# Effective bilipschitz bounds
on drilling and filling

David Futer  Department of Mathematics, Temple University, Philadelphia, PA
19122, USA<EMAIL_ADDRESS>,  Jessica S. Purcell  School of Mathematics,
Monash University, VIC 3800, Australia<EMAIL_ADDRESS>and  Saul
Schleimer  Department of Mathematics, University of Warwick, Coventry CV4 7AL,
UK<EMAIL_ADDRESS>
###### Abstract.

This paper proves explicit bilipschitz bounds on the change in metric between
the thick part of a cusped hyperbolic $3$–manifold $N$ and the thick part of
any of its long Dehn fillings. Given a bilipschitz constant $J>1$ and a
thickness constant $\epsilon>0$, we quantify how long a Dehn filling suffices
to guarantee a $J$–bilipschitz map on $\epsilon$–thick parts. A similar
theorem without quantitative control was previously proved by Brock and
Bromberg, applying Hodgson and Kerckhoff’s theory of cone deformations. We
achieve quantitative control by bounding the analytic quantities that control
the infinitesimal change in metric during the cone deformation.

Our quantitative results have two immediate applications. First, we relate the
Margulis number of $N$ to the Margulis numbers of its Dehn fillings. In
particular, we give a lower bound on the systole of any closed $3$–manifold
$M$ whose Margulis number is less than $0.29$. Combined with Shalen’s upper
bound on the volume of such a manifold, this gives a procedure to compute the
finite list of $3$–manifolds whose Margulis numbers are below $0.29$.

Our second application is to the cosmetic surgery conjecture. Given the
systole of a one-cusped hyperbolic manifold $N$, we produce an explicit upper
bound on the length of a slope involved in a cosmetic surgery on $N$. This
reduces the cosmetic surgery conjecture on $N$ to an explicit finite search.

###### 2020 Mathematics Subject Classification:

57K32, 30F40, 57K10

###### Contents

  1. 1 Introduction
    1. 1.1 Prior work on cone deformations
    2. 1.2 Effective bilipschitz bounds
    3. 1.3 How to prove bilipschitz bounds
    4. 1.4 Application to Margulis numbers
    5. 1.5 Application to cosmetic surgeries
    6. 1.6 Organization
    7. 1.7 Acknowledgements
  2. 2 Cone-manifold basics
    1. 2.1 Covers and deck transformations
    2. 2.2 Injectivity radii, cusps, and tubes
  3. 3 Distance estimates in tubes and cusps
    1. 3.1 Tube radii
    2. 3.2 Distances between tori of fixed injectivity radius
    3. 3.3 Euclidean bounds
  4. 4 Maximal tubes and multi-tubes
    1. 4.1 Areas of maximal tubes
    2. 4.2 Injectivity radii
  5. 5 Existence of cone deformations
    1. 5.1 Background on cone deformations
    2. 5.2 Visual area and maximal tubes
    3. 5.3 Decreasing cone angles to $0$
    4. 5.4 Increasing cone angles from $0$
  6. 6 Bounding the boundary terms
    1. 6.1 Definitions and setup
    2. 6.2 Controlling length and visual area
    3. 6.3 Boundary terms for general tubes
    4. 6.4 Boundary terms along thin tubes
  7. 7 Short geodesics in a cone-manifold
    1. 7.1 Hyperbolic distance between lengths
    2. 7.2 The change in length
    3. 7.3 Application to cosmetic surgery
    4. 7.4 Controlling multiple geodesics
  8. 8 Bilipschitz estimates in the thick part
    1. 8.1 Bounding the pointwise norm
    2. 8.2 Corollaries and variations
  9. 9 Margulis numbers for cone-manifolds
    1. 9.1 Tubes realizing injectivity radii
    2. 9.2 Thick parts stay almost as thick
    3. 9.3 Applications
  10. A Hyperbolic trigonometry

##  1\. Introduction

Dehn filling is the process of changing a compact $3$–manifold by attaching
solid tori to some number of its torus boundary components. For each boundary
torus $T$, the choice of filling is determined by a _slope_ : that is, an
isotopy class of simple closed curve on $T$ that will bound a disk in the
attached solid torus. In the 1960s, Wallace and Lickorish showed that any
closed, orientable $3$–manifold is obtained by Dehn filling a link complement
in $S^{3}$ [60, 42]. This established Dehn filling as an important technique
in the study of $3$–manifold topology.

Thurston pioneered the geometric study of Dehn surgery. When a compact
$3$–manifold with torus boundary has interior admitting a complete hyperbolic
structure, the non-compact ends become _cusps_ with torus cross-sections. The
boundary torus of a cusp neighborhood inherits a Euclidean metric, and each
slope inherits a Euclidean length. Thurston showed that complete hyperbolic
structure on the interior of a manifold with torus boundary components can
always be deformed to incomplete hyperbolic structures [59]. The space of such
structures is called _hyperbolic Dehn surgery space_. The completions of such
deformed structures are often not manifolds, but sometimes they are
diffeomorphic to Dehn fillings of the original. When this happens, the
completion is called a _hyperbolic Dehn filling_. Thurston also showed that as
the Euclidean lengths of Dehn filling slopes approach infinity, the
corresponding hyperbolic Dehn fillings approach the original manifold in the
Gromov–Hausdorff topology. It follows that hyperbolic Dehn filling is an
important technique in the study of $3$–manifold geometry.

We are particularly interested in uniform and effective geometric estimates
for Dehn filling. Here, _uniform_ means that constants appearing in the
estimates are independent of the underlying $3$–manifold, while _effective_
means that these constants are explicitly given. Many uniform estimates
controlling fine-scale geometry under Dehn filling have previously been
developed [11, 14, 33]. These estimates have played an important role in
proving theorems about spaces of Kleinian groups [11, 12, 15, 43]; see Section
1.1 for more details. However, apart from theorems establishing the existence
of hyperbolic structures [2, 33, 41] and bounding volume [22, 34], the
previous results have not been effective. Explicit estimates are needed to
apply Dehn filling techniques to the study of individual manifolds. In
particular, where a computer algorithm depends on some theoretical bound in
order to know when to stop searching, only an explicit bound can make the
algorithm implementable. Such explicit bounds on fine-scale geometry are
provided for the first time in this paper.

The difference between effective and ineffective results can be illustrated as
follows. Thurston’s Dehn surgery theorem [59], which says that all but
finitely many surgeries on a hyperbolic manifold yield closed hyperbolic
manifolds, is powerful but not effective or uniform. It does not say which
slopes one needs to exclude, or even how the number of excluded slopes depends
on the manifold. By contrast, the $6$–theorem proved by Agol [2] and Lackenby
[41] in 2000, which says that all surgeries of length greater than $6$ yield
hyperbolic manifolds, is more powerful precisely because it is effective. (The
conclusion that the filled manifold is hyperbolic depends on Perelman’s proof
of the geometrization conjecture, which occurred several years later.) In many
applications, the $6$–theorem is used to break a problem into cases:
hyperbolic geometry handles the “generic” scenario, while ad-hoc methods
handle the small, concrete list of exceptions. Our results have a similar
effect, enabling computer-assisted proofs for _all_ fillings of a manifold.

We present two applications. First, for any hyperbolic knot complement
$S^{3}-K$, we prove an effective upper bound on the length of a cosmetic
surgery on $K$. This means that if two different Dehn fillings on $K$ yield
the same closed $3$–manifold, the pair of fillings must come from an explicit
finite list. See Corollary 1.10 in Section 1.5 for a precise statement. Thus a
finite computer check establishes that knots up to $16$ crossings have no
cosmetic surgeries (Corollary 1.11). Second, Theorems 9.25 and 9.29 stated in
Section 1.4 provide explicit control on the Margulis numbers of closed
hyperbolic $3$–manifolds.

###  1.1. Prior work on cone deformations

In 2002, just before the resolution of the geometrization conjecture, Hodgson
and Kerckhoff proved the first effective, uniform version of Thurston’s Dehn
surgery theorem [33]. They showed that, for all but $60$ choices of slope $s$
on a one-cusped hyperbolic manifold $N$, the filled manifold $N(s)$ is also
hyperbolic. The slopes excluded by their theorem are the ones that have
shortest _normalized length_ ; see Definition 1.3. Their method was to obtain
a hyperbolic metric on $N(s)$ at the end of a one-parameter family of singular
metrics, with cone-singularities of angle $0\leq\alpha\leq 2\pi$ along the
core of the Dehn filling solid torus. (See Section 2 for a careful definition
of cone-manifolds and related notions.) When the cone angle starts at $0$, the
core is not present, and one obtains the complete hyperbolic metric on the
cusped manifold $N$. When the cone angle becomes $2\pi$, the singular solid
torus becomes non-singular, and one one has a complete hyperbolic metric on
$N(s)$. Thus one has succeeded in performing hyperbolic Dehn filling.

The technique of deformation through cone structures, initiated by Hodgson and
Kerckhoff [31], has been highly useful. In addition to proving uniform bounds
on Dehn filling, Hodgson and Kerckhoff also gave bounds on volume change under
Dehn filling, on the lengths of core geodesics [33], and on the shape of
hyperbolic Dehn surgery space [34]. Purcell extended their techniques to give
bounds on the change of cusp shape under cone deformation, applying the result
to the geometry of knots in $S^{3}$ [53, 52]. Bromberg applied their methods
to study deformations that run from cone angle $2\pi$ to $0$, a process called
_drilling_. Bromberg also extended their results from finite-volume to
infinite-volume manifolds, and gave bounds on the change in length of a short,
nonsingular geodesic [14]. We remark that the above-mentioned results bounding
the change in length of a closed geodesic [14, 33] are uniform (independent of
manifold) but not effective. We prove and apply effective versions of these
results; see Corollary 6.13 and Corollary 7.20, which are also stated later in
the introduction.

The application of cone deformations most relevant to this paper is the
bilipschitz drilling theorem of Brock and Bromberg [11]. Building on Hodgson
and Kerckhoff’s methods, Brock and Bromberg obtained uniform bilipschitz
bounds relating the hyperbolic metrics at the two ends of the deformation. In
the following theorem, $\mu_{3}$ is the $3$–dimensional Margulis constant. See
Definition 1.4 for a review of the thick-thin decomposition and Definition 2.7
for a review of rank-one and rank-two cusps. The hypothesis that $M$ is
geometrically finite means that the convex core of $M$ has finite volume. In
particular, finite-volume manifolds are geometrically finite and have no rank
one cusps.

######  Theorem 1.1 (Drilling theorem, [11]).

Fix $0<\epsilon\leq\mu_{3}$ and $J>1$. Then there is a number
$\ell_{0}=\ell_{0}(\epsilon,J)>0$ such that the following holds for every
geometrically finite hyperbolic $3$–manifold $M$ without rank-one cusps.
Suppose that $\Sigma\subset M$ is a link composed of closed geodesics, whose
total length is less than $\ell_{0}$. Then the inclusion

$\iota\colon(M-\Sigma)\hookrightarrow M$

restricts to a $J$–bilipschitz diffeomorphism on the complement of
$\epsilon$–thin tubes about $\Sigma$.

Theorem 1.1 has had several important applications. Using earlier work of
Bromberg [15], Brock and Bromberg used this result to prove the
Bers–Sullivan–Thurston density conjecture for freely indecomposable Kleinian
groups without parabolics [11]. (The proof of the full density conjecture
relies upon the Ending Lamination Theorem, as in Ohshika [51] and Namazi–Souto
[47].) In further applications, Bromberg [16] and Magid [43] used the drilling
theorem to show that deformation spaces of Kleinian surface groups are not
locally connected. Purcell and Souto used the drilling theorem to show that a
large class of hyperbolic manifolds occurs as geometric limits of knot
complements in $S^{3}$ [54]. Cooper, Futer, and Purcell used it to show that
there are knots in $S^{3}$ with long, geodesic unknotting tunnels [18]. For
each of these applications, it was important that the length cutoff $\ell_{0}$
is independent of the manifold $M$.

However, the drilling theorem also has limitations. In particular, the
constants are not effective: the dependence of the length cutoff
$\ell_{0}(\epsilon,J)$ on the thickness constant $\epsilon$ and the
bilipschitz constant $J$ is not quantified. This means that, while Theorem 1.1
can be used in geometric limit arguments as in the previous paragraph, it is
less suitable for studying individual manifolds. This is because it is never
clear whether a given $M$ satisfies the hypotheses. Furthermore, the
ineffective form cannot be used in algorithms.

###  1.2. Effective bilipschitz bounds

One of the most important results of this paper is Theorem 9.30, which
effectivizes Theorem 1.1. In the following corollary, $M^{\geq\epsilon}$
denotes the $\epsilon$–thick part of $M$; that is, all points of injectivity
radius at least $\epsilon/2$. See Definition 2.14 for full details.

######  Theorem 1.2.

Fix $0<\epsilon\leq\log 3$ and $J>1$. Let $M$ be a finite-volume hyperbolic
$3$–manifold and $\Sigma$ a geodesic link in $M$ whose total length $\ell$
satisfies

$\ell\leq\min\left\\{\frac{\epsilon^{5}}{6771\cosh^{5}(0.6\epsilon+0.1475)},\,\frac{\epsilon^{5/2}\log(J)}{11.35}\right\\}.$

Then, setting $N=M-\Sigma$, and equipping it with its complete hyperbolic
metric, there are natural $J$–bilipschitz inclusions

$\varphi\colon M^{\geq\epsilon}\hookrightarrow
N^{\geq\epsilon/1.2},\qquad\psi\colon N^{\geq\epsilon}\hookrightarrow
M^{\geq\epsilon/1.2}.$

Here $M^{\geq\epsilon}$ and $N^{\geq\epsilon}$ are the $\epsilon$–thick parts
of $M$ and $N$, respectively. The compositions $\varphi\circ\psi$ and
$\psi\circ\varphi$ are the identity wherever both maps are defined.
Furthermore, $\varphi$ and $\psi$ are equivariant with respect to the symmetry
group of the pair $(M,\Sigma)$.

Comparing the statements of Theorems 1.1 and 1.2 reveals several differences.
Most notably, Theorem 1.2 is stronger, in that it gives looser hypotheses on
$\epsilon$ as well as quantified hypotheses on
$\ell=\operatorname{len}(\Sigma)$ that ensure a $J$–bilipschitz map. However,
Theorem 1.2 is slightly weaker in two respects. First, it assumes that $M$ has
finite volume. This assumption is convenient for our line of argument, but is
not crucial: using algebraic and geometric limits, we have extended Theorem
1.2 to all hyperbolic $3$–manifolds with finitely generated fundamental groups
[24]. Second, Theorem 1.2 provides bilipschitz control on a smaller
submanifold of $M$. While Theorem 1.1 excludes the $\epsilon$–thin tubes about
$\Sigma$, Theorem 1.2 excludes _all_ the $\epsilon$–thin regions of $M$,
including all cusps as well as $\epsilon$–thin tubes about geodesics that are
not involved in the cone deformation.

While we do not know how to extend Theorem 1.2 into the $\epsilon$–thin
regions of $M$, we do have quantitative control over the change in complex
length of a sufficiently short geodesic. Consider a closed geodesic
$\gamma\subset M$, which corresponds to a loxodromic isometry
$\varphi=\varphi(\gamma)\in{\operatorname{Isom}}^{+}{\mathbb{H}}^{3}$. This
loxodromic isometry $\varphi$ has an invariant axis in ${\mathbb{H}}^{3}$,
which it translates by distance $\lambda$ and rotates by angle $\tau$. We
define the _complex length_ of $\gamma$ to be
$\mathcal{L}(\gamma)=\lambda+i\tau$. The tubular neighborhood of $\gamma$ of
some radius $r$, is determined up to isometry by $r$ and
$\mathcal{L}(\gamma)$; compare Definition 2.3. Thus controlling the change in
complex length is the first step to controlling the geometry of an entire tube
about $\gamma$.

We prove the following effective version of a result of Bromberg [14,
Proposition 4.3].

######  Corollary 7.20.

Let $M$ be a complete, finite volume hyperbolic $3$–manifold. Let
$\Sigma\cup\gamma$ be a geodesic link in $M$, where $\gamma$ is connected. Let
$\mathcal{L}_{M}(\gamma)=\operatorname{len}_{M}(\gamma)+i\tau_{M}(\gamma)$ be
the complex length of $\gamma$ in the complete metric on $M$, and suppose that
$\max(\operatorname{len}_{M}(\Sigma),\operatorname{len}_{M}(\gamma))\leq
0.0735$. Then $\gamma$ is also a geodesic in the complete metric on
$N=M-\Sigma$, of complex length $\mathcal{L}_{N}(\gamma)$. Furthermore,

$1.9793^{-1}\leq\frac{\operatorname{len}_{N}(\gamma)}{\operatorname{len}_{M}(\gamma)}\leq
1.9793\qquad\text{and}\qquad|\tau_{N}(\gamma)-\tau_{M}(\gamma)|\leq 0.05417.$

When either $\Sigma$ or $\gamma$ is much shorter than $0.0735$, the
quantitative control over $\mathcal{L}(\gamma)$ improves dramatically. See
Theorem 7.19 for the exact statement. We note that Corollary 7.20 also has an
extension to all hyperbolic 3-manifolds with with finitely generated
fundamental groups [24].

###  1.3. How to prove bilipschitz bounds

Next, we outline some top-level steps in the proofs of Theorem 1.2 and
Corollary 7.20. We begin by showing the existence of a one-parameter family of
cone-manifolds interpolating between the complete hyperbolic metric on $M$ and
the complete hyperbolic metric on $N=M-\Sigma$.

######  Theorem 5.1.

Let $M$ be a finite volume hyperbolic $3$–manifold. Suppose that
$\Sigma=\sigma_{1}\cup\dots\cup\sigma_{n}$ is a geodesic link in $M$, whose
components have lengths satisfying

$\ell_{j}=\operatorname{len}_{M}(\sigma_{j})\leq
0.0996\qquad\mbox{and}\qquad\ell=\sum_{j=1}^{n}\ell_{j}\leq 0.15601.$

Then the hyperbolic structure on $M$ can be deformed to a complete hyperbolic
structure on $M-\Sigma$ by decreasing the cone angle $\alpha_{j}$ along
$\sigma_{j}$ from $2\pi$ to $0$. The cone angles on all components of $\Sigma$
change in unison.

Theorem 5.1 is due to Hodgson and Kerckhoff [33, Corollary 6.3] in the special
case where $\Sigma$ is connected. We extend the result to a link $\Sigma$ with
an arbitrary number of components. The cone-manifolds along the deformation
are denoted $M_{t}$, where $t\in[0,(2\pi)^{2}]$. Every component of $\Sigma$
in $M_{t}$ has cone angle $\alpha=\sqrt{t}$. Thus $t=0$ corresponds to the
complete metric on $N=M-\Sigma$, while $t=(2\pi)^{2}$ corresponds to the
complete metric on $M$.

In fact, we show more: when $\ell=\operatorname{len}_{M}(\Sigma)$ is small,
every cone-manifold $M_{t}$ has a large embedded tube about $\Sigma$. See
Theorem 5.14 for the full statement. In the work of Hodgson and Kerckhoff [33,
34], the radius of this tube is the key ingredient in a number of analytic
estimates that control the change in geometry. We work out effective versions
of these estimates in Section 6.

