# Effective distance between nested Margulis tubes

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

We give sharp, effective bounds on the distance between tori of fixed
injectivity radius inside a Margulis tube in a hyperbolic $3$–manifold.

###### 2010 Mathematics Subject Classification:

57M50, 30F40

##  1\. Introduction

A key tool in the study of hyperbolic manifolds is the _thick-thin
decomposition_. For any number $\epsilon>0$, a manifold $M$ is decomposed into
the _$\epsilon$ –thin_ part, consisting of points on essential loops of length
less than $\epsilon$, and its complement the _$\epsilon$ –thick_ part.
Margulis proved the foundational result that there is a universal constant
$\epsilon_{3}>0$ such that for any hyperbolic $3$–manifold $M$, the
$\epsilon_{3}$–thin part is a disjoint union of cusps and tubes. Analogous
statements hold in all dimensions, and for more general symmetric spaces. This
result has had numerous important consequences in the study and classification
of hyperbolic $3$–manifolds and Kleinian groups. Thurston and Jorgensen used
the Margulis lemma to describe the structure of the set of volumes of
hyperbolic manifolds, with limit points occurring only via Dehn filling [27].
The Margulis lemma also plays a major role in the construction of model
manifolds used in the proofs of the Ending Lamination Theorem [20, 7] and the
Density Theorem [22, 23].

Since tubes and cusps are well-understood quotients of hyperbolic space by
elementary groups, it seems that the thin parts of manifolds should be easy to
analyze. However, in practice, the thin parts of a manifold are often very
difficult to control. For example, the optimal value for the Margulis constant
$\epsilon_{3}$ is still unknown. The best known estimate is due to Meyerhoff
[19]. Additionally, given an $\epsilon$–thin tube, it is very difficult to
analyze and bound simple quantities such as the radius of the tube in full
generality. This is because the radius depends not only on $\epsilon$, but
also on the rotation and translation — the _complex length_ — at the core of
the tube. Although the radius is a continuous function of these parameters, it
is non-differentiable in many places. See Proposition 3.10 for a formula, and
Figure 2 for a graph.

In this paper, we address the geometry of thin parts of (possibly singular)
hyperbolic $3$–manifolds. Given $0<\delta<\epsilon$ and a tube of injectivity
radius $\epsilon/2$ (the $\epsilon$–tube, for short), we determine sharp,
effective bounds on the distance between the boundaries of the $\epsilon$–tube
and the $\delta$–tube. These bounds are independent of the complex length
$\lambda+i\tau$ of the core of the tube. Our main result is the following.

######  Theorem 1.1.

Suppose that $0<\delta<\epsilon\leq 0.3$. Let $N=N_{\alpha,\lambda,\tau}$ be a
hyperbolic solid torus whose core geodesic has complex length $\lambda+i\tau$
and cone angle $\alpha\leq 2\pi$, where $\lambda\leq\delta$. Then the distance
$d_{\alpha,\lambda,\tau}(\delta,\epsilon)$ between the $\delta$ and $\epsilon$
tubes satisfies

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

We remark that the argument of $\operatorname{arccosh}$ in the lower bound of
Theorem 1.1 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. The real point is that the lower bound is not
very strong (less than $\epsilon/2$) for any pair $(\delta,\epsilon)$ such
that $\epsilon<\sqrt{7.256\,\delta}$. On the other hand, the lower bound of
Theorem 1.1 is sharp up to additive error for any pair $(\delta,\epsilon)$
where $\epsilon\geq\sqrt{7.256\,\delta}$. The upper bound is sharp for every
pair $(\delta,\epsilon)$. See Section 1.3 for more details.

###  1.1. Motivation and applications

Ineffective bounds on the distances between thin tubes were previously
observed by Minsky [20, Lemma 6.1] and Brock–Bromberg [5, Theorem 6.9], who
credit the bound to Brooks and Matelski [8]. Universal bounds of this sort,
depending only on $\delta$ and $\epsilon$, are required in the proof of the
Ending Lamination Theorem, both for punctured tori [20] and for general
surfaces [21, 7]. In particular, Minsky used such bounds in the proof of the
“a priori bounds” theorem [21] that curves appearing in a hierarchy have
universally bounded length. One consequence of “a priori bounds” is Brock’s
volume estimate for quasifuchsian manifolds and for mapping tori [3, 4]. A
second consequence is the result (due to Minsky, Bowditch, and Brock–Bromberg
[2, 6]) that distance in the curve complex of a surface $S$ is coarsely
comparable to electric distance in a $3$–manifold of the form
$S\times{\mathbb{R}}$.

