherical cone-manifold S of dimension n − 1, a point q ∈ C_HS (respectively C_ES or ˚C_SS) and ε > 0 such that Nε(p) ' Nε(q).

For brevity, we call a hyperbolic, Euclidean or spherical manifold just a cone-manifold, even if there is the much more general notion of Riemannian cone-manifold. However, in this thesis we will not give such a definition, referring the interested reader for instance to [McM].

(42)

Assumption 2.1.2. We assume cone-manifolds to be complete. In particular, up to isometry, the connected one-dimensional cone-manifolds are only circles Cλ of length

λ > 0 and the line R.

Remark 2.1.3. To deal with cone-manifolds with boundary it suffices to include in the definition also 1-manifolds with boundary – thus, by our completeness Assumption 2.1.2, one should consider also compact intervals and the half-line R≥0.

The boundary points will be defined in Remark 2.1.9. However, for simplicity of exposition, we will not deal with the boundary case in detail.

Examples Of course, (complete) hyperbolic, Euclidean and spherical manifolds are cone-manifolds (without singular points – see below). More generally, the same holds for orbifolds (in general, there may be boundary).

Every (not necessarily convex) polytope P ⊂ Xn_{is a cone n-manifold with}

bound-ary. Moreover, the boundary ∂P with intrinsic metric is a cone (n − 1)-manifold. In particular, for every compact convex polytope P there is a cone-manifold structure ∂P on the sphere Sn−1_.

Given some polytopes P1, . . . , Pk ⊂ Xn, by identifying all the facets in pairs

through isometries one gets a cone-manifold. In particular, the double of a com-pact convex polytope P is a cone-manifold supported on the sphere Sn_.

Every PL manifold M carries uncountably-many non-isometric structures of hy-perbolic, Euclidean and spherical cone-manifold: for every regular simplex ∆ ⊂ Xn_,

it suffices to identify every simplex of a triangulation of M with ∆. More generally, the same holds for normal pseudomanifolds (see Section 2.2.4).

Hyperbolic, Euclidean and spherical manifolds with totally geodesic boundary (or more generally, piecewise geodesic boundary with corners) are cone-manifolds with boundary.

Equivalent definition Our next goal is to improve our definition of cone-manifold. Recall that by S0 _{we mean the 0-sphere S}0 _{= {1, −1} with the metric d such that}

d(1, −1) = π. The following convention will be convenient in some proofs:

Convention 2.1.4. Henceforth, with a little abuse, by compact connected cone-manifold we mean a compact connected cone-cone-manifold or S0 _{(which is actually}

0-dimensional and disconnected). In this way, the inductive Definition 2.1.1 still holds starting from n = 0.

We begin with a simple “n = 0” lemma for spherical cone-manifolds. Recall the notion of spherical join introduced in Section 1.2.3.

Cone-manifolds and hyperbolic surgeries

Universit`

a degli Studi di Pisa

Corso di Dottorato in Matematica

XXIX Ciclo

Cone-manifolds and hyperbolic

surgeries

Tesi di Dottorato

Candidato

Stefano Riolo

Relatore

Prof. Bruno Martelli

Universit`

a di Pisa

Abstract

Acknowledgements

Contents

Introduction

Chapter 1

Preliminary notions

1.1

Convex polytopes

1.1.1

Generalities

1.1.2

Acute-angled polytopes

1.1.3

The regular 24-cell

1.2

Cones and joins

1.2.1

Geodesics and length spaces

1.2.2

Cones

X

1.2.3

Spherical joins

1.3

Geometric structures

1.3.1

(G, X)-structures

1.3.2

Geometric orbifolds

1.3.3

Deformations

1.3.4

Rigidity of hyperbolic lattices

Chapter 2

Cone-manifolds

2.1

Generalities

2.1.1

Definitions