These analytic estimates allow us to prove Theorem 8.3, which provides
bilipschitz control on submanifolds of $M$ that stay _thick_ throughout the
deformation. In the following corollary of Theorem 8.3, the submanifold
$M_{t}^{\geq\delta}$ is the $\delta$–thick part of the cone-manifold $M_{t}$
in its singular metric $g_{t}$. See Definitions 1.4 and 2.14.

######  Corollary 8.16.

Fix $0<\delta\leq 0.938$ and $J>1$. Let $M$ be a complete, finite volume
hyperbolic $3$–manifold. Let $\Sigma\subset M$ be a geodesic link whose total
length $\ell$ satisfies

$\ell\leq\min\left\\{\frac{\delta^{2}}{17.11},\,\frac{\delta^{5/2}\log(J)}{7.193}\right\\}.$

Let $W\subset M$ be any submanifold such that $W\subset M_{t}^{\geq\delta}$
for all $t$. Then, for all $a,b\in[0,(2\pi)^{2}]$, the identity map
${\mathop{id}}\colon(W,g_{a})\to(W,g_{b})$ is $J$–bilipschitz.

We also prove a version of Corollary 8.16 whose hypotheses are on the cusped
manifold $N=M-\Sigma$ instead of the filled manifold $M$. Stating this version
requires a definition.

######  Definition 1.3.

Let $N$ be a hyperbolic $3$–manifold with rank-two cusps $C_{1},\ldots,C_{n}$.
Choose a slope $s_{j}$ for each cusp torus $\partial C_{j}$. The _normalized
length_ of $s_{j}$ is

$L_{j}=L(s_{j})=\frac{\operatorname{len}(s_{j})}{\sqrt{\operatorname{area}(\partial
C_{j})}},$

where $\operatorname{len}(s_{j})$ is the length of a geodesic representative
of $s_{j}$ on $\partial C_{j}$.

Let $\mathbf{s}=(s_{1},\ldots,s_{n})$ be the vector of all the slopes. We
define the _total normalized length_ $L=L(\mathbf{s})$ via the formula

$\frac{1}{L^{2}}=\sum_{j=1}^{n}\frac{1}{L_{j}^{2}}.$

Observe that each normalized length $L_{j}$ is scale-invariant, hence does not
depend on the choice of horospherical neighborhood of a cusp $C_{j}$.

Hodgson and Kerckhoff [34] proved that if $\mathbf{s}$ is a vector of slopes
in $N$ whose total normalized length is $L(\mathbf{s})\geq 7.5832$, then there
is a family of cone-manifolds $M_{t}$ interpolating from the complete metric
on $N=M_{0}$ to the complete metric on $N(s_{1},\ldots,s_{n})=M_{(2\pi)^{2}}$.
See Theorem 5.17.

Neumann and Zagier [48, Proposition 4.3] showed that (asymptotically, for very
long fillings) the normalized length of a slope $s_{j}\subset N$ closely
predicts the length of the corresponding core curve in the filled manifold
$N(s_{1},\ldots,s_{n})$. Using the work of Hodgson–Kerckhoff [33, 34] and
Magid [43], we make this relationship completely quantitative.

######  Corollary 6.13.

Suppose that $M$ is a complete, finite volume hyperbolic $3$–manifold and
$\Sigma\subset M$ is a geodesic link such that one of the following hypotheses
holds.

  1. (1)

In the complete structure on $N=M-\Sigma$, the total normalized length of the
meridians of $\Sigma$ is $L\geq 7.823$.

  2. (2)

In the complete structure on $M$, each component of $\Sigma$ has length at
most $0.0996$ and the total length of $\Sigma$ is $\ell\leq 0.1396$.

Then

$\frac{2\pi}{L^{2}+16.17}<\ell<\frac{2\pi}{L^{2}-28.78}.$

Using an estimate closely related to Corollary 6.13, we can prove an analogue
of Corollary 8.16 with hypotheses on the cusped manifold $N=M-\Sigma$.

######  Corollary 8.17.

Fix $0<\delta\leq 0.938$ and $J>1$. Let $M$ be a complete, finite volume
hyperbolic $3$–manifold and $\Sigma$ a geodesic link in $M$. Suppose that in
the complete structure on $N=M-\Sigma$, the total normalized length $L$ of the
meridians of $\Sigma$ satisfies

$L^{2}\geq\max\left\\{\frac{107.6}{\delta^{2}}+14.41,\,\frac{45.20}{\delta^{5/2}\log(J)}+14.41\right\\}.$

Let $W\subset M$ be any submanifold such that $W\subset M_{t}^{\geq\delta}$
for all $t$. Then, for all $a,b\in[0,(2\pi)^{2}]$, the identity map
${\mathop{id}}\colon(W,g_{a})\to(W,g_{b})$ is $J$–bilipschitz.

Now, to derive Theorem 1.2 from Corollary 8.16, we need a way to ensure (using
only hypotheses on $M$ or only hypotheses on $N=M-\Sigma$) that a given
submanifold $W$ remains in the thick part of a cone-manifold throughout a cone
deformation. We do so via the following result.

######  Theorem 9.15.

Fix $0<\epsilon\leq\log 3$ and $1<J\leq e^{1/5}$. Let $M$ be a complete,
finite volume hyperbolic $3$–manifold and $\Sigma\subset M$ a geodesic link.
Suppose that $\ell=\operatorname{len}(\Sigma)$ is bounded as follows:

$\ell\leq\frac{\epsilon^{5}\log J}{496.1\,J^{5}\cosh^{5}(J\epsilon/2+0.1475)}$

Then, for every $a,t\in[0,(2\pi)^{2}]$, the manifolds $M_{a}$ and $M_{t}$ in
the deformation from $M-\Sigma$ to $M$ satisfy

$M_{a}^{\geq\epsilon}\subset M_{t}^{>\epsilon/J}.$

The proof of Theorem 9.15 combines our previous work [26] with a close
analogue of Corollary 8.16 (which provides stronger estimates under stronger
hypotheses) to control the injectivity radius at a point $x\in M_{t}$ for a
sub-interval of time. Then, it uses a delicate _crawling argument_ (a
continuous analogue of induction) to show that this sub-interval must be the
entire time interval $[0,(2\pi)^{2}]$.

Setting $J_{0}=1.2<e^{1/5}$ in Theorem 9.15, we learn that under appropriate
hypotheses on $\ell$, the containment $M^{\geq\epsilon}\subset
M_{t}^{\geq\epsilon/1.2}$ holds for all $t$. Thus, on the submanifold
$W=M^{\geq\epsilon}$, Corollary 8.16 gives bilipschitz control for all $t$.
This proves Theorem 1.2.

A similar crawling argument, using the analytic estimates of Section 6, also
proves Corollary 7.20. See also Corollary 7.24 for a very similar statement
with hypotheses on $N=M-\Sigma$ rather than $M$.

###  1.4. Application to Margulis numbers

Observe that Brock and Bromberg’s Theorem 1.1 requires the thickness constant
$\epsilon$ to be less than the Margulis constant $\mu_{3}$, whose value is
currently unknown. (See Theorem 1.5 for the current state of knowledge.) By
contrast, Theorem 1.2 makes no hypotheses regarding the Margulis constant. In
fact, information flows in the opposite direction: we are able to use
Corollary 8.16 and Theorem 9.15 to control the topology of the thin parts of
the cone-manifolds $M_{t}$ occurring during the deformation. This provides a
strong application to Margulis numbers of (complete, non-singular) hyperbolic
manifolds.

######  Definition 1.4.

Let $M$ be a hyperbolic $3$–manifold. The _$\epsilon$ –thin part of $M$_ is
$M^{<\epsilon}$, which consists of all points of $M$ lying on essential loops
of length less than $\epsilon$. We say that $\epsilon>0$ is _a Margulis
number_ for $M$ if every component of $M^{<\epsilon}$ is isometric to either a
horocusp or an equidistant tube about a closed geodesic.

Note that that if $\epsilon$ is a Margulis number for $M$, then so is every
$\delta<\epsilon$, although $M^{<\delta}$ may have fewer components than
$M^{<\epsilon}$. The _optimal Margulis number_ of $M$ is

$\mu(M)=\sup\\{\epsilon:\epsilon\text{ is a Margulis number for }M\\}.$

The ($3$–dimensional) _Margulis constant_ is

$\mu_{3}=\inf\\{\mu(M):M\text{ is a hyperbolic $3$--manifold}\\}.$

Margulis proved that $\mu_{3}>0$, but the exact value is unknown. The
following theorem summarizes current knowledge about Margulis numbers and the
Margulis constant.

######  Theorem 1.5.

Suppose that $M$ is a non-singular hyperbolic $3$–manifold.

  1. (1)

$\mu(M)\geq 0.104$ for every $M$, hence $\mu_{3}\geq 0.104$.

  2. (2)

$\mu(M_{W})\leq 0.776$ for the Weeks manifold $M_{W}$, hence $\mu_{3}\leq
0.776$.

  3. (3)

$\mu(M)\geq\log 3=1.098\ldots$ for every $M$ that has infinite volume, and for
every $M$ such that $\dim H_{1}(M,{\mathbb{Q}})\geq 3$.

  4. (4)

$\mu(M)\geq 0.292$ for every $M$ such that $\dim H_{1}(M,{\mathbb{Q}})\geq 1$.
This includes all non-closed hyperbolic $3$–manifolds.

  5. (5)

$\mu(M)\geq 0.29$ for every $M$ with $\operatorname{vol}(M)\geq 52.78$.

  6. (6)

$\mu(M)\geq 0.29$ for all but finitely many hyperbolic $3$–manifolds $M$.

###### Proof by references.

Conclusion (1) is a theorem of Meyerhoff [44, Section 9]. Conclusion (2) is
the result of a computation by Yarmola.

Conclusion (3) is a consequence of the “$\log 3$ theorem” of Culler and Shalen
[19, Theorem 9.1], combined with the Tameness and Density theorems for
Kleinian groups [4, 17, 47, 50]. See Shalen [57, Proposition 3.12] for the
derivation.

Conclusion (4) is a theorem of Culler and Shalen [20]. Conclusion (5) is a
special case of a theorem of Shalen [57, Theorem 7.1], substituting
$\lambda=0.29$. Finally, (6) is a theorem of Shalen [56], proved using (4) and
an algebraic limit argument. ∎

Observe that Theorem 1.5.(6) is an ineffective statement: the algebraic limit
argument does not give any way to find the finite list of manifolds with
$\mu(M)<0.29$. On the other hand, combining Theorem 1.5.(5) with our results
gives the following effective theorem. In this theorem,
${\operatorname{sys}}(M)$ denotes the _systole_ of $M$, namely the length of
the shortest closed geodesic in $M$.

######  Theorem 9.29.

Let $M$ be a non-singular hyperbolic $3$–manifold.

  1. (1)

If $\mu(M)\leq 0.2408$, then $M$ is closed and $\operatorname{vol}(M)\leq
36.12$. Furthermore, ${\operatorname{sys}}(M)\geq 2.93\times 10^{-7}$.

  2. (2)

If $\mu(M)\leq 0.29$, then $M$ is closed and $\operatorname{vol}(M)\leq
52.78$. Furthermore, ${\operatorname{sys}}(M)\geq 2.73\times 10^{-8}$.

  3. (3)

If $\mu(M)\leq 0.9536$, then $M$ has finite volume and $k\in\\{0,1,2\\}$
cusps. The $(3-k)$ shortest geodesics in $M$ have total length at least
$5.561\times 10^{-5}$.

We emphasize that parts (1) and (2) of the above statement are completely
effective, because there exist algorithms to produce the finite list of
manifolds with volume bounded above and systole bounded below by the given
numbers. See Kobayashi and Rieck [40] for the details.

The proof of Theorem 9.29 uses many of the ingredients that were mentioned
above. We will prove the contrapositive. For example, in the second case,
suppose $M$ contains a geodesic $\sigma$ whose length is $\ell<2.73\times
10^{-8}$. Then there is a cone-deformation between $M$ and $N=M-\sigma$.
Theorem 9.15 says that for every $t$, the thin part $M_{t}^{<0.29}$ must be
contained in the thin part $N^{<0.292}$, which is a union of tubes and cusps
by Theorem 1.5.(4). Then, a somewhat delicate argument using immersed tubes
(see Theorem 9.1) shows that $M_{t}^{<0.29}$ is also a union of tubes and
cusps. In particular, $\mu(M)\geq 0.29$.

By a similar argument, we can show that long Dehn fillings of a cusped
$3$–manifold $N$ have Margulis numbers similar to those of $N$.

######  Theorem 9.25.

Fix $0<\epsilon\leq\log 3$ and $1<J\leq e^{1/5}$. Let $N$ be a cusped
hyperbolic $3$–manifold such that $\epsilon$ is a Margulis number of $N$. Let
$\mathbf{s}$ be a tuple of slopes on cusps of $N$ whose normalized length
$L=L(\mathbf{s})$ satisfies

$L(\mathbf{s})^{2}\geq\frac{2\pi\cdot
496.1\,J^{5}\cosh^{5}(J\epsilon/2+0.1475)}{\epsilon^{5}\log J}+11.7.$

Then $\delta=\min\\{\epsilon/J,\,0.962\\}$ is a Margulis number for
$M=N(\mathbf{s})$.

###  1.5. Application to cosmetic surgeries

The next application of our results is topological: we control cosmetic
surgeries on $3$–manifolds.

######  Definition 1.6.

Let $N$ be a compact oriented $3$–manifold whose boundary is a single torus.
Let $s_{1},s_{2}$ be distinct slopes on $\partial N$. We call $(s_{1},s_{2})$
a _cosmetic surgery pair_ if there is a homeomorphism $\varphi\colon
N(s_{1})\to N(s_{2})$. The pair is called _chirally cosmetic_ if $\varphi$ is
orientation-reversing, and is called _purely cosmetic_ if $\varphi$ is
orientation-preserving.

There are many examples of chirally cosmetic surgeries where $N$ is Seifert
fibered. See Bleiler–Hodgson–Weeks [6] for a survey, and Ni–Wu [49] for more
examples. There is also one known example of a chirally cosmetic surgery pair
where $N$ and $N(s_{i})$ are hyperbolic, discovered by Ichihara and Jong [36].
By contrast, no purely cosmetic surgeries are known apart from the case where
$N$ is a solid torus. This has led Gordon [29] to propose

######  Conjecture 1.7 (Cosmetic surgery conjecture).

Let $N$ be a compact, oriented $3$–manifold such that $\partial N$ is an
incompressible torus. If $s_{1},s_{2}$ are a purely cosmetic pair of slopes on
$\partial N$, then $s_{1}=s_{2}$.

A well-known classical argument, recorded by Bleiler, Hodgson, and Weeks [6],
implies that Conjecture 1.7 holds for long fillings on a hyperbolic manifold.

######  Theorem 1.8 (Bleiler–Hodgson–Weeks [6]).

Let $N$ be a one-cusped hyperbolic $3$–manifold. Then there is a number $E>0$,
such that Conjecture 1.7 holds for all pairs of slopes longer than $E$.

This useful but ineffective result is a fairly direct application of
Thurston’s Dehn surgery theorem and Mostow rigidity. By contrast, we prove the
following effective result.

######  Theorem 7.30.

Let $N$ be a one-cusped hyperbolic $3$–manifold. Suppose that $s_{1}$ and
$s_{2}$ are distinct slopes, such that the normalized length of each $s_{i}$
satisfies

$L(s_{i})\geq\max\left\\{10.1,\sqrt{\frac{2\pi}{{\operatorname{sys}}(N)}+58}\right\\}.$

Then $(s_{1},s_{2})$ cannot be a purely cosmetic pair. If $(s_{1},s_{2})$ is a
chirally cosmetic pair, then there is a homeomorphism of $N$ sending $s_{1}$
to $s_{2}$. In particular, Conjecture 1.7 holds if
${\operatorname{sys}}(N)\geq 0.1428$ and $L(s_{i})\geq 10.1$ for $i=1,2$.

In fact, Theorem 7.30 is a special case of a theorem that also holds for
tuples of slopes. Our result addresses the following generalization of
Conjecture 1.7.

######  Conjecture 1.9 (Hyperbolic cosmetic surgery conjecture).

Let $N$ be a finite-volume hyperbolic $3$–manifold with one or more cusps. Let
$\mathbf{s}_{1}$ and $\mathbf{s}_{2}$ be tuples of slopes on the cusps of $N$.
If there is an orientation-preserving homeomorphism $\varphi\colon
N(\mathbf{s}_{1})\to N(\mathbf{s}_{2})$, and this manifold is hyperbolic, then
$\varphi$ restricts (after an isotopy) to a homeomorphism $N\to N$ sending
$\mathbf{s}_{1}$ to $\mathbf{s}_{2}$.

Compare Kirby [39, Problem 1.81(B)] and Jeon [38, Section 1.1 and Theorem 1.6]
for related statements. The above-mentioned result of Ichihara and Jong [36]
shows that restricting to purely cosmetic surgeries is necessary, even when
$N$ has a single cusp.

We prove Conjecture 1.9 for sufficiently long tuples of slopes, where “long”
is explicitly quantified. For these long tuples of slopes, the only purely or
chirally cosmetic surgeries come from symmetries of $N$ itself.

######  Theorem 7.29.

Let $N$ be a hyperbolic $3$–manifold with cusps. Suppose that
$\mathbf{s}_{1},\mathbf{s}_{2}$ are distinct tuples of slopes on the cusps of
$N$, whose normalized length satisfies

$L(\mathbf{s}_{i})\geq\max\left\\{10.1,\,\sqrt{\frac{2\pi}{{\operatorname{sys}}(N)}+58}\right\\}.$

Then any homeomorphism $\varphi\colon N(\mathbf{s}_{1})\to N(\mathbf{s}_{2})$
restricts (after an isotopy) to a self-homeomorphism of $N$ sending
$\mathbf{s}_{1}$ to $\mathbf{s}_{2}$.

Given any lower bound on the systole of $N$, Theorems 7.29 and 7.30 provide an
effective estimate on the normalized length after which the cosmetic surgery
conjecture holds. However, even if $N$ has just one cusp, there could
hypothetically be infinitely many purely cosmetic pairs, where $s_{1}$ is
short but $s_{2}$ is arbitrarily long. This possibility is ruled out in
Theorem 1.13 below. Before setting up that result, we treat the special case
where $N$ is the complement of a knot in $S^{3}$. The following result follows
by combining Theorem 7.30 with the work of Ni and Wu [49].

######  Corollary 1.10.

Let $K\subset S^{3}$ be a hyperbolic knot. Let $\mu,\lambda$ be the meridian
and longitude of $K$. Suppose that $s,s^{\prime}$ are a purely cosmetic pair
on the cusp of $S^{3}-K$. Then, after possibly swapping $s$ and $s^{\prime}$,
the following holds.

  1. (1)

$\displaystyle{L(s)<\max\left\\{10.1,\,\sqrt{\frac{2\pi}{{\operatorname{sys}}(S^{3}-K)}+58}\right\\}.}$

  2. (2)

If $s=p\mu+q\lambda$, then $s^{\prime}=p\mu-q\lambda$, and furthermore $p$
divides $q^{2}+1$.