In a slightly different direction, Brock and Bromberg applied the ineffective
bounds on distances between tubes to cone-manifolds, establishing uniform
bilipschitz estimates between the thick part of a cusped $3$–manifold and the
thick parts of its long Dehn fillings [5]. This application requires a version
of Theorem 1.1 for solid tori with a cone-singularity at the core. In turn,
the Brock–Bromberg result has been combined with the Ending Lamination Theorem
to show that geometrically finite hyperbolic $3$–manifolds are dense in the
space of all (tame) hyperbolic $3$–manifolds [22, 23].

The past few years have witnessed an intense effort to make theorems in coarse
geometry _effective_ , that is, to make the constants explicit. Recent
effective results include, for instance, [1, 14, 16]. Finding an effective
version of the distance between thin tubes has been a major obstacle to
extending those efforts. Theorem 1.1 provides such an effective result.

Theorem 1.1 is already being applied to obtain effective versions of several
results mentioned above. Futer and Taylor have outlined an effective “a priori
bounds” theorem, combining Theorem 1.1 with sweepout arguments [14] and
effective results about hierarchies [1]. Aougab, Patel, and Taylor have found
an effective “electric distance” theorem, again using a combination of Theorem
1.1 and sweepout arguments. Finally, the authors of this paper have used
Theorem 1.1 in combination with a number of cone-manifold estimates due to
Hodgson and Kerckhoff [17, 18] to effectivize the Brock–Bromberg bilipschitz
theorem [13]. Our effective results on cone-manifolds require Theorem 1.1 to
hold for singular solid tori.

Finally, Theorem 1.1 offers a useful step toward finding the Margulis constant
$\epsilon_{3}$. The current state of knowledge is that
$0.104\leq\epsilon_{3}\leq 0.775$, with the lower bound due to Meyerhoff [19]
and the upper bound realized by the Weeks manifold. Furthermore, a theorem of
Shalen [24, 25], building on earlier work of Culler and Shalen [12], says that
$0.29$ is a Margulis number for all but finitely many hyperbolic
$3$–manifolds. That is, for all but finitely many choices of $M$, the
$0.29$–thin part of $M$ is a disjoint union of cusps and tubes. Any manifold
$M$ failing this property must be closed and must have
$\operatorname{vol}(M)<52.8$. By combining Theorem 1.1 with our work on cone-
manifolds, we produce an explicit lower bound on the injectivity radius of any
exception to Shalen’s theorem. This makes it theoretically feasible (although
computationally impractical) to enumerate all manifolds with
$\operatorname{vol}(M)<52.8$ and injectivity radius bounded below, and to
determine their Margulis numbers [13].

###  1.2. Distance between tubes, as a function

Let $0<\delta\leq\epsilon$ be injectivity radii, and consider a solid torus
$N=N_{\alpha,\lambda,\tau}$ whose core geodesic has complex length
$\lambda+i\tau$ and cone angle $\alpha\leq 2\pi$. The distance between the
$\epsilon$– and $\delta$–tubes in $N$ is defined carefully in Definition 3.1.
We denote this distance by $d_{\alpha,\lambda,\tau}(\delta,\epsilon)$. For our
current discussion, it helps to note that
$d_{\alpha,\lambda,\tau}(\delta,\epsilon)$ is the difference of tube radii of
the $\epsilon$–tube and $\delta$–tube, and that each tube radius is determined
by taking a maximum of many smooth functions. See Proposition 3.10 for an
exact formula. As a consequence, $d_{\alpha,\lambda,\tau}(\delta,\epsilon)$ is
a continuous but only piecewise smooth function of the quantities $\delta$,
$\epsilon$, $\lambda$, and $\tau$.

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\end{overpic} Figure 1. In each region, all complex lengths $\lambda+i\tau$
have the indicated _power_ for $\epsilon=0.2$. That is: every based loop of
length $\epsilon$ that realizes injectivity radius is freely homotopic to this
power of the core. This figure was inspired by [11, Figure 2].

