HOLED CONE STRUCTURES ON 3-MANIFOLDS KENICHI YOSHIDA Abstract. We introduce holed cone structures on 3-manifolds to generalize

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HOLED CONE STRUCTURES ON 3-MANIFOLDS

KEN’ICHI YOSHIDA

Abstract. We introduce holed cone structures on 3-manifolds to generalize

cone structures. In the same way as a cone structure, a holed cone struc-

ture induces the holonomy representation. We consider the deformation space

consisting of the holed cone structures on a 3-manifold whose holonomy rep-

resentations are irreducible. This deformation space for positive cone angles

is a covering space on a reasonable subspace of the character variety.

1. Introduction

A cone-manifold is a generalization of a Riemannian manifold of constant sec-

tional curvature, allowed to have cone singularity. The rotational angle around cone

singularity is not equal to 2π, and it is called the cone angle. While a ﬁnite volume

complete hyperbolic structure on a 3-manifold admits no continuous deformation

by the Mostow rigidity, it can often be deformed via hyperbolic cone-manifolds. For

example, a hyperbolic Dehn surgery (in the strong sense) gives continuous defor-

mation from a cusped hyperbolic 3-manifold to a hyperbolic 3-manifold obtained

by gluing solid tori via hyperbolic cone-manifolds whose singular locus consists of

the cores of glued solid tori. If the cone angles of a cone-manifold are equal to

2π/nifor ni∈Z>0, this cone-manifold can be regarded as an orbifold. Proofs of

the geometrization of 3-orbifolds in [3, 6] are based on this fact.

Local and global rigidities of hyperbolic cone-manifolds are known as follows. For

an oriented 3-manifold Xand an n-component link Σ in X, let C[0,θ](X, Σ) denote

the space of hyperbolic cone structures on (X, Σ) with cone angles at most θ. Let

Θ: C[0,θ](X, Σ) →[0, θ]ndenote the map assigning the cone angles. The local rigid-

ity by Hodgson and Kerckhoﬀ [14] states that Θ: C[0,2π](X, Σ) →[0,2π]nis a local

homomorphism. The global rigidity by Kojima [19] states that Θ: C[0,π](X, Σ) →

[0, π]nis injective. The global rigidity is not known if cone angles exceed π. Iz-

mestiev [16] gave examples that the global rigidity does not hold and cone angles

exceed 2π. The proof in [19] is based on the fact that two cone loci with cone an-

gles less than πare not close. A cone structure in C[0,π](X, Σ) can be continuously

deformed to the cusped hyperbolic structure on X\Σ. However, this argument

does not work if cone angles exceed π. Cone structures may degenerate by meeting

cone loci even if the cone angles decrease [23].

In this paper, we introduce the notion of holed cone structures to generalize

cone structures. A holed cone structure on (X, Σ) is deﬁned as an equivalence

class of cone metrics on outside balls in X. In the same way as a non-holed

one, a holed cone structure induces the holonomy representation of π1(X\Σ)

to Isom+(H3) up to conjugate. We consider the deformation space HCirr(X, Σ)

consisting of the holed cone structures on (X, Σ) whose holonomy representations

are irreducible. In Theorem 3.13, we will show that HCirr(X, Σ) is Hausdorﬀ.

We will deﬁne a subspace Xcone(X, Σ) of the character variety X(π1(X\Σ)).

2020 Mathematics Subject Classiﬁcation. 57M50, 57K31, 57K32.

Key words and phrases. cone-manifolds, holonomy representations.

arXiv:2210.14765v1 [math.GT] 26 Oct 2022

2 KEN’ICHI YOSHIDA

The b

Xcone(X, Σ) consists of the pairs of elements in Xcone(X, Σ) and compat-

ible cone angles. The map c

hol: HCirr(X, Σ) →b

Xcone(X, Σ) assigns the holo-

nomy representation and the cone angles. In Theorem 3.12, we will show that

the map c

hol: HCirr

+(X, Σ) →b

Xcone

+(X, Σ) is a regular covering map to each path-

connected component of b

Xcone

+(X, Σ) that contains an image, where the map is

restricted to the elements with positive cone angles. Lemma 3.8 implies that the

map c

hol: HCirr(X, Σ) →b

Xcone(X, Σ) is not injective. Consequently, global rigidity

for holed cone structures does not hold even if X\Σ admits a hyperbolic structure.

Section 4 concerns (non-holed) cone structures in the space of holed cone struc-

tures. In Theorem 4.1, we will show that cone metrics gand g0with c

hol(g) = c

hol(g0)

are equivalent. By Corollary 4.4, we may regard a cone structure as a holed cone

structure. We expect that the notion of holed cone structures is useful to con-

sider global rigidity for cone-manifolds. It should be worthwhile to ask whether a

cone structure can be deformed to the cusped hyperbolic structure via holed cone

structures.

In Section 5, we will introduce the volume of a holed cone structure. This is

deﬁned as the sum of the volume of a holed cone metric and the volume enclosed

by the holes. After all, this volume is equal to the volume of the holonomy repre-

sentation by Theorem 5.3.