###### Proof.

Conclusion (1) is a restatement of Theorem 7.30. Meanwhile, Conclusion (2) is
part of the statement of [49, Theorem 1.2]. ∎

To verify Conjecture 1.7 for a given manifold $N=S^{3}-K$, it suffices to
check the finitely many pairs $(s,-s)$ where $L(s)$ satisfies (1). This is a
practical computational task [25]. In practice, the vast majority of knot
complements enumerated by Hoste, Thistlethwaite, and Weeks [35] have systole
greater than $0.15$, which means that the normalized length cutoff in the
corollary is $10.1$. By work of Agol [2, Lemma 8.2], there are at most 104
slopes on $\partial N$ of normalized length less than $10.1$. Among those
short slopes, there are typically at most $8$ slopes that have the form
$s=p\mu+q\lambda$ where $p$ divides $q^{2}+1$. Thus checking the cosmetic
surgery conjecture for a typical knot $K$ amounts to distinguishing $8$ or
fewer pairs of closed manifolds. We ran a computer program to show the
following:

######  Corollary 1.11.

The cosmetic surgery Conjecture 1.7 holds for all prime knots with at most 16
crossings.

Contemporaneously with our work, Hanselman has obtained an independent proof
of Corollary 1.11 [30]. To do this, he proved a finiteness theorem in the same
spirit as Corollary 1.10, constraining the slopes to check to a short (and
frequently empty) list that depends on the knot genus $g(K)$ and the thickness
of the knot Floer homology of $K$. For knots up to $16$ crossings, his
criterion only requires checking slopes $\pm 1$ and $\pm 2$ for 337 knots.

Very recently, Detcherry discovered a criterion on the Jones polynomial at the
fifth root of unity that severely constrains the slopes involved in cosmetic
surgeries [21]. By combining his criterion with Hanselman’s results, he
verified Conjecture 1.7 for knots up to 17 crossings.

###### Sketch proof of Corollary 1.11.

For each of the 1,701,935 prime knots with at most 16 crossings, we begin by
computing the (symmetrized) Alexander polynomial $\Delta_{K}(t)$ and its
second derivative. By a theorem of Boyer and Lines [9, Proposition 5.1], any
knot $K$ such that $\Delta_{K}^{\prime\prime}(1)\neq 0$ has no purely cosmetic
surgery. This criterion eliminates 1,513,776 knots, roughly $89\%$ of the
total.

For the remaining knots, we compute the Jones polynomial $V_{K}(t)$ and its
third derivative. By a theorem of Ichihara and Wu [37, Theorem 1.1], any knot
$K$ such that $V_{K}^{\prime\prime\prime}(1)\neq 0$ has no purely cosmetic
surgery. This criterion eliminates another 152,740 knots, roughly $9\%$ of the
total.

The remaining 35,419 knots are all hyperbolic. For each knot $K$ on the
remaining list, we compute
${\operatorname{sys}}(N)={\operatorname{sys}}(S^{3}-K)$ and calculate the list
of short slopes satisfying the conclusion of Corollary 1.10. For each short
slope $s$, we compute the hyperbolic structures on $N(s)$ and $N(-s)$ using
SnapPy and compare the verified volume and verified Chern–Simons invariants.
In each case, these invariants distinguish the pair. Code will be included
with [25]. ∎

Returning to the setting of a general one-cusped hyperbolic manifold, we
describe a practical finiteness theorem that can be used to verify whether $N$
has any cosmetic surgeries at all. We need the following definition.

######  Definition 1.12.

Let $N$ be a one-cusped hyperbolic $3$–manifold. Define the finite set of
slopes

$\mathcal{S}_{1}(N)=\left\\{s\>\bigg{|}\>L(s)<\max\left(10.1,\,\sqrt{\frac{2\pi}{{\operatorname{sys}}(N)}+58}\right)\right\\}.$

Next, define

$V(N)=\max\big{\\{}\operatorname{vol}(N(s))\mid
s\in\mathcal{S}_{1}(N)\big{\\}}.$

Here, we employ the convention is that the volume of any non-hyperbolic
manifold is $0$, hence will not realize the maximum. Using the theorem of
Gromov and Thurston [59, Theorem 6.5.6] that $V(N)<\operatorname{vol}(N)$,
define the finite set of slopes

$\mathcal{S}_{2}(N)=\left\\{s\>\bigg{|}\>\operatorname{len}(s)\leq
2\pi\left(1-\left(\frac{V(N)}{\operatorname{vol}(N)}\right)^{2/3}\right)^{-1/2}\right\\}.$

Here, $\operatorname{len}(s)$ is ordinary Euclidean length on the boundary of
the maximal cusp in $N$.

######  Theorem 1.13.

Let $N$ be a one-cusped hyperbolic $3$–manifold. Then each of Conjecture 1.7
and Conjecture 1.9 holds for $N$ if and only if it holds for all pairs of
slopes in the finite set $\mathcal{S}_{1}(N)\times\mathcal{S}_{2}(N)$.

###### Proof.

The “only if” direction is obvious. For the “if” direction, suppose that
$(s_{1},s_{2})$ are a (purely or chirally) cosmetic pair for $N$. Assume,
without loss of generality, that $L(s_{1})\leq L(s_{2})$. Then Theorem 7.30
implies $s_{1}\in\mathcal{S}_{1}(N)$. With $V(N)$ as above, Futer,
Kalfagianni, and Purcell proved [22, Theorem 1.1] that if some Dehn filling
$N(s)$ satisfies $\operatorname{vol}(N(s))\leq V(N)$, then
$s\in\mathcal{S}_{2}(N)$. Thus any potential counterexample to Conjectures 1.7
or 1.9 must lie in $\mathcal{S}_{1}(N)\times\mathcal{S}_{2}(N)$. ∎

For manifolds with reasonable systole, the set
$\mathcal{S}_{1}(N)\times\mathcal{S}_{2}(N)$ is practical to compute using
SnapPy, and not too large in size. In ongoing work [25], we use Theorem 1.13
to verify Conjecture 1.7 for a large subset of the SnapPy census.

###  1.6. Organization

In Section 2, we review background on cone-manifolds and their properties.
Section 3 reviews a number of results on tubes in cone-manifolds, as well as
distances between nested tubes; many of these results were proved in [26]. In
Section 4, we control the areas of embedded multi-tubes, analogous to similar
results in [33, 34], which will ensure that cone deformations exist. We also
prove Theorem 4.26, which controls the injectivity radius on the tube boundary
and may be of independent interest.

The above results are combined in Section 5 to produce cone deformations that
maintain a large embedded tube about the singular locus $\Sigma$. The
technical Section 6 presents results (phrased as bounds on so-called _boundary
terms_) that will be needed in future sections to control the change in
geometry during the cone deformation.

Section 7 contains the first pair of our main results: Theorem 7.19 and
Theorem 7.21, which bound the change in length of short geodesics under cone
deformation. At the end of Section 7, we apply these results to the cosmetic
surgery conjecture, proving Theorems 7.29 and 7.30.

Section 8 proves Theorem 8.3, which provides effective bilipschitz bounds on
submanifolds of $M$ that stay thick throughout the cone deformation.

Section 9 contains results related to the thick-thin decomposition and
Margulis numbers. A main result, Theorem 9.15, ensures that submanifolds of
$M$ stay thick throughout the deformation. This result is applied to show
Theorem 9.29 about Margulis numbers, as well as Theorem 9.30, which provides
bilipschitz bounds without any hypotheses on cone-manifolds.

Finally, there is a short appendix, on hyperbolic trigonometry, that we use
throughout the paper.

###  1.7. Acknowledgements

As should be clear, this paper owes an enormous debt to the ideas of Jeff
Brock, Ken Bromberg, Craig Hodgson, and Steve Kerckhoff. We thank all four of
them for sharing their insights and explaining many technical points.

We thank Steve Boyer, Jonathan Hanselman, Kazuhiro Ichihara, Tye Lidman, Tim
Morris, and Matthew Stover for many enlightening suggestions regarding
cosmetic surgeries. We thank Marc Culler, David Gabai, Peter Shalen, and
Andrew Yarmola for helpful input about Margulis numbers. Finally, we thank the
intrepid referee for carefully reading the paper and making a number of
suggestions that improved our exposition.

During this project, Futer was supported by the NSF and the Simons Foundation.
Purcell was supported by the NSF and the ARC. Schleimer was supported by
EPSRC. All three authors were able to gather in a common location on a few
occasions, thanks to support from MSRI and the Institute for Advanced Study,
as well as support from NSF grants DMS–1107452, 1107263, 1107367, “RNMS:
Geometric Structures and Representation Varieties” (the GEAR Network).

##  2\. Cone-manifold basics

In this section we set up notation and definitions about cone-manifolds,
geodesics, and injectivity radii.

######  Definition 2.1.

Let $\sigma\subset{\mathbb{H}}^{3}$ be a bi-infinite geodesic. Let
${\hat{{\mathbb{H}}}^{3}}$ denote the metric completion of the universal cover
of $({\mathbb{H}}^{3}-\sigma)$. Let $\hat{\sigma}$ be the set of points added
in the completion.

The space ${\hat{{\mathbb{H}}}^{3}}$ can be regarded as an infinite cyclic
branched cover of ${\mathbb{H}}^{3}$, branched over $\sigma$. The branch set
$\hat{\sigma}\subset{\hat{{\mathbb{H}}}^{3}}$ is a singular geodesic of cone
angle $\infty$.

There is a natural action of ${\mathbb{C}}$ (thought of as an additive group)
on ${\hat{{\mathbb{H}}}^{3}}$, where $z=\zeta+i\theta\in{\mathbb{C}}$
translates $\hat{\sigma}$ by distance $\zeta$ and rotates by angle $\theta$.
Since $\hat{\sigma}$ has cone angle $\infty$, angles of rotation are indeed
real-valued. Conversely, every isometry $\varphi$ of
${\hat{{\mathbb{H}}}^{3}}$ that preserves orientation on both
${\hat{{\mathbb{H}}}^{3}}$ and $\hat{\sigma}$ comes from this action, and has
a well-defined _complex length_ $z=\zeta+i\theta$. We can therefore write
$\varphi=\varphi_{\zeta+i\theta}$.

We endow ${\hat{{\mathbb{H}}}^{3}}$ with a system of cylindrical coordinates
$(r,\zeta,\theta)$, as follows. Choose a ray perpendicular to $\hat{\sigma}$,
and let the points of this ray have coordinates $(r,0,0)$, where $r\geq 0$
measures distance from $\hat{\sigma}$. Then, let $(r,\zeta,\theta)$ be the
image of $(r,0,0)$ under the isometry $\varphi_{\zeta+i\theta}$. The distance
element in these coordinates is

(2.2) $ds^{2}=dr^{2}+\cosh^{2}r\,d\zeta^{2}+\sinh^{2}r\,d\theta^{2}.$

######  Definition 2.3.

Consider a group $G={\mathbb{Z}}\times{\mathbb{Z}}$ of isometries of
${\hat{{\mathbb{H}}}^{3}}$, generated by an elliptic $\psi_{i\alpha}$ and a
loxodromic $\varphi=\varphi_{\lambda+i\tau}$, where $\alpha>0$ and
$\lambda>0$. The quotient space $N_{\alpha,\lambda,\tau}$ is an open solid
torus whose core curve is a closed geodesic of length $\lambda$, and with a
cone singularity of angle $\alpha$ at the core. We call
$N=N_{\alpha,\lambda,\tau}$ a _model solid torus_.

For $r>0$, a _model tube_ $U_{\alpha,\lambda,\tau}$ is the open
$r$–neighborhood of the core curve of $N_{\alpha,\lambda,\tau}$. We note that
the closure $\overline{U}_{\alpha,\lambda,\tau}$ is compact.

######  Definition 2.4.

A _hyperbolic cone-manifold_ $(M,\Sigma)$ is a metric space where every point
has a neighborhood isometric to a ball in a model solid torus. More precisely,
$M$ is a topological $3$–manifold and $\Sigma$ is a link in $M$, such that
every component of $\Sigma$ has a neighborhood isometric to a model tube
$U_{\alpha,\lambda,\tau}$. Meanwhile, every point $x\in M-\Sigma$ has a
neighborhood isometric to a ball in ${\mathbb{H}}^{3}$.

We allow components of $\Sigma$ to be non-singular, i.e. have cone angle
$2\pi$. When the link $\Sigma$ is clear from context, we will often suppress
it from the notation.

###  2.1. Covers and deck transformations

######  Definition 2.5.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold. Then the _universal branched
cover_ of $(M,\Sigma)$, denoted $\hat{M}$, is the metric completion of
$\widetilde{X}$, where $\widetilde{X}$ is the universal cover of $X=M-\Sigma$.
Every component of $\Sigma$ lifts to a disjoint union of singular geodesics in
$\hat{M}$, with cone angle $\infty$. Thus $\hat{M}$ is locally modeled on
${\hat{{\mathbb{H}}}^{3}}$, as in Definition 2.1. The deck transformation
group for $\hat{M}$ is isomorphic to $\pi_{1}(M-\Sigma)$.

Let $\widetilde{\sigma}\subset\hat{M}$ be a lift of $\sigma\subset\Sigma$.
Then there is a map

$D\colon\hat{M}\to{\hat{{\mathbb{H}}}^{3}},$

which shares some features of an exponential map based on the normal bundle to
a geodesic. Let $\widetilde{U}\subset\hat{M}$ be a regular $r$–neighborhood of
$\widetilde{\sigma}$, and let
$D\colon\widetilde{U}\hookrightarrow{\hat{{\mathbb{H}}}^{3}}$ be an isometric
embedding sending $\widetilde{\sigma}$ to
$\widehat{\sigma}\subset{\hat{{\mathbb{H}}}^{3}}$. Then, extend the map along
geodesic rays: if $\gamma\subset\hat{M}$ is a geodesic ray orthogonal to
$\widetilde{\sigma}$, then $D|_{\gamma}$ is an isometry to a geodesic ray
orthogonal to $\hat{\sigma}$.

In the special case where $M$ is a model solid torus with core curve $\Sigma$,
we have $\hat{M}={\hat{{\mathbb{H}}}^{3}}$ by Definition 2.3, hence $D$ is a
global isometry. On the other hand, if $(M,\Sigma)$ is non-elementary, hence a
singular geodesic $\sigma\subset M$ has multiple preimages in $\hat{M}$, the
map $D$ will fail to be even a _local_ isometry outside a neighborhood of
$\widetilde{\sigma}$. Indeed, given a geodesic segment
$\gamma_{0}\subset\hat{M}$ that runs from $\widetilde{\sigma}$ to another
singular geodesic, there is a one-parameter family of distinct geodesic rays
in $\hat{M}$ that all contain $\gamma_{0}$ and then separate; these rays will
be mapped to the unique ray in ${\hat{{\mathbb{H}}}^{3}}$ containing
$D(\gamma_{0})$.

An important property of the universal branched cover is

######  Proposition 2.6.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold. Then $\hat{M}$ is a complete
${\operatorname{CAT}(-1)}$ space.

###### Proof.

This is a consequence of the Cartan–Hadamard theorem [10, Chapter II.4,
Theorem 4.1(2)]. See Soma [58, Lemma 1.2] for the derivation. ∎

If $M$ is a finite-volume hyperbolic manifold and $\Sigma\subset M$ is a
geodesic link, Kerckhoff showed that $M-\Sigma$ admits a complete metric of
negative sectional curvature. (See Agol [3] for a summary and for more details
of the construction.) The same construction applies if $(M,\Sigma)$ is a cone-
manifold of finite volume. By Thurston’s hyperbolization, this implies
$M-\Sigma$ admits a complete hyperbolic metric.

######  Definition 2.7.

Let $G\subset{\operatorname{Isom}}^{+}({\mathbb{H}}^{3})$ be a discrete, free
abelian group of parabolic isometries of ${\mathbb{H}}^{3}$. The quotient
${\mathbb{H}}^{3}/G$ is called a _model cusp_. If $H\subset{\mathbb{H}}^{3}$
is an open horoball stabilized by $G$, the quotient $U=H/G$ is called a
_horocusp_. A horocusp or model cusp is called _rank one_ if
$G\cong{\mathbb{Z}}$ and _rank two_ if $G\cong{\mathbb{Z}}^{2}$.

######  Definition 2.8.

Let $(M,\Sigma)$ be a complete, finite volume hyperbolic cone-manifold. Let
$\varphi\in\pi_{1}(M-\Sigma)$ be a non-trivial element. We say that $\varphi$
is _peripheral_ if a loop representing $\varphi$ is freely homotopic into a
horocusp of $M-\Sigma$.

######  Lemma 2.9.

Let $(M,\Sigma)$ be a complete, finite volume hyperbolic cone-manifold, with
universal branched cover $\hat{M}$. Let $\varphi\in\pi_{1}(M-\Sigma)$ be a
non-trivial deck transformation of $\hat{M}$. Then the following are
equivalent:

  1. (1)

$\varphi$ corresponds to a peripheral homotopy class.

  2. (2)

$\varphi$ stabilizes either a horoball in $\hat{M}$ or a singular geodesic
covering a component of $\Sigma$.

  3. (3)

$\varphi$ does not act by translation on any non-singular geodesic in
$\hat{M}$.

###### Proof.

$\eqref{Itm:Peripheral}\Rightarrow\eqref{Itm:ParabolicOrSingular}$: Suppose
$\varphi$ is peripheral. Then a loop representing $\varphi$ is homotopic into
the neighborhood of some cusp of $M-\Sigma$. In the cone-metric on
$(M,\Sigma)$, this cusp of $M-\Sigma$ either stays a cusp or becomes a
neighborhood of some component $\sigma_{i}\subset\Sigma$. In the first case,
the deck transformation $\varphi$ stabilizes the universal cover of a horocusp
in $\hat{M}$. In the second case, the deck transformation $\varphi$ stabilizes
some preimage of $\sigma_{i}$ in $\hat{M}$.

$\eqref{Itm:ParabolicOrSingular}\Rightarrow\eqref{Itm:Peripheral}$: Suppose
that the deck transformation $\varphi$ stabilizes a horoball $H\subset\hat{M}$
that covers a horocusp in $(M,\Sigma)$. Then, for an appropriate choice of
basepoint, a path-lift $\widetilde{\varphi}$ of the loop $\varphi$ starts and
ends in $H$. After a free homotopy of $\varphi$, we may assume that the entire
path-lift $\widetilde{\varphi}$ lies in a sub-horoball of $H$ that covers a
horocusp of $(M,\Sigma)$, hence $\varphi$ is peripheral. The case of tubes is
similar.

$\eqref{Itm:ParabolicOrSingular}\Leftrightarrow\eqref{Itm:NotLoxodromic}$: The
deck transformation $\varphi$ must act on $\hat{M}$ by isometry. This isometry
is either _elliptic_ (meaning it has fixed points), _parabolic_ (meaning it
has no fixed points, but the infimal translation is $0$), or _hyperbolic_
(meaning that the infimal translation distance is $d>0$ and is realized). If
$\varphi$ is elliptic, then recalling that it is a deck transformation implies
that it must rotate about a singular axis; hence (2) and (3) both hold. If
$\varphi$ is parabolic, then it stabilizes a horoball, hence (2) and (3) both
hold.