The failure of global smoothness can be explained as follows. For each value
of $\epsilon>0$ and each complex length $\lambda+i\tau$ of the core of $N$,
the radius of the $\epsilon$–tube is determined by some power of the generator
of $\pi_{1}(N)$. This power can change as the data $\lambda,\tau,\epsilon$
changes. For instance, Figure 1 shows the power of the core when
$\epsilon=0.2$ is fixed but $\lambda+i\tau$ varies. Meanwhile, Example 3.9
shows how the power can depend on $\epsilon$.

\begin{overpic}[width=361.34999pt]{juarez.png} \put(89.0,15.0){$\lambda$}
\put(36.0,2.0){$\tau$} \end{overpic} Figure 2. The graph of
$d_{2\pi,\lambda,\tau}(\delta,\epsilon)$ when $\epsilon=0.2$ and $\delta=0.05$
are fixed, while $\lambda$ and $\tau$ are varying. The values of
$d_{2\pi,\lambda,\tau}(\delta,\epsilon)$ range from approximately $0.117$ to
$2.065$. By comparison, Theorem 1.1 gives $\frac{\epsilon-\delta}{2}=0.075$ as
a fairly good lower bound and $2.0650\ldots$ as a sharp upper bound.

Figure 2 shows the graph of $d_{2\pi,\lambda,\tau}(\delta,\epsilon)$ when
$\delta=0.05$ and $\epsilon=0.2$ are fixed but $(\lambda,\tau)$ vary. Since
the graph is the difference of a pair of wildly varying, piecewise-smooth
functions, it is an extremely complicated terrain of deep valleys, narrow
ridges, and sharp peaks. The sharp ridges are points of non-differentiabilty,
and occur where the power of the core for $\delta$ changes. Other points of
non-differentiability, where the power for $\epsilon$ changes, occur in the
valleys. Even though $\delta$ and $\epsilon$ are fixed, the value of
$d_{\alpha,\lambda,\tau}(\delta,\epsilon)$ ranges a great deal: from
approximately $0.117$ to $2.065$. Nevertheless, Theorem 1.1 gives upper and
lower bounds that depend only on $\delta$ and $\epsilon$.

###  1.3. Sharpness and numerical constants

The hypothesis $\epsilon\leq 0.3$ in Theorem 1.1 is slightly arbitrary. This
hypothesis is not needed at all in the upper bound (see Proposition 5.7). In
the lower bound, our line of argument requires $\epsilon$ to be bounded in
some way, with the choice of bound affecting the additive constant ($-0.0424$)
in the statement. (See Theorem 8.8 for a generalized statement that holds for
larger $\epsilon$.) We chose the value $0.3$ because of its connection to
currently available estimates on the Margulis constant $\epsilon_{3}$. In
particular, as described in Section 1.1, applying Theorem 1.1 with
$\epsilon=0.29$ can help determine the finite list of manifolds for which
$0.29$ fails to be a Margulis number.

There are interesting examples illustrating the sharpness of both the upper
and lower bounds of Theorem 1.1. As Proposition 5.7 will show, the upper bound
of Theorem 1.1 is sharp for every pair $(\delta,\epsilon)$. It is attained if
and only if $N$ is a nonsingular tube whose core has complex length
$\lambda+i\tau=\delta+0$; that is, the core has length $\delta$ and trivial
rotational part.

The lower bound of Theorem 1.1 is sharp up to additive error, which can be
seen as follows. For every pair $(\delta,\epsilon)$ such that
$0<\sqrt{7.256\delta}\leq\epsilon\leq 0.3$, Theorem 4.6 constructs a solid
torus $N=N_{2\pi,\lambda,\tau}$ such that

(1.2)
$d_{2\pi,\lambda,\tau}(\delta,\epsilon)\>\leq\>\operatorname{arccosh}\left(1.116\frac{\epsilon}{\sqrt{\delta}}\right).$

The core of $N$ has complex length $\lambda+i\tau=1/n^{2}+2\pi i/n$, where $n$
is the least natural number such that $1/n^{2}\leq\delta$. Meanwhile, Theorem
1.1 gives

(1.3)
$d_{2\pi,\lambda,\tau}(\delta,\epsilon)\>\geq\>\operatorname{arccosh}\left(\frac{\epsilon}{\sqrt{7.256\,\delta}}\right)-0.0424.$

Since $\operatorname{arccosh}(x)\sim\log(2x)$ for large $x$, the expressions
in (1.2) and (1.3) differ by an additive error. In fact, the additive
difference is less than $2.2$.