In Section 6, we will give an explicit example of holed cone structures. This is

consistent with the construction of cone structures by the author [23]. Holed cone

structures enable us to avoid degeneration with meeting cone loci.

Euclidean and spherical holed cone structures can be deﬁned in the same manner.

To show the corresponding results, it is necessary to take care of the topologies of

quotients of representation spaces.

2. Definition of holed cone structures

Hyperbolic metrics on a manifold are Riemannian metrics with constant sec-

tional curvature −1. Equivalently, they are modeled by the hyperbolic space Hnof

constant sectional curvature −1. According to [6], a (hyperbolic) cone-manifold is

a topological manifold with a complete path metric (called a cone metric) which

can be triangulated into hyperbolic simplices. (Although it does not matter in 3

dimensions, the link of each simplex in this triangulation is needed to be piecewise

linearly homeomorphic to a standard sphere.)

The singular locus of a cone-manifold consists of the points with no neighborhood

isometric to a hyperbolic ball. We consider 3-dimensional cone-manifolds whose

singular locus consists of closed geodesics. Then locally a cone metric has the form

dr2+ sinh2rdθ2+ cosh2rdz2

in cylindrical coordinates around an axis, where ris the distance from the axis, z

is the distance along the axis, and θis the angle measured modulo the cone angle

θ0>0. If a cone angle is equal to 2π, then the metric is smooth around the

point. By generalizing the notion, a cusp of a hyperbolic 3-manifold is regarded as

a component of the singular locus with cone angle zero.

Let Xbe an oriented 3-manifold, and let Σ be a union of disjoint circles in X. A

cone metric on (X, Σ) is a metric on Xsuch that Xis a cone-manifold with singular

locus Σ. We allow that a cone angle is equal to 0 or 2π. We call a component of

Σ a cone locus. A cone structure on (X, Σ) is an isometry class of a cone metric

on (X, Σ), where two metrics are equivalent if there is an isometry isotopic to the

identity between them.

We introduce holed cone structures on (X, Σ) as a generalization of cone struc-

tures.

HOLED CONE STRUCTURES ON 3-MANIFOLDS 3

Deﬁnition 2.1. Let Bbe a union of ﬁnitely many (possibly empty) disjoint closed

3-balls in X\Σ. A holed cone metric on (X, Σ) is a cone metric on (X\int(B),Σ)

with smooth boundary ∂B. We call each component of Bahole. We call the metric

space (X\int(B),Σ; g) a holed cone-manifold.

Deﬁnition 2.2. Let gand g0be holed cone metrics on (X, Σ) respectively with

holes Band B0. The metrics gand g0are equivalent if there are holed cone metrics

giwith holes Bion (X, Σ) for 0 ≤i≤nsuch that g0=g,gn=g0, and for each

0≤i≤n−1 either

(1) there is a map f: (X, Σ) →(X, Σ) isotopic to the identity (preserving Σ)

such that the restriction of fto (X\int(Bi),Σ; gi) is an isometry onto

(X\int(Bi+1),Σ; gi+1),

(2) Bi⊂Bi+1, and gi+1 is the restriction of gito X\int(Bi+1), or

(3) Bi+1 ⊂Bi, and giis the restriction of gi+1 to X\int(Bi).

We call an equivalent class [g] a holed cone structure. The relation (1) will not be

mentioned explicitly.

A cone metric is a holed cone metric by deﬁnition. Moreover, we can say that a

cone structure is a holed cone structure. We will prove it in Corollary 4.4.

We give some elementary examples of holed cone metrics. If we remove an em-

bedded ball disjoint from the cone loci in a holed cone-manifold, then we obtain an

equivalent holed cone metric. Conversely, if a lift of the boundary of a hole is em-

bedded by the developing map (which we will deﬁne in Section 3) and neighborhood

of this boundary extends outward, then we can ﬁll the hole by gluing the bounded

ball in H3. In general, however, the boundary of a hole may not be embedded by

the developing map.

Cone loci may prevent a hole from expanding to a ﬁllable one as indicated in

Figure 1. This enables us to deform holed cone structures when cone structures

degenerate with meeting cone loci. An explicit example will be given in Section 6.

Figure 1. A non-ﬁllable hole between two cone loci

Let (X, Σ; g) and (X0,Σ0;g0) be holed cone-manifolds. By making holes in (X, Σ)

and (X0,Σ0) and attaching a 1-handle, we obtain a holed cone metric on the con-

nected sum (X#X0,ΣtΣ0). By making two holes in (X, Σ) attaching a 1-handle,

we obtain a holed cone metric on the connected sum (X#S2×S1,Σ). Since met-

rics on a 1-handle can be arbitrarily deformed, the local rigidity for holed cone

structures does not hold in general.