Finally, suppose $\varphi$ is hyperbolic. Since $\hat{M}$ is a complete
${\operatorname{CAT}(-1)}$ space by Proposition 2.6, $\varphi$ must translate
along a unique geodesic axis. If this axis is singular, then (2) and (3) both
hold. If the geodesic is non-singular, then (2) and (3) both fail. ∎

###  2.2. Injectivity radii, cusps, and tubes

We will be looking at tubes with injectivity radius less than some value
$\epsilon>0$ or $\delta>0$. At times we will need to discuss the injectivity
radius over an entire cone-manifold. At other times we only need to consider
injectivity radius within a tube. We encapsulate these separate notions in
Definition 2.10 and Definition 2.17.

######  Definition 2.10.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold and $x\in M$. Then the
_injectivity radius_ , denoted ${\operatorname{injrad}}(x)$, is the supremal
radius $r$ such that a metric $r$–ball about $x$ is isometric to a ball
$B_{r}(y)\subset{\mathbb{H}}^{3}$. (Since we are using open balls, the
supremal radius is attained, unless $M={\mathbb{H}}^{3}$, in which case
${\operatorname{injrad}}(x)=\infty$.) If $x$ lies on the singular set of $M$,
we set ${\operatorname{injrad}}(x)=0$.

######  Lemma 2.11.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold, where every component of
$\Sigma$ is singular. Choose a point $x\in M-\Sigma$ and a lift
$\widetilde{x}\in\hat{M}$. Then ${\operatorname{injrad}}(x)$ can be
characterized as follows:

(2.12) $\displaystyle 2\,{\operatorname{injrad}}(x)$
$\displaystyle=\inf\\{\operatorname{len}(\gamma):\gamma\mbox{ is a non-trivial
loop in $M-\Sigma$ based at }x\\}$ (2.13)
$\displaystyle=\min\\{d(\widetilde{x},\varphi\widetilde{x}):1\neq\varphi\in\pi_{1}(M-\Sigma)\\}.$

Furthermore, the infimum in (2.12) is realized by a pointed geodesic, unless
$d(x,\Sigma)={\operatorname{injrad}}(x)$ and the isometry $\varphi$ in (2.13)
is elliptic.

In the case where $M$ is a non-singular hyperbolic manifold, hence
$\Sigma=\emptyset$, the result of Lemma 2.11 is well-known. In the case where
$M$ is a model solid torus, the result is contained in [26, Lemma 2.5]. Thus
Lemma 2.11 generalizes those previously known cases to general cone-manifolds.

###### Proof of Lemma 2.11.

Let $\epsilon=2{\operatorname{injrad}}_{\Sigma}(x)$. Then, for an arbitrary
$y\in{\mathbb{H}}^{3}$, there is an isometric embedding $f\colon
B_{\epsilon/2}(y)\to M$, such that $f(y)=x$. It follows that any non-trivial
loop through $x$ must have length at least $\epsilon$. Similarly, any non-
trivial element $\varphi\in\pi_{1}(M-\Sigma)$ must translate
$B_{\epsilon/2}(\widetilde{x})$ by distance at least $\epsilon$. Thus both
(2.12) and (2.13) give lower bounds on $\epsilon$.

Next, we show that these expressions give upper bounds on $\epsilon$. Since
${\operatorname{injrad}}(x)=\epsilon/2$, the continuous extension of $f$ to
$\overline{B_{\epsilon/2}(y)}$ either hits $\Sigma$ or fails to be $1$–$1$. We
consider these cases in turn.

First, suppose that the image ball $f(\overline{B_{\epsilon/2}(y)})$ has a
point of self-tangency in $M-\Sigma$. This means that two distinct lifts of
this ball, namely $B_{\epsilon/2}(\widetilde{x})$ and
$B_{\epsilon/2}(\varphi\widetilde{x})$, are tangent in $\hat{M}$, which means
that $d(\widetilde{x},\varphi\widetilde{x})=\epsilon$. The geodesic
$\widetilde{\gamma}$ connecting $\widetilde{x}$ to $\varphi\widetilde{x}$
projects to a geodesic loop $\gamma\subset M-\Sigma$ of length exactly
$\epsilon$. This means that (2.12) is an equality, hence the infimum is
realized in this case.

Next, suppose there is a point $z\in
f(\overline{B_{\epsilon/2}(y)})\cap\Sigma$. Then we construct a closed loop
$\gamma$ of length $\epsilon+\delta$, for arbitrarily small $\delta$. This
closed loop has the form of an “eyeglass”: walk from $x$ to a point near $z$,
walk around a loop of length $\delta$ about $\Sigma$, and then return to back
to $x$. Thus (2.12) is an upper bound on $\epsilon$. The homotopy class
$[\gamma]\in\pi_{1}(M-\Sigma,x)$ corresponds to an elliptic isometry of
$\hat{M}$, which fixes a lift $\widetilde{z}$ of $z$. This elliptic isometry
must move the ball $B_{\epsilon/2}(\widetilde{x})$ to a disjoint ball
$B_{\epsilon/2}(\varphi\widetilde{x})$, with the two balls tangent at
$\widetilde{z}$. Thus $d(\widetilde{x},\varphi\widetilde{x})=\epsilon$. ∎

######  Definition 2.14.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold. For $\epsilon>0$, the
_$\epsilon$ –thick part_ of $M$ is

$M^{\geq\epsilon}=\\{x\in M:{\operatorname{injrad}}(x)\geq\epsilon/2\\}.$

The _$\epsilon$ –thin part_ is $M^{<\epsilon}=M-M^{\geq\epsilon}$. We define
$M^{\leq\epsilon}$ and $M^{>\epsilon}$ similarly.

We emphasize that our definition of the _$\epsilon$ –thick part_ corresponds
to injectivity radius $\epsilon/2$ (hence, translation length $\epsilon$)
rather than injectivity radius $\epsilon$. Both choices seem to be common in
the literature on Kleinian groups. Our convention agrees with that of Minsky
[45, 46] and Brock–Canary–Minsky [13], while differing from the convention of
Brock–Bromberg [11] and Namazi–Souto [47].

######  Definition 2.15.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold. We say that $\epsilon>0$ is a
_Margulis number_ for $M$ if every component of the $\epsilon$–thin part
$M^{<\epsilon}$ is isometric to either a model tube (Definition 2.3) or a
horocusp (Definition 2.7). The _optimal Margulis number_ $\mu(M)$ is the
supremum of all Margulis numbers for $M$.

In Theorem 9.27, we will prove an effective Margulis lemma for cone-manifolds:
$0.29$ is a Margulis number for all cone-manifolds satisfying certain
hypotheses. See also Theorem 9.26.

In addition to studying embedded tubes and cusps in a cone-manifold $M$, we
will study their immersed analogues.

######  Definition 2.16.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold. An _immersed tube in $M$_ is a
local isometry $f\colon U\to M$, where $U$ is a model tube, and furthermore
$f^{-1}(\Sigma)$ is either the core of $U$ or $\emptyset$. (The “furthermore”
condition is automatic when each component of $\Sigma$ is singular.)
Similarly, an _immersed horocusp in $M$_ is a local isometry $f\colon U\to M$,
where $U$ is a horocusp.

If $f$ is an embedding, we refer to the image $f(U)$ as an _embedded tube_ in
$M$. In this case, we will often conflate the domain $U$ with the image
$f(U)$.

######  Definition 2.17.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold. Let $U\subset M$ be an
embedded tube or horocusp in $M$. Let $\pi\colon\hat{U}\to U$ be the universal
covering map. Thus $\pi$ is an ordinary cover with deck group
$G\cong{\mathbb{Z}}$ if $U\cap\Sigma=\emptyset$, and a branched cover with
deck group $G\cong{\mathbb{Z}}\times{\mathbb{Z}}$ if
$U\cap\Sigma\neq\emptyset$. If $U$ is a horocusp, then $\hat{U}$ is the usual
universal cover with deck group $G\in\\{{\mathbb{Z}},{\mathbb{Z}}^{2}\\}$.

Let $x\in U$. Let $\hat{x}$ be a lift of $x$ in $\hat{U}$, and consider all
translates of $\hat{x}$ under the action of $G$. Define

(2.18)
${\operatorname{injrad}}(x,U)=\frac{1}{2}\min\\{d(\hat{x},\varphi(\hat{x})):1\neq\varphi\in
G\\}.$

For $x\in\partial U$, we may define ${\operatorname{injrad}}(x,U)$ by
extending (2.18) by continuity.

When every component of $\Sigma$ is singular, the injectivity radius is well-
behaved under immersions of tubes or cusps.

######  Lemma 2.19.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold, where every component of
$\Sigma$ is presumed to be singular. Let $f\colon U\to M$ be an immersed tube
or horocusp. Then, for all $x\in\overline{U}$,

${\operatorname{injrad}}(f(x))\leq{\operatorname{injrad}}(x,U).$

###### Proof.

By continuity, it suffices to assume that $x\in U$. As in Definition 2.17, let
$\hat{U}$ be the universal cover of $U$, which is branched if $U$ is singular.
Let $G$ be the group of deck transformations of $\hat{U}$.

The local isometry $f\colon U\to M$ gives an elevation, a local isometry
$\hat{f}\colon\hat{U}\to\hat{M}$. We claim that $\hat{f}$ must be one-to-one:
since $\hat{U}$ is convex, any pair of points are connected by a geodesic
segment. The image of this segment is a geodesic segment in $\hat{M}$, which
necessarily has distinct endpoints by Proposition 2.6. Since $\hat{f}$ is one-
to-one, we get an inclusion $f_{*}\colon G\hookrightarrow\pi_{1}(M-\Sigma)$.
(Compare Baker and Cooper [5, Propositions 2.1 and 2.2].) Thus the minimum in
(2.18) is taken over a smaller set than the minimum in (2.13), hence
${\operatorname{injrad}}(f(x))\leq{\operatorname{injrad}}(x,U)$. ∎

##  3\. Distance estimates in tubes and cusps

This section contains several estimates about tubes and cusps that will be
needed in subsequent arguments. Most of the results listed here are proved in
[26]. We begin with a general estimate that applies to all cone-manifolds.

######  Lemma 3.1.

Let $M$ be a hyperbolic cone-manifold. Let $x,y$ be points of $M$ such that
$2{\operatorname{injrad}}(x)=\delta>0$ and
$2{\operatorname{injrad}}(y)=\epsilon>\delta$. Then

$d(x,y)\geq\frac{\epsilon-\delta}{2}.$

###### Proof.

This was observed in [26, Lemma 5.1] in the case where $M$ is a model tube.
The same proof works in general.

Let $h=d(x,y)$. If $h\geq\epsilon/2$, there is nothing to prove. Thus we may
assume that $h<\epsilon/2$. By Definition 2.10, there is an embedded ball
$B=B_{\epsilon/2}(y)$ that is isometric to a ball in ${\mathbb{H}}^{3}$. Since
$h<\epsilon/2$, we have $x\in B$. By the triangle inequality, there is an
embedded ball $B_{\epsilon/2-h}(x)$ contained in $B$, implying

${\operatorname{injrad}}(x)=\delta/2\geq\epsilon/2-h.\qed$

###  3.1. Tube radii

######  Definition 3.2.

Let $N=N_{\alpha,\lambda,\tau}$ be a model solid torus, as in Definition 2.3.
For $\epsilon>\lambda$, let $U^{\epsilon}$ be a component of $N^{<\epsilon}$.
Then $U^{\epsilon}$ is a tube about a core geodesic $\gamma$, and
$T^{\epsilon}=\partial U^{\epsilon}$ is a torus consisting of points whose
injectivity radius is exactly $\epsilon/2$. All of the points of
$T^{\epsilon}$ lie at the same radius from $\gamma$. We denote this radius

$r(\epsilon)=r_{\alpha,\lambda,\tau}(\epsilon).$

We let $T_{r}$ denote the equidistant torus at radius $r$ from the core of
$N$. Subscripts denote radius, while superscripts denote thinness. Thus

$T^{\epsilon}=T_{r(\epsilon)}.$

If $N$ is a model solid torus, modeling a neighborhood of $\sigma_{i}$, a
component of $\Sigma$ in $M$, we will often write $r(\epsilon,\sigma_{i})$,
$T^{\epsilon}(\sigma_{i})$, $U^{\epsilon}(\sigma_{i})$ to refer to the radius,
equidistant torus and tubes about a particular component of $\Sigma$.

######  Lemma 3.3.

Let $T^{\epsilon}=\partial U^{\epsilon}$, where $U^{\epsilon}$ is a tube about
a singular geodesic $\sigma$ with cone angle $\alpha<2\pi$. Then

$\operatorname{area}(T^{\epsilon})\geq\frac{\sqrt{3}}{2}\epsilon^{2}.$

###### Proof.

This follows by a standard packing argument, because $T^{\epsilon}$ contains
an embedded disk of radius $\epsilon/2$. See [26, Equation (7.4)]. Compare
Proposition 4.3 and Theorem 4.16 for a much more involved packing argument. ∎

######  Lemma 3.4.

Let $N=N_{\alpha,\lambda,\tau}$ be a model solid torus whose core has cone
angle $\alpha<2\pi$. Then

$\sinh
2r_{\alpha,\lambda,\tau}(\epsilon)\>\geq\>\frac{\sqrt{3}\,\epsilon^{2}}{\alpha\lambda}\>>\>\frac{\sqrt{3}\,\epsilon^{2}}{2\pi\lambda}.$

###### Proof.

This follows from Lemma 3.3 and (2.2). See also [26, Lemma 7.2]. ∎

###  3.2. Distances between tori of fixed injectivity radius

######  Definition 3.5.

Let $0<\delta\leq\epsilon$. For a model solid torus
$N=N_{\alpha,\lambda,\tau}$ with $\lambda\leq\delta$, the distance between
equidistant tori $T^{\delta}=T_{r(\delta)}$ and $T^{\epsilon}=T_{r(\epsilon)}$
is

$d_{\alpha,\lambda,\tau}(\delta,\epsilon)=d(N^{\leq\delta},N^{\geq\epsilon})=r_{\alpha,\lambda,\tau}(\epsilon)-r_{\alpha,\lambda,\tau}(\delta).$

For a model horocusp $N$, we similarly define

$d_{N}(\delta,\epsilon)=d(N^{\leq\delta},N^{\geq\epsilon}).$

If $N$ is a tube, the distance $d_{\alpha,\lambda,\tau}(\delta,\epsilon)$
depends on the parameters of the tube. Nevertheless, we have upper and lower
bounds on $d_{\alpha,\lambda,\tau}(\delta,\epsilon)$ that hold independent of
the parameters $\alpha,\lambda,\tau$.

######  Theorem 3.6.

Suppose that $0<\delta<\epsilon\leq\log 3$. Let $N=N_{\alpha,\lambda,\tau}$ be
a model solid torus with cone angle $\alpha\leq 2\pi$ and core geodesic of
length $\lambda\leq\delta$, or a model horocusp whose $\epsilon$–thick part is
not empty. Then

$\max\left\\{\frac{\epsilon-\delta}{2},\>\operatorname{arccosh}\left(\frac{\epsilon}{\sqrt{7.256\,\delta}}\right)-0.1475\right\\}\leq
d_{\alpha,\lambda,\tau}(\delta,\epsilon)\leq\operatorname{arccosh}\sqrt{\frac{\cosh\epsilon-1}{\cosh\delta-1}}.$

We remark that the argument of $\operatorname{arccosh}$ in the lower bound of
Theorem 3.6 may be less than $1$, hence $\operatorname{arccosh}(\cdot)$ is
undefined. To remedy this, we employ the convention that an undefined value
does not realize the maximum. Observe that the lower bound
$\frac{\epsilon-\delta}{2}$ follows by Lemma 3.1.

###### Proof.

If $N$ is a model solid torus, this is a special case of [26, Theorem 8.8]. In
the notation of that theorem, substituting $\epsilon_{\max}=\log(3)$ implies a
value $j_{\max}=0.14798\ldots$, which gives an additive constant
$\operatorname{arcsinh}(j_{\max})\leq 0.1475$ in the lower bound on
$d_{\alpha,\lambda,\tau}(\delta,\epsilon)$.

If $N$ is a horocusp, the estimate follows by taking a geometric limit of
model solid tori converging to $N$. ∎

Under stronger hypotheses on $\epsilon$, we obtain a stronger lower bound on
$d_{\alpha,\lambda,\tau}(\delta,\epsilon)$.

######  Theorem 3.7.

Suppose that $0<\delta<\epsilon\leq 0.3$. Let $N=N_{\alpha,\lambda,\tau}$ be a
model solid torus with cone angle $\alpha\leq 2\pi$ and core geodesic of
length $\lambda\leq\delta$, or a model horocusp whose $\epsilon$–thick part is
not empty. Then

$\max\left\\{\frac{\epsilon-\delta}{2},\>\operatorname{arccosh}\left(\frac{\epsilon}{\sqrt{7.256\,\delta}}\right)-0.0424\right\\}\leq
d_{\alpha,\lambda,\tau}(\delta,\epsilon)\leq\operatorname{arccosh}\sqrt{\frac{\cosh\epsilon-1}{\cosh\delta-1}}.$

###### Proof.

If $N$ is a model solid torus, this is [26, Theorem 1.1]. If $N$ is a cusp,
take a geometric limit of tubes. ∎

###  3.3. Euclidean bounds

Consider an equidistant torus $T_{r}=\partial U_{r}$. Then the Euclidean path-
metric on $T_{r}$ lifts to a Euclidean metric on $\widetilde{T}_{r}$, which we
denote $d_{E}$.

######  Lemma 3.8.

Let $\widetilde{T}_{r}\subset{\hat{{\mathbb{H}}}^{3}}$ be a plane at fixed
distance $r>0$ from the singular geodesic $\hat{g}$. Let
$p,q\in\widetilde{T}_{r}$ be points whose $\theta$–coordinates differ by at
most $A\leq\pi$ and whose $\zeta$–coordinates differ by at most $B$. Then

$\frac{1-\cos A}{A^{2}}\,d_{E}(p,q)^{2}\leq\cosh d(p,q)-1\leq\frac{\cosh
B-1}{B^{2}}\,d_{E}(p,q)^{2}.$

###### Proof.

See [26, Lemma 6.2]. ∎

If an equidistant plane in ${\hat{{\mathbb{H}}}^{3}}$ is replaced by a
horosphere in ${\mathbb{H}}^{3}$, Lemma 3.8 becomes the following (well-known)
statement.

######  Lemma 3.9.

Let $\widetilde{T}\subset{\mathbb{H}}^{3}$ be a horosphere, and let
$p,q\in\widetilde{T}$. Let $d_{E}(p,q)$ be the distance between $p$ and $q$ in
the Euclidean metric on $\widetilde{T}$. Then

$2\sinh\frac{d(p,q)}{2}=d_{E}(p,q),\qquad\text{or equivalently}\qquad\cosh
d(p,q)-1=\frac{1}{2}d_{E}(p,q)^{2}.$

###### Proof.

See [18, Lemma A.2]. ∎

We observe that as $A,B\to 0$, the upper and lower bounds in Lemma 3.8 both
approach $\frac{1}{2}d_{E}(p,q)^{2}$. Thus Lemma 3.9 realizes this limiting
value.

##  4\. Maximal tubes and multi-tubes

The goal of this section is to control the area and injectivity radius of
maximal tubes in cone-manifolds. The main results are Theorem 4.16, giving a
lower bound on area, and Theorem 4.26, giving a lower bound on injectivity
radius. Theorem 4.16 is essentially due to Hodgson and Kerckhoff [33, Theorem
4.4], while Theorem 4.26 is new. Before getting to those results, we must
carefully construct maximal tubes of many components.

######  Definition 4.1.

Let $(M,\Sigma)$ be a hyperbolic cone-manifold. Let $\Sigma^{+}$ be a geodesic
link in $M$, such that $\Sigma\subset\Sigma^{+}$. Let
$\sigma_{1},\ldots,\sigma_{n}$ be the components of $\Sigma^{+}$. For a
positive vector ${\mathbf{r}}=(r_{1},\ldots,r_{n})$, let
$U_{i}=U_{r_{i}}(\sigma_{i})$ be the set of all points whose distance to
$\sigma_{i}$ is less than $r_{i}$. Let

$U_{\mathbf{r}}=\cup_{i=1}^{n}U_{r_{i}}(\sigma_{i}).$

We say that $U_{\mathbf{r}}$ is a _multi-tube_ if every $U_{i}$ is isometric
to a model tube as in Definition 2.3, and the $U_{i}$ are pairwise disjoint.

We choose a particular construction of maximal tubes.

######  Definition 4.2.

Let $M$ be a non-elementary hyperbolic cone-manifold, and $\Sigma^{+}$ a
geodesic link containing the singular locus. We construct a maximal multi-tube
about $\Sigma^{+}$, as follows:

  1. 1.

For a sufficiently small $r>0$, choosing a constant vector
${\mathbf{r}}=(r,\ldots,r)$ produces an embedded multi-tube $U_{\mathbf{r}}$.
Let $R_{1}$ be the largest value of $r$ for which this holds. The hypothesis
that $M$ is non-elementary ensures that such an $R_{1}$ exists.

Setting $r=R_{1}$, we have an (open) multi-tube $U_{\mathbf{r}}$ whose closure
$\overline{U}_{\mathbf{r}}$ is not a disjointly embedded union of closed
tubes. In other words, either some tube has bumped into itself, or some number
of tubes have bumped into one another. Any tube $U_{i}$ that cannot be
expanded further without intersecting itself or another tube is declared
_maximal_ , and its radius will remain fixed for the rest of the construction.

  2. 2.

Suppose, after relabeling, that $U_{1},\ldots,U_{k}$ are maximal, but
$U_{k+1},\ldots,U_{n}$ are not. We expand the radii $r_{k+1},\ldots,r_{n}$ at
a uniform rate until some tube $U_{i}$ for $i\geq k+1$ bumps into some other
tube $U_{j}$ (it may happen that $i=j$ or $j\leq k$). We then declare the
tubes that have just bumped to be _maximal_ , and freeze their radii.

  3. 3.

Repeat step 2 as needed until no tube can be expanded further. We call the
resulting union of maximal tubes the _maximal multi-tube_ , and denote it
$U_{\max}(\Sigma^{+})$ or $U_{\max}$ for short. We order the components of
$\Sigma^{+}$ so that the vector of radii ${\mathbf{r}}=(R_{1},\ldots,R_{n})$
appears in non-decreasing order, with $R_{1}$ the smallest radius.

######  Proposition 4.3.

Let $M$ be a non-elementary hyperbolic cone-manifold, and $\Sigma^{+}$ a
geodesic link containing the singular locus $\Sigma$. Let
$U_{\max}(\Sigma^{+})$ be the maximal multi-tube about $\Sigma^{+}$, with
smallest tube radius $R=R_{1}$. Suppose that a component $U_{i}\subset
U_{\max}$ that becomes maximal by bumping into a component $U_{j}$. Then the
radii of these tubes satisfy $R_{1}\leq R_{j}\leq R_{i}$ and $T_{i}=\partial
U_{i}$ contains an embedded ellipse whose semi-major axes are

(4.4) $a(R_{i},R_{j})=\frac{\cosh R_{i}\sinh
R_{j}}{S(R_{j})\cosh(R_{i}+R_{j})}\quad\mbox{and}\quad
b(R_{i},R_{j})=\frac{\sinh R_{i}\sinh R_{j}}{\sinh(R_{i}+R_{j})}.$

Here $S(R_{j})$ is defined via

(4.5)
$S(r)=\begin{dcases}\dfrac{\sqrt{2}/4}{\operatorname{arcsinh}(\sqrt{2}/4)}=1.02013\ldots,&\mbox{
if }\sinh r\leq\dfrac{1}{\sqrt{2}},\\\ \dfrac{\sinh
r/\cosh(2r)}{\operatorname{arcsinh}\left(\sinh r/\cosh(2r)\right)},&\mbox{ if
}\sinh r\geq\dfrac{1}{\sqrt{2}}.\end{dcases}$

Furthermore, if the tube $U_{i}$ became maximal by bumping into itself,
$T_{i}=\partial U_{i}$ contains two disjoint ellipses as in (4.4), with
parallel major axes.

###### Proof.

This result is essentially due to Hodgson and Kerckhoff [33, Section 4]. They
prove an identical statement when $\Sigma^{+}$ is connected, and briefly
mention that the argument extends to multiple tubes. We make small
modifications to their argument in order to handle maximal multi-tubes of
disparate radii. It is worth noting that our construction of maximal multi-
tubes differs somewhat from that of Hodgson and Kerckhoff.

As in Definition 2.5, let $\hat{M}$ be the universal branched cover of $M$,
branched over $\Sigma^{+}$. Every component of $\Sigma^{+}$ lifts to a
singular geodesic in $\hat{M}$, with cone angle $\infty$, and the space is
locally modeled on ${\hat{{\mathbb{H}}}^{3}}$, as in Definition 2.1. For any
singular basepoint in $\hat{M}$, we have the exponential-like map
$D\colon\hat{M}\to{\hat{{\mathbb{H}}}^{3}}$.

Consider the tube $U_{i}\subset U_{\max}$. Since $U_{i}$ is maximal, the
expansion of $U_{i}$ came to a halt because $U_{i}$ bumped into some other
tube $U_{j}$, which became maximal no later than $U_{i}$ did. By the
construction of Definition 4.2, we have

(4.6) $R_{i}\geq R_{j}\geq R_{1}=R.$

Since $\partial U_{i}$ is tangent to $\partial U_{j}$, there is a geodesic arc
$\gamma$ of length $R_{i}+R_{j}$ that travels radially outward from the core
circle $\sigma_{i}$, enters into $U_{j}$ at a point of tangency, and meets the
core circle $\sigma_{j}$ perpendicularly at its endpoint. This arc $\gamma$
lifts to an arc $\widetilde{\gamma}\subset\hat{M}$ from a lift
$\widetilde{\sigma_{i}}$ of $\sigma_{i}$ to a lift $\widetilde{\sigma_{j}}$ of
$\sigma_{j}$. By construction, $\widetilde{\gamma}$ is a shortest geodesic
from $\widetilde{\sigma_{i}}$ to any other singular geodesic in $\hat{M}$. In
other words, we have:

######  Claim 4.7.

The geodesic $\widetilde{\sigma_{i}}\subset\hat{M}$ has an embedded
neighborhood $\widetilde{V_{i}}$ of radius $R_{i}+R_{j}$. If we choose a
basepoint on $\widetilde{\sigma_{i}}$, the map
$D:\widetilde{V_{i}}\to{\hat{{\mathbb{H}}}^{3}}$ is an isometric embedding. ∎

Let $\Gamma$ be the set of all lifts of $\gamma$ starting at
$\widetilde{\sigma_{i}}$, oriented outward from $\widetilde{\sigma_{i}}$. Let
$Q$ be the set of their forward endpoints. As in Definition 2.4, there is a
${\mathbb{Z}}\times{\mathbb{Z}}$ group of deck transformations of $\hat{M}$
acting effectively and transitively on $\Gamma$, hence on $Q$.

######  Claim 4.8.

Let $q,q^{\prime}\in Q$ be endpoints of distinct lifts of $\gamma$. Then

$d(D(q),D(q^{\prime}))=d(q,q^{\prime})\geq 2R_{j}.$

The equality of distances holds by Claim 4.7. The inequality holds because $q$
and $q^{\prime}$ lie on distinct lifts of $\sigma_{j}$, and $\sigma_{j}$ has
an embedded tube of radius $R_{j}$. See [33, Lemma 4.1] for more details.

For each $q\in Q$, let $B(q)\subset{\hat{{\mathbb{H}}}^{3}}$ be a ball of
radius $R_{j}$ centered at $D(q)\in{\hat{{\mathbb{H}}}^{3}}$. By Claim 4.8,
these balls are disjointly embedded in ${\hat{{\mathbb{H}}}^{3}}$. In other
words, each $B(q)$ is disjoint from its translates under
${\mathbb{Z}}\times{\mathbb{Z}}$.

Let $(r,\theta,\zeta)$ be cylindrical coordinates on
${\hat{{\mathbb{H}}}^{3}}$, as in Definition 2.1. We normalize things so that
the geodesic arc $D(\widetilde{\gamma})$ lies on the geodesic ray
$\\{(r,0,0):r>0\\}$. Let $B(q)$ be the ball of radius $R_{j}$ centered at
$(R_{i}+R_{j},0,0)$. By [33, Lemma 4.3], the projection of $B(q)$ to the
$(\theta,\zeta)$-plane consists of all points $(\theta,\zeta)$ that satisfy
the equation

(4.9)
$\sinh^{2}\zeta\cosh^{2}(R_{i}+R_{j})+\sin^{2}\theta\sinh^{2}(R_{i}+R_{j})\leq\sinh^{2}R_{j}.$

######  Claim 4.10.

Let $(\theta,\zeta)$ be a point in the projection of $B(q)$ to the
$(\theta,\zeta)$-plane. Setting $S(R_{j})$ is as in (4.5), we have
$\left|\sinh\zeta\right|\leq S(R_{j})\left|\zeta\right|$. Furthermore,
$S(R_{j})$ is a decreasing function of $R_{j}$.

To see this, observe that (4.9), combined with (4.6), implies

(4.11) $|\sinh\zeta|\leq\frac{\sinh R_{j}}{\cosh(R_{i}+R_{j})}\leq\frac{\sinh
R_{j}}{\cosh(2R_{j})}.$

Now, setting $x=\sinh r$, observe that the function

$\frac{\sinh r}{\cosh(2r)}=\frac{x}{2x^{2}+1}$

reaches a global maximum value of $\sqrt{2}/4$ when $x=1/\sqrt{2}$, and
declines to $0$ thereafter. Since $\left|\sinh\zeta/\zeta\right|$ is
increasing in $\left|\zeta\right|$, we have

$\left|\frac{\sinh\zeta}{\zeta}\right|\leq\frac{\sinh\left(\operatorname{arcsinh}(\sqrt{2}/4)\right)}{\operatorname{arcsinh}(\sqrt{2}/4)}=S(0)=1.02013\ldots\quad\text{for
all values of $R_{j}$},$

as observed by Hodgson–Kerckhoff [33, Pages 401–402]. Furthermore, when
$x=\sinh R_{j}\geq 1/\sqrt{2}$, we have

$\left|\frac{\sinh\zeta}{\zeta}\right|\leq\dfrac{\sinh
R_{j}/\cosh(2R_{j})}{\operatorname{arcsinh}\left(\sinh
R_{j}/\cosh(2R_{j})\right)}=S(R_{j}),$

which is increasing in $\sinh R_{j}/\cosh(2R_{j})$, hence decreasing in
$R_{j}$. This proves the claim.

Combining (4.9), Claim 4.10, and the standard fact
$\left|\sin\theta\right|\leq|\theta|$ gives:

######  Claim 4.12.

Let $B(q)$ be the ball of radius $R_{j}$ centered at $(R_{i}+R_{j},0,0)$. Then
the projection of $B(q)$ to the $(\theta,\zeta)$-plane contains the elliptical
region consisting of points $(\theta,\zeta)$ that satisfy

(4.13)
$\zeta^{2}\,S(R_{j})^{2}\cosh^{2}(R_{i}+R_{j})+\theta^{2}\sinh^{2}(R_{i}+R_{j})\leq\sinh^{2}R_{j}.$

Now, recall the tube $U_{i}$ about $\sigma_{i}$, with radius $R_{i}$. This
tube lifts to a tube $\widetilde{U}_{i}$ about $\widetilde{\sigma}_{i}$, which
isometrically embeds in ${\hat{{\mathbb{H}}}^{3}}$ by Claim 4.7. We consider
the shadow of $B(q)$ on the Euclidean plane $D(\partial\widetilde{U}_{i})$.
Since $(\zeta\cosh R_{i})$ and $(\theta\sinh R_{i})$ are Euclidean coordinates
on the plane at radius $R_{i}$ from the singular geodesic of
${\hat{{\mathbb{H}}}^{3}}$, the elliptical region of (4.13) can be rewritten
in coordinates as

$\left(\frac{S(R_{j})\cosh(R_{i}+R_{j})}{\cosh R_{i}\sinh
R_{j}}\right)^{\\!2}\\!(\zeta\cosh
R_{i})^{2}+\left(\frac{\sinh(R_{i}+R_{j})}{\sinh R_{i}\sinh
R_{j}}\right)^{\\!2}\\!(\theta\sinh R_{i})^{2}\leq 1.$

Since the elliptical region is disjoint from its translates under
${\mathbb{Z}}\times{\mathbb{Z}}$, it follows that the quotient torus
$T_{i}=\partial U_{i}$ contains an embedded ellipse whose semi-major axes are

$a(R_{i},R_{j})=\frac{\cosh R_{i}\sinh
R_{j}}{S(R_{j})\cosh(R_{i}+R_{j})}\quad\mbox{and}\quad
b(R_{i},R_{j})=\frac{\sinh R_{i}\sinh R_{j}}{\sinh(R_{i}+R_{j})},$

as required in (4.4).

If the tube $U_{i}$ became maximal by bumping into itself, the arc $\gamma$
must have both of its endpoints on $\sigma_{i}$. This means there are two
distinct ${\mathbb{Z}}\times{\mathbb{Z}}$ orbits of lifts of $\gamma$ with an
endpoint on $\widetilde{\sigma_{i}}$, giving rise to two orbits of balls
$B(q)$ and two disjoint ellipses on $\partial U_{i}$. This is the case that
Hodgson and Kerckhoff analyze in [33, Theorem 4.4]. ∎

######  Remark 4.14.

It follows from Equations (4.13) and (4.6) that the ellipse constructed in the
last proof has $\theta$–coordinate bounded as follows:

$|\theta|\leq\frac{\sinh R_{j}}{\sinh(R_{i}+R_{j})}\leq\frac{\sinh
R_{j}}{\sinh(2R_{j})}=\frac{1}{2\cosh R_{j}}<\frac{1}{2}.$

###  4.1. Areas of maximal tubes

We present two applications of Proposition 4.3 that will be crucial in the
sequel. The first application, developed by Hodgson and Kerckhoff [33, 34],
concerns the area of maximal multi-tubes. We will use the area of the ellipse
constructed in Proposition 4.3 to get a lower bound on
$\operatorname{area}(T_{i})$. To do this, we need to remove the dependence on
the tube $T_{j}$.

######  Lemma 4.15.

Let $a(R_{i},R_{j})$ and $b(R_{i},R_{j})$ be as in Equation (4.4). Then the
function that is their product

$ab(R_{i},R_{j})=\frac{\sinh R_{i}\cosh
R_{i}\sinh^{2}R_{j}}{S(R_{j})\sinh(R_{i}+R_{j})\cosh(R_{i}+R_{j})}$

is increasing in both variables.

###### Proof.

Since $S(R_{j})^{-1}$ is increasing in $R_{j}$ by Claim 4.10, it suffices to
show that $S(R_{j})ab(R_{i},R_{j})$ is increasing. To that end, we substitute
the variable names $x=R_{i}$ and $y=R_{j}$, and simplify:

$Sab(x,y)=\frac{(\sinh x\cosh
x)\sinh^{2}y}{\sinh(x+y)\cosh(x+y)}=\frac{\sinh(2x)\cdot\frac{1}{2}(\cosh(2y)-1)}{\sinh(2x+2y)}\,.$

Now, we can compute the partial derivatives:

$\displaystyle\frac{\partial Sab}{\partial x}$
$\displaystyle=\frac{2\sinh(2x+2y)\cosh(2x)-2\sinh(2x)\cosh(2x+2y)}{\sinh^{2}(2x+2y)}\cdot\frac{1}{2}(\cosh(2y)-1)$
$\displaystyle=\frac{\sinh(2y)}{\sinh^{2}(2x+2y)}\cdot(\cosh(2y)-1)>0$

when $y>0$. Similarly,

$\displaystyle\frac{\partial Sab}{\partial y}$
$\displaystyle=\frac{\sinh(2x+2y)\sinh(2y)-(\cosh(2y)-1)\cosh(2x+2y)}{\sinh^{2}(2x+2y)}\cdot\sinh(2x)$
$\displaystyle=\frac{[\sinh(2x+2y)\sinh(2y)-\cosh(2y)\cosh(2x+2y)]+\cosh(2x+2y)}{\sinh^{2}(2x+2y)}\cdot\sinh(2x)$
$\displaystyle=\frac{\cosh(2x+2y)-\cosh(2y)}{\sinh^{2}(2x+2y)}\cdot\sinh(2x)>0.$

when $x>0$ and $y>0$. ∎

We can now show the following.

######  Theorem 4.16.

Let $M$ be a non-elementary hyperbolic cone-manifold, and $\Sigma^{+}$ a
geodesic link containing the singular locus $\Sigma$. Let
$U_{\max}(\Sigma^{+})$ be the maximal multi-tube about $\Sigma^{+}$, with
smallest tube radius $R$. Let $T_{i}=\partial U_{i}$ be the boundary torus of
any component of $U_{\max}$. Then

(4.17)
$\operatorname{area}(T_{i})\geq\frac{\sqrt{3}\sinh^{2}R}{S(R)\cosh(2R)}\geq
1.69785\frac{\sinh^{2}R}{\cosh(2R)}.$

If the tube $U_{i}$ became maximal by bumping into itself (for instance, if
$\Sigma^{+}$ is connected), then $\operatorname{area}(T_{i})$ is bounded below
by _twice_ the estimate of (4.17).

###### Proof.

This result is essentially [33, Theorem 4.4]. To derive the theorem from
Proposition 4.3, perform an area-preserving affine transformation on $T_{i}$
that turns the ellipse of Proposition 4.3 into a circle. The area of this
circle is

$\pi ab(R_{i},R_{j})\geq\pi ab(R,R)=\frac{\pi\sinh R\cosh
R\sinh^{2}R}{S(R)\sinh(2R)\cosh(2R)}=\frac{\pi\sinh^{2}R}{2S(R)\cosh(2R)},$

where the inequality is Lemma 4.15. By a theorem of Böröczky [8], the maximal
density of a circle packing in the torus is $\frac{\pi}{2\sqrt{3}}$.
Therefore,

$\operatorname{area}(T_{i})\geq\frac{2\sqrt{3}}{\pi}\pi
ab(R,R)=2\sqrt{3}\,ab(R,R)=\frac{\sqrt{3}\sinh^{2}R}{S(R)\cosh(2R)}.$

Recall from Claim 4.10 that $S(R)\leq S(0)=1.02013\ldots$, hence
$\sqrt{3}/S(R)\geq 1.69785$.