4 KEN’ICHI YOSHIDA

3. Deformation space of holed cone structures

Let Σ = Fn

i=1 Σi⊂Xbe a link in an oriented 3-manifold. Let gbe a (hyperbolic)

holed cone metric on (X, Σ) with holes B. Let Γ = π1(X\Σ) = π1(X\(Σ ∪

int(B))). Suppose that Γ is ﬁnitely generated. Note that the holes do not aﬀect the

fundamental group. Let f

Mdenote the universal cover of the incomplete hyperbolic

3-manifold M= (X\(Σ ∪int(B)); g). The developing map devg:f

M→H3is

deﬁned in an ordinary way as in [3, 5, 6].

Let G= Isom+(H3) denote the group consisting of the orientation-preserving

isometries of H3. The holonomy representation ρg: Γ →Gis also deﬁned so that

devgis equivariant, i.e. devg(γ·x) = ρg(γ)·devg(x) for any γ∈Γ and x∈f

M. The

map devgand the representation ρgare unique up to conjugate in G. Since the

restriction and extension in Deﬁnition 2.2 do not aﬀect the holonomy representation,

the holonomy representation ρgis well-deﬁned for a holed cone structure [g]. In

other words, we have ρg=ρg0for equivalent holed cone metrics gand g0. The

developing map is an immersion. Conversely, an equivariant immersion induces

a metric. The restriction of the developing map to the boundary of a hole is an

immersion, but it is not an embedding in general.

Consider the representation space Hom(Γ, G) endowed with the compact-open

topology. The group Gacts on Hom(Γ, G) by the conjugation. Let [ρ] denote

the conjugacy class of ρ∈Hom(Γ, G). The topological quotient Hom(Γ, G)/G

consisting of such [ρ] is not Hausdorﬀ in general. For instance, if ρ0: Γ →Z→G

whose non-trivial images are parabolic, conjugate elements of ρ0accumulate the

trivial representation. To obtain a manageable deformation space, we need to

restrict the space of holonomy representations.

The group G= Isom+(H3)∼

=PSL(2,C)∼

=SO(3,C) is a complex algebraic

group. Hence the space Hom(Γ, G) admits a structure of a complex aﬃne algebraic

set. The character variety X(Γ) = Hom(Γ, G)//G is deﬁned as the GIT quotient in

the category of algebraic varieties. Consider the Euclidean topology of the aﬃne

variety X(Γ), which is induced as a subset in CNfor some N. It is known that the

space X(Γ) is the largest Hausdorﬀ quotient of Hom(Γ, G)/G (see [21]).

A representation in Hom(Γ, G) is irreducible if there is no proper Γ-invariant

subspace of C2in the action as G= PSL(2,C). Let Homirr(Γ, G) denote the set

of irreducible representations. Let t: Hom(Γ, G)→ X(Γ) denote the projection.

Any representation ρ∈Homirr(Γ, G) satisﬁes that the set t−1(t(ρ)) consists of the

conjugates of ρ(see [4, 8, 12, 22] for details). Hence we may regard Xirr(Γ) =

Homirr(Γ, G)/G as a subset of X(Γ). In particular, Xirr(Γ) is Hausdorﬀ. We write

[ρ] = t(ρ)∈ Xirr(Γ).

We deﬁne the deformation space of holed cone structures. Let f

HC(X, Σ) denote

the space of holed cone metrics on (X, Σ) endowed with the C∞-topology induced

by the metric tensors. Let HC(X, Σ) denote the space of holed cone structures on

(X, Σ) endowed with the quotient topology from f

HC(X, Σ). The continuous map

hol: HC(X, Σ) → X(Γ) is deﬁned by hol([g]) = [ρg]. Let HCirr(X, Σ) ⊂ HC(X, Σ)

and f

HCirr(X, Σ) ⊂f

HC(X, Σ) denote the preimages of Xirr(Γ) by hol. Note that

a quotient space of a Hausdorﬀ space is not Hausdorﬀ in general. We show that

HCirr(X, Σ) is Hausdorﬀ in Theorem 3.13.

Let us consider a condition for the holonomy of a neighborhood of cone loci.

For each 1 ≤i≤n, let N(Σi) be a regular neighborhood of the cone locus Σi

in X. The meridian for Σ is (the homotopy class of) a simple closed curve on

∂N (Σi) that is contractible in N(Σi). Fix a longitude for Σ, which is a simple

closed curve on ∂N(Σi) intersecting the meridian at exactly once. Fix orientations

of the meridian and longitude compatible with the orientation of X. Fix commuting

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HOLEDCONESTRUCTURESON3-MANIFOLDSKEN'ICHIYOSHIDAAbstract.Weintroduceholedconestructureson3-manifoldstogeneralizeconestructures.Inthesamewayasaconestructure,aholedconestruc-tureinducestheholonomyrepresentation.Weconsiderthedeformationspaceconsistingoftheholedconestructuresona3-manifoldwhoseholonomyrep...

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HOLED CONE STRUCTURES ON 3-MANIFOLDS KENICHI YOSHIDA Abstract. We introduce holed cone structures on 3-manifolds to generalize

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