If the tube $U_{1}$ became maximal by bumping into itself, the two ellipses
guaranteed by Proposition 4.3 become two circles of identical radius. Thus the
estimate of (4.17) becomes doubled. This is the case that Hodgson and
Kerckhoff analyze in [33, Theorem 4.4]. ∎

We apply Theorem 4.16 to bound the visual area of $\Sigma^{+}$.

######  Definition 4.18.

Let $M$ be a non-elementary hyperbolic cone-manifold, and let
$\Sigma^{+}=\sigma_{1}\cup\ldots\cup\sigma_{n}$ be a geodesic link containing
the singular locus. Let $\alpha_{j}$ be the cone angle along $\sigma_{j}$, and
let $\lambda_{j}=\operatorname{len}(\sigma_{j})$ be the length of
$\sigma_{j}$. We define the _visual area of $\sigma_{j}$_ to be

$\mathcal{A}_{j}=\alpha_{j}\lambda_{j}.$

The _visual area of $\Sigma^{+}$_ is defined by summation:
$\mathcal{A}=\sum_{j}\mathcal{A}_{j}$. Note that if $T_{j}$ is the boundary of
some tube $U_{j}\subset U_{\max}$, then (2.2) implies

(4.19) $\operatorname{area}(T_{j})=\mathcal{A}_{j}\sinh R_{j}\cosh
R_{j}=\mathcal{A}_{j}\sinh(2R_{j})/2.$

######  Definition 4.20.

Define a function

$h(r)=3.3957\,\frac{\tanh(r)}{\cosh(2r)}=3.3957\,\frac{z(1-z^{2})}{1+z^{2}},$

where $z=\tanh r$.

######  Theorem 4.21.

Let $M$ be a non-elementary hyperbolic cone-manifold, and $\Sigma^{+}$ a
geodesic link containing the singular locus. Let $U_{\max}(\Sigma^{+})$ be the
maximal multi-tube about $\Sigma^{+}$, with smallest tube radius $R$. Then

$\displaystyle\mathcal{A}\geq h(R).$

###### Proof.

This result is due to Hodgson and Kerckhoff [34, Theorem 5.6]. We repeat the
short proof for completeness. If $T_{i}$ is a boundary torus of some component
of $U_{\max}$, equation (4.19) gives

$\operatorname{area}(T_{i})=\mathcal{A}_{i}\sinh R_{i}\cosh R_{i}.$

If $U_{1}$ became maximal by bumping into itself, Theorem 4.16 guarantees

$\mathcal{A}\geq\mathcal{A}_{1}=\frac{\operatorname{area}(T_{1})}{\sinh R\cosh
R}\geq 3.3957\,\frac{\sinh^{2}R/\cosh(2R)}{\sinh R\cosh R}=h(R),$

as desired. Meanwhile, if $U_{1}$ became maximal by bumping into another tube
$U_{2}$, then $R_{1}=R_{2}=R$, and Theorem 4.16 bounds the area of each of
$T_{1}=\partial U_{1}$ and $T_{2}=\partial U_{2}$. Therefore,

$\mathcal{A}\geq\mathcal{A}_{1}+\mathcal{A}_{2}=\frac{\operatorname{area}(T_{1})+\operatorname{area}(T_{2})}{\sinh
R\cosh R}\geq 3.3957\,\frac{\sinh^{2}R/\cosh(2R)}{\sinh R\cosh R}=h(R).\qed$

In Section 5, we will apply Theorem 4.21 to prove the existence of cone
deformations maintaining a given tube radius about $\Sigma$. To set up this
application, we need to establish some important properties of $h(r)$.

######  Lemma 4.22 (Lemma 5.2 of [33]).

For $r>0$, the function $h(r)$ of Definition 4.20 has a unique critical point
at $r_{0}=\operatorname{arctanh}\sqrt{\sqrt{5}-2}\approx 0.5306375$. This
critical point is a global maximum, hence $h(r)$ is strictly decreasing when
$r\geq 0.531$. ∎

######  Remark 4.23.

Recall from Definition 4.20 that $h(r)$ can be expressed as a rational
function of $z=\tanh r$. This leads us to define

${\operatorname{haze}}(z)=h(\tanh^{-1}(z))=3.3957\,\frac{z(1-z^{2})}{1+z^{2}}.$

By Lemma 4.22, ${\operatorname{haze}}(z)$ is decreasing and invertible in the
range $z\in\big{[}\sqrt{\sqrt{5}-2},\,1\big{)}$. The functions
${\operatorname{haze}}$ and ${\operatorname{haze}}^{-1}$ will play an
important role in Section 6 and Section 7. Because inverting
${\operatorname{haze}}(z)$ amounts to solving a cubic equation, Cardano’s
Formula can be used to obtain a closed-form expression for
${\operatorname{haze}}^{-1}(h)$:

(4.24)
${\operatorname{haze}}^{-1}(3.3957x)=\frac{2\sqrt{x^{2}+3}}{3}\cos\left(\frac{\pi}{3}+\frac{1}{3}\tan^{-1}\\!\left(\frac{-3\sqrt{-3x^{4}-33x^{2}+3}}{x^{3}+18x}\right)\right)-\frac{x}{3}.$

Returning to the function $h(r)$, we define ${h_{\max}}=h(0.531)\approx
1.01967$. By Lemma 4.22, this is slightly less than the true maximal value of
$h$. Now, Theorem 4.21 and Lemma 4.22 have the following immediate corollary.

######  Corollary 4.25.

The function $h$ of Definition 4.20 has a well-defined inverse

$h^{-1}\colon(0,{h_{\max}}]\to[0.531,\infty),$

which can be computed via (4.24). Furthermore, $h^{-1}$ is a decreasing
function such that the maximal tube radius satisfies

$R\geq h^{-1}(\mathcal{A}),$

provided $R\geq 0.531$. ∎

###  4.2. Injectivity radii

Recall the definition of ${\operatorname{injrad}}(x,U_{i})$ from Definition
2.17.

######  Theorem 4.26.

Let $M$ be a non-elementary hyperbolic cone-manifold, and $\Sigma^{+}$ a
geodesic link containing the singular locus. Let $U_{\max}(\Sigma^{+})$ be the
maximal multi-tube about $\Sigma^{+}$, with smallest tube radius $R$.

Then, for every tube $U_{i}\subset U_{\max}$ and every $x\in\partial U_{i}$,

(4.27) $\displaystyle 2\,{\operatorname{injrad}}(x,U_{i})$ $\displaystyle\geq
1.361\sqrt{1-\cos\left(\frac{1}{\cosh(R)}\right)}\cdot\frac{\sinh R}{S(R)}$
(4.28) $\displaystyle>1.1227\tanh R-0.1604.$

where $S(R)$ is defined in (4.5). This functions on the right-hand side of
(4.27) and (4.28) are increasing in $R$.

###### Proof.

First, we check that the function expressing the lower bound in (4.27) is
increasing in $R$, and calculate its limit as $R\to\infty$. We define

$f_{1}(R)=\sqrt{1-\cos\left(\frac{1}{\cosh(R)}\right)}\cdot\cosh(R),\qquad
f_{2}(R)=\frac{1}{S(R)},\qquad f_{3}(R)=\tanh R,$

so that the lower-bound function in (4.27) becomes
$f(R)=1.361\,f_{1}(R)\,f_{2}(R)\,f_{3}(R)$. The first non-constant term in the
product can be written as

$f_{1}(R)=\sqrt{1-\cos\left(\frac{1}{\cosh(R)}\right)}\cdot\cosh(R)=\sqrt{\frac{1-\cos(A)}{A^{2}}}\qquad\text{for}\qquad
A=\frac{1}{\cosh(R)}.$

For $A$ between $0$ and $1$, the function $f_{1}(R)$ is decreasing in $A$,
hence increasing in $R$. It satisfies
$\displaystyle{\lim_{R\to\infty}f_{1}(R)=\tfrac{1}{\sqrt{2}}}$. The second
term is $f_{2}(R)=S(R)^{-1}$, which is increasing by Claim 4.10 and approaches
$1$ as $R\to\infty$. Finaly, the third term is $f_{3}(R)=\tanh R$, which is
also increasing in $R$ and approaches $1$ as $R\to\infty$. Thus
$f(R)=1.361\,f_{1}(R)\,f_{2}(R)\,f_{3}(R)$ increases at least as fast as
$Z=\tanh R$, and satisfies

(4.29)
$\lim_{R\to\infty}f(R)=1.361\,\lim_{R\to\infty}f_{1}(R)\,f_{2}(R)\,f_{3}(R)=\frac{1.361}{\sqrt{2}}=0.96237\ldots$

Next, we check that the function in (4.27) is larger than the one (4.28). Set
$Z=\tanh R$, as above. When $Z\in[0.99995,\,1)$, the increasing function in
(4.27) is bounded below by $0.9623$, whereas $1.1227Z-0.1604$ is bounded above
by $0.9623$. Meanwhile, when $Z\in[0,0.99995]$, we check using interval
arithmetic in Sage that the function in (4.27) is larger than
$1.1227Z-0.1604$. This is established by breaking the domain $[0,0.99995]$
into small intervals and checking the desired inequality on each sub-interval.
See the ancillary files [23] for details.

Now, we proceed to the main portion of the proof: the lower bound on
${\operatorname{injrad}}(x,U_{i})$ expressed in (4.27). Consider the torus
$T_{i}=\partial U_{i}$. Proposition 4.3 has the following consequence.

######  Claim 4.30.

The torus $T_{i}=\partial U_{i}$ contains an embedded open disk of radius

$\frac{b(R,R)}{S(R)}=\frac{\sinh^{2}R}{S(R)\sinh(2R)}=\frac{\tanh
R}{2S(R)}\,,$

where $b(\cdot,\cdot)$ is a semi-major axis as in (4.4), and $S(\cdot)$ is as
in (4.5).

This can be seen as follows. By Proposition 4.3, torus $T_{i}$ contains an
embedded ellipse, whose semi-major axes are

$a(R_{i},R_{j})=\frac{\cosh R_{i}\sinh
R_{j}}{S(R_{j})\cosh(R_{i}+R_{j})}\quad\mbox{and}\quad
b(R_{i},R_{j})=\frac{\sinh R_{i}\sinh R_{j}}{\sinh(R_{i}+R_{j})}.$

This ellipse contains a disk of radius $\min\\{a,b\\}$. We would like to
determine this minimum. By Lemma A.2,

$\frac{\cosh R_{i}}{\cosh(R_{i}+R_{j})}>\frac{\sinh
R_{i}}{\sinh(R_{i}+R_{j})},\quad\mbox{hence}\quad
S(R_{j})a(R_{i},R_{j})>b(R_{i},R_{j}).$

Since $S(R_{j})>1$, it follows that

$\min\left\\{a(R_{i},R_{j}),\,b(R_{i},R_{j})\right\\}\geq\min\left\\{a(R_{i},R_{j}),\,\frac{b(R_{i},R_{j})}{S(R_{j})}\right\\}=\frac{b(R_{i},R_{j})}{S(R_{j})}\geq\frac{b(R,R)}{S(R)}.$

Here, the last inequality follows because $b(R_{i},R_{j})$ is monotonically
increasing in both variables, by a calculation similar to Lemma 4.15.
Meanwhile, $S(R)$ is monotonically decreasing by Claim 4.10, hence the
quotient is increasing. This proves the claim.

Proceeding toward the main proof, let $\hat{M}$ be the universal branched
cover of $M$, branched over $\Sigma^{+}$. Choose a preimage
$\widetilde{U}_{i}$ of $U_{i}$. Then
$\widetilde{T}_{i}=\partial\widetilde{U}_{i}$ is a Euclidean plane that covers
$T_{i}$. Our goal, following Definition 2.17, is to give a lower bound on the
distance between a lift $\widetilde{x}$ of $x$ and any of its translates under
$\pi_{1}(T_{i})={\mathbb{Z}}\times{\mathbb{Z}}$.

By Claim 4.30, $\widetilde{T_{i}}$ contains a
${\mathbb{Z}}\times{\mathbb{Z}}$–equivariant family of disjoint disks, of
radius $b(R,R)/S(R)$. Fix $p=\widetilde{x}$, and let
$q=\varphi(\widetilde{x})$ be the closest translate of $p$. Since
${\operatorname{injrad}}(x,U_{i})$ is constant over points of $T_{i}$, we may
assume that $p$ and $q$ lie at centers of disks in this family. Thus
$d_{E}(p,q)\geq 2b(R,R)/S(R)$, where $d_{E}$ denotes the Euclidean distance on
$\widetilde{T_{i}}$, as in Section 3.3.

######  Claim 4.31.

We have $d(p,q)\geq
f(R)=\displaystyle{1.361\sqrt{1-\cos\left(\frac{1}{\cosh(R)}\right)}\cdot\frac{\sinh
R}{S(R)}}$.

Before proving this claim, we make some quick reductions. First, as we
computed in (4.29), the function $f(R)$ is bounded above by $0.9625$. Thus it
suffices to assume $d(p,q)\leq 0.9625$. Second, it suffices to assume that the
disks of radius $b/S$ centered at $p$ and $q$ are tangent, because any lower
bound on distance for tangent disks will still apply as $p,q$ are moved
further apart.

Our lower bound on $d(p,q)$ will come from Lemma 3.8. In preparation for
applying that lemma, we note that since $d(p,q)\leq 0.9625$, we have

(4.32) $\frac{\cosh d(p,q)-1}{d(p,q)^{2}}\leq\frac{\cosh
0.9625-1}{0.9625^{2}}=0.53981\ldots.$

By Remark 4.14, the $\theta$–coordinates of an ellipse centered at $(r,0,0)$
must satisfy

(4.33)
$|\theta|\leq\frac{1}{2\cosh(R_{j})}\leq\frac{1}{2\cosh(R)}=\frac{A}{2},$

where recall that we defined $A=1/\cosh(R)$. This means that the
$\theta$–coordinates of $p$ and $q$, whose disks are assumed to be tangent,
must differ by at most $A$, which is at most $1<\pi$. Finally, if the disks
centered at $p,q$ are tangent, Claim 4.30 implies

(4.34) $d_{E}(p,q)=\frac{2b(R,R)}{S(R)}=\frac{\tanh R}{S(R)}.$

Now, we may plug (4.32) and (4.34) into the lower bound of Lemma 3.8. Using
the upper bound from (4.32), we obtain

$0.53982\,d(p,q)^{2}\geq\cosh d(p,q)-1\geq\frac{1-\cos
A}{A^{2}}d_{E}(p,q)^{2}=\frac{1-\cos
A}{A^{2}}\cdot\frac{\tanh^{2}R}{S(R)^{2}}.$

Using the value $A=1/\cosh R$ from (4.33), this simplifies to

$\displaystyle d(p,q)$
$\displaystyle\geq\sqrt{\frac{1}{0.53982}}\cdot\frac{\sqrt{1-\cos
A}}{A}\cdot\frac{\tanh R}{S(R)}$
$\displaystyle=1.36105\ldots\sqrt{1-\cos\left(\frac{1}{\cosh(R)}\right)}\cdot\cosh
R\cdot\frac{\tanh R}{S(R)},$

proving the claim. Since $q$ was assumed to be the closest translate of
$p=\widetilde{x}$, Claim 4.31 proves the theorem. ∎

######  Remark 4.35.

If the cone manifold $M$ has cusps, the constructions and results of this
section also apply to a maximal neighborhood consisting of tubes and horocusps
in $M$. To extend Definition 4.2, first construct a maximal multi-tube as in
that definition. Then, choose any ordering on the cusps and expand each cusp
neighborhood until it bumps into a tube or a previously expanded cusp.

After such a construction, Theorems 4.16 and 4.26 hold for the boundary tori
of both tubes and horocusps. One way to see this is to view horocusps as
limiting cases of tubes with radius $R_{i}\to\infty$. A key point in the
proofs of both Theorem 4.16 and Theorem 4.26 is that the relevant estimates
are monotonically increasing in $R_{i}$. Thus they will also hold if $R_{i}$
is replaced by $\infty$.

If there are no compact tubes at all, but only a union of maximal cusps, both
theorems become well-known statements from the literature. Theorem 4.16
becomes the well-known estimate due to Meyerhoff [44, Section 5] that every
cusp torus $T_{i}=\partial U_{i}$ satisfies

$\operatorname{area}(T_{i})\geq\frac{\sqrt{3}}{2}=\lim_{R\to\infty}\frac{\sqrt{3}\sinh^{2}R}{S(R)\cosh(2R)}.$

Meanwhile, Theorem 4.26 becomes a well-known estimate observed by Adams [1,
Lemma 2.4]: every non-trivial element $\varphi\in\pi_{1}(T_{i})$ corresponds
to a horocycle of length $\geq 1$. In other words, for every
$\widetilde{x}\in\widetilde{T}_{i}$ and every $1\neq\varphi\in\pi_{1}(T_{i})$,
we have $d_{E}(\widetilde{x},\varphi\widetilde{x})\geq 1$. By Lemma 3.9, this
implies

$2{\operatorname{injrad}}(x,U_{i})\>=\>\min\\{d(\widetilde{x},\varphi(\widetilde{x})):\varphi\neq
1\\}\>\geq\>2\operatorname{arcsinh}(1/2)\>=\>0.96242\ldots,$

which is nearly the same as the asymptotic limit computed in (4.29).

##  5\. Existence of cone deformations

This section proves that if $M$ is a hyperbolic manifold and $\Sigma\subset M$
is a geodesic link that is sufficiently short, then there exists a cone
deformation interpolating between $M$ and $M-\Sigma$. See Theorem 5.1 for a
precise statement. This result is closely related to theorems of Hodgson and
Kerckhoff [33] and Bromberg [14] showing that cone deformations exist under
certain conditions. However, we need a version that has explicitly quantified
hypotheses, allows for multiple components of $\Sigma$, and allows $M$ to be a
cusped manifold. Such a version did not previously appear in the literature.
Still, our proof in this section relies heavily on the cone deformation theory
developed by Hodgson and Kerckhoff [31, 33, 34]. In order to explain the
statement and set up the proof, we review necessary background material from
their work. Reviewing background from cone deformation theory will also allow
us to define several important quantities and set up notation that will be
used in the subsequent sections.

On the way to proving Theorem 5.1, we will establish Theorem 5.14, which
provides quantitative control on the radius of a maximal multi-tube about
$\Sigma$. This result will be used repeatedly in the sequel.

A related theorem of Hodgson and Kerckhoff [34, Theorem 1.2] provides an
interpolation by cone manifolds from $M-\Sigma$ to $M$ (i.e. in the opposite
direction of Theorem 5.1), provided that all meridians on the cusps of
$M-\Sigma$ are sufficiently long. We recall their result below, in Theorem
5.17, again adding quantitative control over the radius of a multi-tube about
$\Sigma$.

In this section, and in the sequel,
$\Sigma=\sigma_{1}\cup\ldots\cup\sigma_{n}$ is a geodesic link. We use the
notation $\ell_{j}=\operatorname{len}(\sigma_{j})$ to denote the initial
length of $\sigma_{j}$ in a non-singular metric, and
$\lambda_{j}=\lambda_{j}(t)=\operatorname{len}_{t}(\sigma_{j})$ to denote the
length of $\sigma_{j}$ in a changing metric $g_{t}$.

######  Theorem 5.1.

Let $M$ be a finite volume hyperbolic $3$–manifold. Suppose that
$\Sigma=\sigma_{1}\cup\dots\cup\sigma_{n}$ is a geodesic link in $M$, whose
components have lengths satisfying

$\ell_{j}=\operatorname{len}(\sigma_{j})\leq 0.0996\>\>\mbox{for all
$j$}\qquad\mbox{and}\qquad\ell=\sum_{j=1}^{n}\ell_{j}\leq 0.15601.$

Then the hyperbolic structure on $M$ can be deformed to a complete hyperbolic
structure on $M-\Sigma$ by decreasing the cone angle $\alpha_{j}$ along
$\sigma_{j}$ from $2\pi$ to $0$. The cone angles on all components of $\Sigma$
change in unison.

Hodgson and Kerckhoff have shown this result in the setting where $M$ is a
closed hyperbolic 3–manifold and $\Sigma$ is connected [33, Corollary 6.3]. In
this special case, it suffices to assume that
$\ell=\operatorname{len}(\Sigma)\leq 0.11058$. Bromberg extended their result
to geometrically finite manifolds without rank one cusps [14, Theorem 1.2].
However, his hypotheses are not explicitly quantified, while we need explicit
bounds under explicit hypotheses.

###  5.1. Background on cone deformations

Hodgson and Kerckhoff [31] show that an infinitesimal deformation of a cone
manifold structure on $M$, with singular locus $\Sigma$, can be represented as
a harmonic $1$–form $\omega$ with values in the bundle $E$ of infinitesimal
isometries of $X=M-\Sigma$. Explicit information about $\omega$ is used to
determine the effect of the deformation on the singular locus.

Since $X$ is a hyperbolic $3$–manifold, its bundle of infinitesimal isometries
can be identified with $TX\otimes{\mathbb{C}}\cong TX\oplus i\,TX$. Here
$(v,iw)\in TX\oplus i\,TX$ corresponds to an infinitesimal translation in the
direction of $v$ and an infinitesimal rotation about an axis in the direction
of $w$. In [34], Hodgson and Kerckhoff show that $\omega$ can be taken to be
_harmonic_ , which means it will have the form

(5.2) $\omega=\eta+i*D\eta$

where $\eta$ is a $TX$–valued $1$–form on $X$, and $*$ is the Hodge star
operator on forms on $X$ that takes the vector valued 2–form $D\eta$ to a
vector valued 1–form. The forms $\eta$ and $*D\eta$ are both symmetric and
traceless. Under an appropriate $L^{2}$ integrability condition, $\omega$ is
the unique closed and co-closed harmonic form in its cohomology class; see
Remark 5.18 for details.

Given any component $\sigma_{j}$ of the singular locus $\Sigma$, Hodgson and
Kerckhoff use cylindrical coordinates about $\sigma_{j}$ to compute two
explicit closed and co-closed forms. The first,
$\omega_{m}=\eta_{m}+i*D\eta_{m}$, represents an infinitesimal deformation
which decreases the cone angle but does not affect the real part of the
complex length of the meridian. The second,
$\omega_{\ell}=\eta_{\ell}+i*D\eta_{\ell}$, stretches the singular locus but
leaves the holonomy of the meridian unchanged. The effects of $\omega_{m}$ and
$\omega_{\ell}$ on the complex length of any peripheral curve were computed in
[31, pages 32–33] and recorded in [33, Lemma 2.1].

In the following lemma, $t$ is a dummy variable expressing the “direction” of
an infinitesimal change of metric. Part of the content of Theorem 5.6 will be
that infinitesimal deformations can actually be promoted to local
deformations, parametrized by $t$.

######  Lemma 5.3 (Lemma 2.1 of [33]).

The effects of $\omega_{m}$ and $\omega_{\ell}$ on the complex length
$\mathcal{L}$ of any peripheral curve are as follows.

  1. (1)

If $\omega=\omega_{m}$, then $\frac{d}{dt}(\mathcal{L})=-2\mathcal{L}$.

  2. (2)

If $\omega=\omega_{\ell}$, then
$\frac{d}{dt}(\mathcal{L})=2\rm{Re}(\mathcal{L})$.

Any harmonic infinitesimal deformation affecting $\sigma_{j}$ alone can be
written in terms of these forms:

(5.4)
$\displaystyle\omega=s_{j}\,\omega_{m}+(x_{j}+i\,y_{j})\,\omega_{\ell}+\omega_{c},$

where $s_{j}$, $x_{j}$, and $y_{j}$ are real constants, and $\omega_{c}$ is an
infinitesimal deformation that does not affect the holonomy of the meridian
and longitude on the torus $T_{j}$ of distance $R$ from $\sigma_{j}$. We
define $\omega_{0}$ to be $\omega-\omega_{c}$.

Because only $\omega_{m}$ affects the cone angle, the coefficient $s_{j}$
determines the change in cone angle at $\sigma_{j}$ for our given
parametrization.

Lemma 5.3 implies that the effect of $\omega_{0}$ on complex length
$\mathcal{L}_{j}$ of $\sigma_{j}$ is given by

(5.5)
$\displaystyle\frac{d}{dt}(\mathcal{L}_{j})=-2s_{j}\mathcal{L}_{j}+2(x_{j}+iy_{j}){\rm{Re}}(\mathcal{L}_{j})$

A central result of Hodgson and Kerckhoff [31] is that there always exists a
local cone deformation that changes the cone angle on each component of
$\Sigma$ at the desired rate. In fact, we may let the deformation preserve
some number of closed geodesics whose cone angle is not changing. The
following is a special case of [31, Theorem 4.8], with parametrization
information added as in [34, Page 1073].

######  Theorem 5.6.

Let $M$ be a finite volume hyperbolic cone manifold with singular locus
$\Sigma=\sigma_{1}\cup\ldots\cup\sigma_{n}$, such that each component of
$\Sigma$ has cone angle $\alpha_{j}\leq 2\pi$. Let
$\Sigma^{+}=\sigma_{1}\cup\ldots\cup\sigma_{m}$ be a geodesic link containing
$\Sigma$. Pick a vector $(s_{1},\ldots,s_{m})\in{\mathbb{R}}^{m}$, where
$s_{j}=0$ for $j>n$. Then there is a local cone deformation
$(M,\Sigma^{+},g_{t})$, parametrized by $t$, such that

(5.7) $\frac{d\alpha_{j}}{dt}=-2\alpha_{j}s_{j}\,.$

Furthermore, the metric $g_{t}$ is determined up to isometry by the vector
$(\alpha_{1}(t),\ldots,\alpha_{n}(t),\alpha_{n+1},\ldots,\alpha_{m})$.

In our setting, we will consider deformations where each component of $\Sigma$
has the same cone angle. As in [34], we choose the parametrization
$t=\alpha^{2}$ for $0\leq\alpha\leq 2\pi$, and insist that $\alpha_{j}=\alpha$
for every $j\leq n$. When $\alpha>0$, equation (5.7) becomes

(5.8) $\displaystyle
s_{j}=-\frac{1}{2\alpha}\frac{d\alpha}{dt}=\frac{-1}{4\alpha^{2}}\qquad\mbox{for
all $j\leq n$}.$

###  5.2. Visual area and maximal tubes

Recall from Definition 4.18 that the visual area of the $j^{\rm th}$ component
of $\Sigma$ is $\mathcal{A}_{j}=\alpha_{j}\lambda_{j}$, and the total visual
area is $\mathcal{A}=\sum\mathcal{A}_{j}$. Recall as well the notion of a
maximal multi-tube from Definition 4.2. Our goal is to ensure that the radius
of $U_{\max}$ does not degenerate to $0$ during the course of the cone
deformation. We do this by showing that $\mathcal{A}$ is monotonic in $\alpha$
(Lemma 5.9) and that small visual area implies deep tubes (Corollary 4.25).

######  Lemma 5.9.

Consider a local cone deformation $(M,\Sigma,g_{t})$, parametrized by
$t=\alpha^{2}$. Let $U_{\max}(\Sigma)$ be the maximal multi-tube about
$\Sigma$, and let $R$ be the smallest radius of the tubes in $U_{\max}$. Let
$\operatorname{len}_{t}(\Sigma)=\sum_{j}\lambda_{j}$ denote the total length
of $\Sigma$ in the cone-metric $g_{t}$. If $Z=\tanh R\geq 1/\sqrt{3}$, and
$t>0$, then

(5.10)
$\frac{d}{dt}\operatorname{len}_{t}(\Sigma)\geq\frac{\operatorname{len}_{t}(\Sigma)}{2t}\cdot\frac{3Z^{2}-1}{Z^{2}(3-Z^{2})}\geq
0.$

Furthermore,

(5.11)
$\frac{d\mathcal{A}}{dt}\geq\frac{\mathcal{A}}{2t}\left(\frac{3Z^{2}-1}{Z^{2}(3-Z^{2})}+1\right)\geq\frac{\mathcal{A}}{2t}>0.$

###### Proof.

In our setting, every component of $\Sigma$ has the same angle
$\alpha_{j}=\alpha$. Define
$v=\mathcal{A}/\alpha^{2}=\operatorname{len}(\Sigma)/\alpha$. Then

$\operatorname{len}(\Sigma)=\sum\lambda_{j}=\frac{\mathcal{A}}{\alpha}=\alpha\,v=\sqrt{t}\,v.$

Consequently,

$\frac{d\operatorname{len}(\Sigma)}{dt}=\frac{d(\sqrt{t}\,v)}{dt}=\sqrt{t}\,\frac{dv}{dt}+\frac{v}{2\sqrt{t}}=\frac{v}{2\sqrt{t}}\left(\frac{2t}{v}\frac{dv}{dt}+1\right)=\frac{\operatorname{len}(\Sigma)}{2t}\left(\frac{2t}{v}\frac{dv}{dt}+1\right).$

By [34, Proposition 5.5], the hypothesis
$R\geq\operatorname{arctanh}(1/\sqrt{3})$ implies

$\displaystyle\frac{1}{v}\frac{dv}{dt}\geq-\frac{1}{\sinh^{2}R}\left(\frac{2\cosh^{2}R-1}{2\cosh^{2}R+1}\right)\frac{1}{\alpha}\frac{d\alpha}{dt}=-\left(\frac{1-Z^{4}}{Z^{2}(3-Z^{2})}\right)\frac{1}{2t},$

where the last equality uses $Z=\tanh R$ and $t=\alpha^{2}$. Then

(5.12)
$\displaystyle\frac{2t}{v}\frac{dv}{dt}+1\geq-\frac{1-Z^{4}}{Z^{2}(3-Z^{2})}+1=\frac{3Z^{2}-1}{Z^{2}(3-Z^{2})}.$

Since $Z=\tanh R<1$, the denominator of the last expression is always
positive. The numerator will be non-negative whenever $Z\geq 1/\sqrt{3}$,
hence the whole expression in (5.12) is non-negative. Thus

$\frac{d\operatorname{len}(\Sigma)}{dt}=\frac{\operatorname{len}(\Sigma)}{2t}\left(\frac{2t}{v}\frac{dv}{dt}+1\right)\geq\frac{\operatorname{len}_{t}(\Sigma)}{2t}\cdot\frac{3Z^{2}-1}{Z^{2}(3-Z^{2})}\geq
0,$

establishing (5.10). For (5.11), we recall that $\mathcal{A}=\alpha^{2}v=tv$.
Thus

$\frac{d\mathcal{A}}{dt}=t\frac{dv}{dt}+v=\frac{v}{2}\left(\frac{2t}{v}\frac{dv}{dt}+2\right)=\frac{\mathcal{A}}{2t}\left(\frac{2t}{v}\frac{dv}{dt}+2\right)\geq\frac{\mathcal{A}}{2t}\left(\frac{3Z^{2}-1}{Z^{2}(3-Z^{2})}+1\right).$

Since the expression in (5.12) is non-negative, (5.11) follows. ∎

Our goal is to bound the tube radius throughout a cone deformation. Following
Hodgson and Kerckhoff, we do this by using Corollary 4.25. That corollary can
be rephrased as follows: if the tube radius at some initial time $t$ is larger
than $0.531$, and $\mathcal{A}(t)$ remains smaller than ${h_{\max}}\approx
1.0196$ throughout the cone deformation, then the tube radius will remain
large.

Note that to apply Lemma 5.9, we need to ensure that the cone-locus $\Sigma$
has a tube of radius $R\geq\operatorname{arctanh}(1/\sqrt{3})>0.531$. This
minimal assumption on tube radius will appear in many results below.

###  5.3. Decreasing cone angles to $0$

Recall that by Theorem 5.6, there always exists a local cone deformation on
$(M,\Sigma)$ that decreases the cone angle on each component of $\Sigma$ from
$\alpha$ to $\alpha-\epsilon$, for some small $\epsilon>0$. To show that the
cone deformation can be continued, we apply a result of Hodgson and Kerckhoff.

######  Theorem 5.13 (Theorem 3.12 of [33]).

Suppose $M_{t}$, $t\in[0,t_{\infty})$, is a smooth path of finite volume
hyperbolic cone manifold structures on $(M,\Sigma)$ with cone angle
$\alpha_{j}(t)$ along the $j^{\rm th}$ component of the singular locus
$\Sigma$. Suppose $\alpha_{j}(t)\in[0,2\pi]$ for all $t$, and
$\alpha_{j}(t)\to a_{j}$ as $t\to t_{\infty}$. Suppose there is a constant
${R_{\min}}>0$ such that there is an embedded tube of radius at least
${R_{\min}}$ around $\Sigma$ for all $t$. Then the path extends continuously
to $t=t_{\infty}$ so that as $t\to t_{\infty}$, $M_{t}$ converges in the
bilipschitz topology to a cone manifold structure $M_{\infty}$ on $M$ with
cone angle $a_{j}$ along the $j^{\rm th}$ component of $\Sigma$.

###### Proof.

This theorem is exactly [33, Theorem 3.12], except for three minor differences
in the statement:

  1. (1)

The result [33, Theorem 3.12] is stated for closed manifolds rather than
finite-volume manifolds.

  2. (2)

It is stated for cone structures where all cone angles around the singular
locus agree. In fact, we can be more flexible with parametrizing the
deformation.

  3. (3)

It is stated for cone-manifolds satisfying a uniform upper volume bound,
independent of $t$.

Hypothesis (3) can be omitted because it holds automatically. This follows
from a construction of Agol [3], as follows. Agol uses the cone-metric $g_{t}$
(which is non-singular outside a tube about $\Sigma$) to construct a complete
metric of pinched negative curvature on $M-\Sigma$, which we denote $h_{t}$.
The sectional curvatures of this metric are bounded in terms of the constant
${R_{\min}}>0$, while $\operatorname{vol}(h_{t})$ differs from
$\operatorname{vol}(g_{t})$ by a multiplicative factor that depends only on
${R_{\min}}$. Furthermore, by a result of Boland, Connell, and Souto [7],
$\operatorname{vol}(h_{t})$ differs by a bounded multiplicative factor from
the volume of the complete hyperbolic metric, denoted
$\operatorname{vol}(M-\Sigma)$. Consequently, Agol’s work gives a uniform
upper bound on $\operatorname{vol}(M_{t})$ as a function of ${R_{\min}}$ and
$\operatorname{vol}(M-\Sigma)$.111The main result of Agol’s paper [3, Theorem
2.1] uses this construction to bound the ratio
$\operatorname{vol}(M-\Sigma)/\operatorname{vol}(M_{t})$ from above. However
the same ingredients also suffice to bound
$\operatorname{vol}(M-\Sigma)/\operatorname{vol}(M_{t})$ from below.

Hypothesis (2) can be omitted because it is never used in the proof of [33,
Theorem 3.12]. The proof goes through verbatim without this assumption.

Issue (1) can now be resolved by an appeal to (2). Let $\Sigma^{+}$ consist of
geodesics and cusps, where the cusps have cone angle $0$. Now apply [33,
Theorem 3.12] to $\Sigma^{+}$, so that the cone angle remains $0$ on all cusps
that remain unfilled. This immediately gives the result for finite volume
manifolds. ∎

######  Theorem 5.14.

Suppose $M$ is a finite volume hyperbolic $3$–manifold, and
$\Sigma=\sigma_{1}\cup\dots\cup\sigma_{n}$ is a geodesic link in $M$, of total
length $\ell=\operatorname{len}(\Sigma)\leq 0.15601\approx
h(\operatorname{arctanh}(1/\sqrt{3}))/(2\pi)$. Let $R$ be the radius of a
maximal embedded tube about $\Sigma$, and assume $R\geq 0.531$. Define
${R_{\min}}=h^{-1}(2\pi\ell)$, and note that this value exists by Corollary
4.25.

Then the hyperbolic structure on $M$ can be deformed to the complete
hyperbolic structure on $M-\Sigma$ by decreasing the cone angles on $\Sigma$
from $2\pi$ to $0$ in such a way that at any time $t$,

  1. (1)

Every component of $\Sigma$ has cone angle $\alpha=\sqrt{t}$,

  2. (2)

The tube radius in $M_{t}$ about $\Sigma$ is
$R(t)\geq{R_{\min}}\geq\operatorname{arctanh}(1/\sqrt{3})$,

  3. (3)

If $t>0$, we have $\mathcal{A}^{\prime}(t)>0$.

###### Proof.

By Theorem 5.6, there exists a cone deformation with cone angles near $2\pi$,
parametrized by $t=\alpha^{2}$. At the maximal value of $t$, namely
$t=(2\pi)^{2}$, we have

$\mathcal{A}(t)=2\pi\ell=h({R_{\min}})\leq
h(\operatorname{arctanh}(1/\sqrt{3}))<{h_{\max}},$

hence $R\geq{R_{\min}}\geq\operatorname{arctanh}(1/\sqrt{3})$ by Corollary
4.25. By Lemma 5.9, we have $\mathcal{A}^{\prime}((2\pi)^{2})>0$.

Let $I\subset[0,(2\pi)^{2}]$ be the maximal subinterval containing
$(2\pi)^{2}$, such that conclusions (1), (2), and (3) all hold for $t\in I$.
In the previous paragraph, we checked that $I$ contains $(2\pi)^{2}$, so is
not empty.

Next, we show that $I$ is open. Suppose $t_{0}\in I$, so there exists a
hyperbolic cone manifold structure on $M$ with cone angles
$\alpha_{0}=\sqrt{t_{0}}$. On a small neighborhood of $t_{0}$, condition (1)
holds as a consequence of Theorem 5.6: there exists a local cone deformation
with cone angles near $\alpha_{0}$, parametrized by $t=\alpha^{2}$ for $t$
near $t_{0}$. Condition (3) is an open condition, hence
$\mathcal{A}^{\prime}(t)>0$ in a small neighborhood of $t_{0}$ in
$[0,(2\pi)^{2}]$. Therefore, in the union of $I$ and this small neighborhood,
we have $\mathcal{A}(t)\leq\mathcal{A}((2\pi)^{2})\leq h({R_{\min}})$, hence
$R\geq{R_{\min}}\geq\operatorname{arctanh}(1/\sqrt{3})$ by Corollary 4.25. So
condition (2) is satisfied as well in this neighborhood, and $I$ is open.

Now, we show that $I$ is closed. Let $t_{\infty}=\inf I$. By Theorem 5.13, the
assumption that $R(t)\geq\operatorname{arctanh}(1/\sqrt{3})$ for $t\in I$
implies that the cone deformation extends to time $t_{\infty}$, hence (1)
holds. Second, note that (2) is a closed condition, hence
$R(t_{\infty})\geq{R_{\min}}$ by continuity. Third, by Lemma 5.9,
$R(t_{\infty})\geq\operatorname{arctanh}(1/\sqrt{3})$ implies that if
$t_{\infty}>0$, then $\mathcal{A}^{\prime}(t_{\infty})>0$, hence condition (3)
holds. Finally, if $t_{\infty}=0$, then condition (3) holds vacuously.
Therefore, $I$ is closed.

Since $I$ is open, closed, and non-empty, it follows that $I=[0,(2\pi)^{2}]$,
hence the desired cone deformation interpolates all the way between cone angle
$2\pi$ and $0$. ∎

The style of argument in the above proof will be employed several more times
in the paper. Conditions (1)–(3) are mutually reinforcing, with the property
that if they hold on an interval $I$, then they also hold on a slightly larger
interval. If $I$ is closed, the conclusions hold on a neighborhood of the
endpoint; if $I$ is open, they hold on the closure. This continuous analogue
of induction will be called a _crawling argument_.

We will prove Theorem 5.1 by applying Theorem 5.14. Theorem 5.14 needs a
hypothesis on the length $\ell$ and a hypothesis on the radius of the maximal
tube. Meanwhile, Theorem 5.1 only has hypotheses on length. It turns out that
for non-singular manifolds, the tube radius can be estimated from length
alone.

######  Lemma 5.15.

Let $M$ be a hyperbolic $3$–manifold. Let $\Sigma\subset M$ be a geodesic link
with components $\sigma_{1},\dots,\sigma_{n}$, such that
$\operatorname{len}(\sigma_{j})\leq 0.0996$ for every $j$. Then the maximal
embedded tube about $\Sigma$ has radius $R>0.531$.

See [28, Proposition 1.10] for a very similar statement, with slightly
different numbers in the hypotheses and the conclusion. Our proof, using
results of Meyerhoff [44], is based on the proof of that proposition.

###### Proof of Lemma 5.15.

This follows from a theorem of Meyerhoff [44, Section 3]. For each $j$, let
$\mathcal{L}_{j}=\ell_{j}+i\tau_{j}$ be the complex length of $\sigma_{j}$.
For each $j$, Meyerhoff constructs an embedded tube about $\sigma_{j}$ whose
radius $r_{j}$ satisfies

$\sinh^{2}r_{j}=\frac{\sqrt{1-2k(\mathcal{L}_{j})}}{2k(\mathcal{L}_{j})}-\frac{1}{2},\quad\text{where}\quad
k(\mathcal{L}_{j})=\min_{m\in{\mathbb{N}}}\\{\cosh(m\ell_{j})-\cos(m\tau_{j})\\}.$

Furthermore, the tubes about different components are disjoint [44, Section
7].

Observe that $\frac{\sqrt{1-2k}}{2k}$ is a decreasing function of $k$ when
$k\in(0,\sqrt{2}-1)$, and that $r_{j}=0.531$ when
$k(\mathcal{L}_{j})=0.34932\ldots$. Thus it remains to show that
$k(\mathcal{L}_{j})\leq 0.34932$ for all $\ell_{j}\in[0,0.0996]$ and
$\tau_{j}\in[0,2\pi]$. Since $\cosh(m\ell_{j})$ is an increasing function of
$\ell_{j}$, it suffices to set $\ell_{j}=0.0996$. Since $\cos(m\tau_{j})$ is
an even function of $\tau_{j}$, it suffices to consider values
$\tau_{j}\in[0,\pi]$.

Finally, we claim that for every $\tau_{j}\in[0,\pi]$, there is an integer
$m\in\\{1,\ldots,8\\}$ such that $\cosh(m\cdot 0.0996)-\cos(m\tau_{j})\leq
0.34932$. This is verified using interval arithmetic in Sage; see the
ancillary files [23] for details. ∎

###### Proof of Theorem 5.1.

Suppose that $\Sigma=\sigma_{1}\cup\dots\cup\sigma_{n}$ is a geodesic link in
$M$, such that each component has length $\operatorname{len}(\sigma_{j})\leq
0.0996$ and $\sum\operatorname{len}(\sigma_{j})\leq 0.15601$. Since
$\operatorname{len}(\sigma_{j})\leq 0.0996$ for each $j$, Lemma 5.15 says the
maximal tubular neighborhood of $\Sigma$ has radius $R>0.531$. Since
$\ell=\sum\operatorname{len}(\sigma_{j})\leq 0.15601$, Theorem 5.14 implies
that we may deform the cone angles on $\sigma_{j}$ downward from $2\pi$ to
$0$. ∎

###  5.4. Increasing cone angles from $0$

Next, we present a companion result to Theorem 5.14, whose hypotheses are on
the drilled manifold $M-\Sigma$ rather than the on the filled manifold $M$
where $\Sigma$ is non-singular. Recall that normalized length was defined in
Definition 1.3. If the total normalized length of all meridians in $M-\Sigma$
is sufficiently large, one obtains a cone deformation from $M-\Sigma$ to $M$,
with control on tube radii.

######  Definition 5.16.

Define a function $I\colon(0,1)\to{\mathbb{R}}$ as follows:

$I(z)=\frac{(2\pi)^{2}}{3.3957\,(1-z)}\,\exp\left(\int_{z}^{1}\frac{1+4w+6w^{2}+w^{4}}{(1+w)(1+w^{2})^{2}}dw\right),$

where $z=\tanh r$ as usual. This function has unique critical point: a global
minimum when $z=\sqrt{\sqrt{5}-2}$, with minimum value $56.469\ldots$. The
function is monotonically increasing for larger $z$, hence for $r\geq 0.531$.
It blows up as $z\to 1$. See [33, Pages 409–410].

Hodgson and Kerckhoff proved the following result.

######  Theorem 5.17.

Let $M$ be a $3$–manifold with empty or toroidal boundary, and $\Sigma$ a
smoothly embedded link in $M$. Suppose that $M-\Sigma$ is a cusped hyperbolic
manifold such that the total normalized length of the meridians of $\Sigma$
satisfies

$L^{2}\geq
I({Z_{\min}}),\quad\mbox{where}\quad{Z_{\min}}=\tanh({R_{\min}})\geq
1/\sqrt{3}.$

Then $M$ admits a hyperbolic metric in which $\Sigma$ is isotopic to a union
of geodesics. Furthermore, the hyperbolic structure on $M-\Sigma$ can be
deformed to that of $M$ via a family of cone-manifolds $M_{t}$, while
maintaining the following properties.

  1. (1)

Every component of $\Sigma$ has the same cone angle in $M_{t}$,

  2. (2)

The tube radius in $M_{t}$ about $\Sigma$ is
$R(t)\geq{R_{\min}}\geq\operatorname{arctanh}(1/\sqrt{3})$,

  3. (3)

$\mathcal{A}(t)<{h_{\max}}$.

###### Proof.

This is essentially [34, Theorem 5.11], with information about tube radius
extracted from the proof. By the remark following [31, Theorem 4.8], there is
a family of cone-manifolds $(M,\Sigma,g_{t})$ for $t\in[0,\epsilon)$, in which
the cone angles on $\Sigma$ agree for each $t$. In [34, Theorem 5.8], Hodgson
and Kerckhoff prove that the deformation can be continued so long as
$R(t)\geq\operatorname{arctanh}(1/\sqrt{3})$. Meanwhile, in [34, Theorem 5.7]
and the discussion preceding the theorem, they show that so long as $L^{2}\geq
I({Z_{\min}})$, and the cone angles are at most $2\pi$, the tube radius $R(t)$
will stay bounded below by ${R_{\min}}\geq\operatorname{arctanh}(1/\sqrt{3})$.
Thus the deformation can be continued all the way up to cone angle $2\pi$,
where we reach the complete hyperbolic metric on $M$. The link $\Sigma$ is
geodesic in each cone metric $g_{t}$, hence is also geodesic in the non-
singular metric at cone angle $2\pi$. ∎

We conclude this section with a particularly natural choice of the harmonic
form $\omega$.

######  Remark 5.18.

Recall from Section 5.1 that an infinitesimal deformation of the cone-metric
$g_{t}$ is determined by a harmonic $1$–form $\omega$ defined on $X=M-\Sigma$,
with values in the bundle $E\cong TX\oplus iTX$ of infinitesimal isometries of
$X$. By Theorem 5.6, the local family of cone-metrics $g_{t}$ is determined up
to isometry by its cone-angles, but different choices of $\omega$ within the
same cohomology class in $H^{1}(X,E)$ lead to different choices of cone-metric
within the same isometry class. In our bilipschitz theorem in Section 8, it
will be important to have a natural way to identify points of
$(M,\Sigma,g_{a})$ with points of $(M,\Sigma,g_{b})$, for the purpose of
comparing the metrics $g_{a}$ and $g_{b}$ at a point $p\in X$. To that end, we
pin down a canonical choice of $\omega$.

Suppose $\widetilde{\omega}$ is a smooth $E$–valued $1$–form on $X=M-\Sigma$.
In [31, Theorem 2.7], Hodgson and Kerckhoff prove that so long as all cone
angles are at most $2\pi$, which is always the case in our setting, there is a
unique closed and co-closed harmonic form $\omega$ such that
$[\widetilde{\omega}]=[\omega]\in H^{1}(X,E)$, and furthermore
$\widetilde{\omega}-\omega=ds$, where $s$ is a globally defined $L^{2}$
section of $E$. This choice of $\omega$ determines the one-parameter family of
cone-metrics $g_{t}$ on the nose, and defines a _natural identity map_
${\mathop{id}}\colon(M-\Sigma,g_{a})\to(M-\Sigma,g_{b})$ that allows us to
compare the metrics at any given point. Because of the canonical way in which
$\omega$ is chosen, the identity map conjugates every isometry of
$(M,\Sigma,g_{a})$ to an isometry of $(M,\Sigma,g_{b})$. We will say, for
short, that the identity map is _equivariant with respect to the symmetry
group of $(M,\Sigma)$_.

In Sections 8 and 9, we will always use this $1$–form $\omega$ and the
accompanying identity map. In Section 7, where we will need the flexibility to
enlarge $\Sigma$ to a larger link $\Sigma^{+}$ containing a non-singular
geodesic, we will accordingly choose a harmonic form $\omega$ with reference
to $\Sigma^{+}$.

##  6\. Bounding the boundary terms

In this section, we will find explicit bounds on certain boundary terms that
arise in the cone deformation. These boundary terms were used in [31] to prove
that there are no infinitesimal deformations of hyperbolic cone manifolds
fixing the cone angles. They have been used in many other applications of cone
deformations to obtain geometric control. We will use boundary terms in
Section 7 to bound the change in length of a non-singular geodesic, and in
Section 8 to get bilipschitz estimates in the thick part of a manifold.

This section is quite technical, reviewing definitions and results from [31,
33, 34] that require significant work from analysis and differential geometry
to state and to prove. For our applications, we need only the results
(technical though they are), and not the analysis. Therefore we will skim over
some of the definitions and results quickly, sweeping the complicated work of
[31, 33, 34] into the references, pointing the reader to statements in those
papers for careful definitions and details. Our goal in being brief is to
attempt to avoid unnecessary complications that are peripheral to our
applications. The reader interested only in the applications can view this
section as a black-box, while the reader with more interest in cone
deformations can still turn to the references for details.

###  6.1. Definitions and setup

Throughout, $(M,\Sigma)$ will be a hyperbolic cone manifold. We will also
consider a submanifold $X\subset M$ with _tubular boundary_ : This means that
$X$ is either a model tube, or the complement of some number of model tubes.
We orient the boundary of $X$ by _inward_ normal vectors. This orientation
will be important, as it affects the signs of our results.

Recall from Section 5.1 that an infinitesimal deformation of a cone manifold
structure can be represented by a harmonic $1$–form $\omega$, and that we made
a canonical choice of $\omega$ in Remark 5.18. The harmonic form $\omega$
decomposes as $\omega=\eta+i*D\eta$, as in (5.2). In [31, Proposition 1.3 and
page 36], Hodgson and Kerckhoff show that integrating by parts over the
submanifold $X$, again oriented by inward normal, gives

(6.1)
$\int_{X}\|\omega\|^{2}dV=\int_{X}||D\eta||^{2}+||\eta||^{2}\,dV=\int_{\partial
X}*D\eta\wedge\eta.$

See also [33, Lemma 2.3] for a formulation of the result in notation that
better matches ours.

The term on the far right of (6.1) is important. Thus Hodgson and Kerckhoff
define the _boundary term_ $b_{X}$ on $TX$–valued 1–forms $\mu$ and $\nu$ as
follows.

(6.2) $b_{X}(\mu,\nu)=\int_{\partial X}*D\mu\wedge\nu.$

Thus the term on the far right of (6.1) becomes $b_{X}(\eta,\eta)$.

Next, recall from (5.4) and the ensuing discussion that $\omega$ can be
written as a sum $\omega=\omega_{0}+\omega_{c}$ where $\omega_{0}$ is written
in terms of the explicit forms $\omega_{m}$ and $\omega_{\ell}$ that affect
meridian and longitude, and $\omega_{c}$ is a correction term. We may write
$\omega_{0}=\eta_{0}+i*D\eta_{0}$ and $\omega_{c}=\eta_{c}+i*D\eta_{c}$. Then
(6.1) becomes

(6.3)
$\int_{X}\|\omega\|^{2}dV=b_{X}(\eta,\eta)=b_{X}(\eta_{0},\eta_{0})+b_{X}(\eta_{c},\eta_{c}),$

using [33, Lemma 2.5] (the cross terms vanish). See [34, Equation (6) and
(7)], where integration is implicit in their definition of the $L^{2}$ norm.

We emphasize that the above formulas (6.1)–(6.3) hold both when $X$ is a model
tube and when $X$ is the complement of some number of model tubes. This
flexibility will be important in Section 7.

For the rest of this section, and in Section 8, boundary terms will appear in
the following specific context. Let ${\mathbf{r}}=(r_{1},\ldots,r_{n})$ be a
vector of positive radii. Suppose that $U_{\mathbf{r}}=U_{\mathbf{r}}(\Sigma)$
is an embedded multi-tube about the singular locus $\Sigma$, as in Definition
4.1, and let $X_{\mathbf{r}}=M-U_{\mathbf{r}}$. The inward normal vectors that
orient $\partial X_{\mathbf{r}}$ point away from $\Sigma$. For any
$TX_{\mathbf{r}}$–valued $1$–forms $\mu$ and $\nu$, define

$b_{\mathbf{r}}(\mu,\nu)=b_{X_{\mathbf{r}}}(\mu,\nu)=\int_{\partial
X_{\mathbf{r}}}*D\mu\wedge\nu.$

######  Lemma 6.4.

Let ${\mathbf{r}}=(r_{1},\dots,r_{n})$, where $\tanh(r_{j})\geq 1/\sqrt{3}$
for all $j$. Then

$b_{\mathbf{r}}(\eta_{c},\eta_{c})\leq 0.$

###### Proof.

Let $U$ be a solid torus of radius $r$. Then the principal curvatures of
$\partial U$ are $k_{1}=\tanh r$ and $k_{2}=\coth r$. See, for instance, [34,
Page 1066]. Thus, under the hypotheses of the lemma, the principal curvatures
along every component $\partial X$ satisfy $1/\sqrt{3}\leq k_{1}\leq
k_{2}\leq\sqrt{3}$.

Under this hypothesis on principal curvatures, Hodgson and Kerckhoff prove in
[34, Theorem 4.2] that $b_{\mathbf{r}}(\eta_{c},\eta_{c})\leq 0$. ∎

We remark that the hypotheses of Lemma 6.4 also imply
$b_{\mathbf{r}}(\eta_{0},\eta_{0})>0$. See [34, Corollary 4.3].

We will need an upper bound on $b_{\mathbf{r}}(\eta,\eta)$. By Lemma 6.4, this
amounts to finding an upper bound on $b_{\mathbf{r}}(\eta_{0},\eta_{0})$.

######  Lemma 6.5.

With the parametrization $t=\alpha^{2}$, the boundary term
$b_{\mathbf{r}}(\eta_{0},\eta_{0})$ satisfies

$b_{\mathbf{r}}(\eta_{0},\eta_{0})\leq\sum_{j=1}^{n}\frac{4(1-z_{j}^{2})}{z_{j}^{2}(3-z_{j}^{2})}\cdot\frac{1}{16\alpha^{4}}\cdot\mathcal{A}_{j}$

where $z_{j}=\tanh(r_{j})$, and $\alpha$ is the cone angle, and
$\mathcal{A}_{j}$ is the visual area.

###### Proof.

This result is contained in the proof of [34, Proposition 5.4]. On the bottom
of page 1074 and the top of page 1075, it is shown that

$b_{\mathbf{r}}(\eta_{0},\eta_{0})\leq\sum_{j=1}^{n}\frac{4a_{j}c_{j}-b_{j}^{2}}{4a_{j}}s^{2}\mathcal{A}_{j}$

where $s=\frac{-1}{2\alpha}\frac{d\alpha}{dt}$, and where where $a_{j}$,
$b_{j}$, and $c_{j}$ are as in [34, equation (32)]:

$\displaystyle
a_{j}=\frac{-\sinh^{2}r_{j}}{\cosh^{2}r_{j}}(2\cosh^{2}r_{j}+1),\qquad
b_{j}=\frac{-2}{\cosh^{2}r_{j}},\qquad
c_{j}=\frac{2\cosh^{2}r_{j}-1}{\sinh^{2}r_{j}\cosh^{2}r_{j}}.$

As in [34, page 1079], we let $t=\alpha^{2}$. Thus, as in (5.8), we have
$s=-1/(4\alpha^{2})$. Rewriting $a_{j}$, $b_{j}$, and $c_{j}$ in terms of
$z_{j}$, using Lemma A.1, gives the result. ∎

###  6.2. Controlling length and visual area

The next several lemmas prove estimates relating how visual area changes under
cone deformations. These results culminate in an estimate relating the
normalized length $L$, measured on cusps in the complete metric on $M-\Sigma$,
to the length $\ell=\operatorname{len}_{4\pi^{2}}(\Sigma)$ in the complete
metric on $M$. This will feed into the bound on boundary terms later in this
section.

######  Remark 6.6.

We recall notation that will be used below. As usual, we are assuming that a