One consequence of the above paragraphs is that Theorem 1.1 is sharpest when
the solid torus $N$ is nonsingular; that is, when $N$ is the quotient of
${\mathbb{H}}^{3}$ by a loxodromic isometry. Thus, while extending Theorem 1.1
to singular tubes introduces a few technical complications (for example see
Propositions 3.10 and 5.6), this extension does not weaken the statement in
any way. We emphasize that the extension to singular tubes is needed for our
forthcoming applications to cone-manifolds and to bounding the Margulis
numbers of (nonsingular) hyperbolic manifolds.

The results of this paper have an interesting relation to the discussion of
Margulis tubes in the work of Minsky (see [20, Section 6] and [21, Section
3.2.2]). On the one hand, Proposition 7.1 confirms and effectivizes Minsky’s
assertion ([20, proof of Lemma 6.1] and [21, Equation (3.6)]) that, for
$\epsilon\leq\epsilon_{3}$, there is a constant $c=c(\epsilon)$ such that the
radius of an $\epsilon$–tube of core length $\lambda+i\tau$ satisfies

$r_{2\pi,\lambda,\tau}(\epsilon)\geq\log\left(\frac{1}{\sqrt{\lambda}}\right)-c.$

On the other hand, the examples of Theorem 4.6 satisfy (1.2) and therefore
contradict [21, Equation (3.7)]. Of course, this does not affect the overall
correctness of [21], as _any_ lower bound depending only on $\epsilon$ and
$\delta$ that grows large as $\delta\to 0$ (for instance, the lower bound of
Theorem 1.1) suffices for Minsky’s work toward the classification of Kleinian
surface groups.

###  1.4. Cusps

Recall that the thin part of a hyperbolic 3–manifold is a disjoint union of
cusps and tubes. Although Theorem 1.1 is only stated for tubes, there is a
simpler and stronger statement for cusps.

######  Proposition 1.4.

Let $0<\delta<\epsilon$. Let $N$ be a horocusp whose $\epsilon$–thick part,
$N^{\geq\epsilon}$, is non-empty. Then the $\delta$–thin and $\epsilon$–thick
parts of $N$ are separated by distance

$d_{N}(\delta,\epsilon)=d(N^{\leq\delta},N^{\geq\epsilon})=\log\left(\frac{\sinh(\epsilon/2)}{\sinh(\delta/2)}\right).$

###### Proof.

Let $x\in N$ be a point such that ${\operatorname{injrad}}(x)=\delta/2$. Then
there must be a lift $\widetilde{x}\in\widetilde{N}$ and a parabolic covering
transformation $\varphi$, such that
$d(\widetilde{x},\varphi\widetilde{x})=\delta$. By [10, Lemma A.2], a
horocyclic segment from $\widetilde{x}$ to $\varphi\widetilde{x}$ has length
$2\sinh(\delta/2)$. Similarly, if $y\in N$ is a point such that
${\operatorname{injrad}}(y)=\epsilon/2$, then there is a horocyclic segment
from $\widetilde{y}$ to $\varphi\widetilde{y}$ of length $2\sinh(\epsilon/2)$.
Now, a standard calculation shows that the horospheres containing
$\widetilde{x}$ and $\widetilde{y}$ are separated by distance

$d_{N}(\delta,\epsilon)=d(N^{\leq\delta},N^{\geq\epsilon})=\log\left(\frac{2\sinh(\epsilon/2)}{2\sinh(\delta/2)}\right).\qed$

We observe that the distance $d_{N}(\delta,\epsilon)$ satisfies both the upper
and lower bounds of Theorem 1.1. Thus Theorem 1.1 also applies to cusps.

###  1.5. Organization

Section 2 lays out definitions and sets up notation that will be used for the
remainder of the paper. Section 3 proves Proposition 3.10, which gives an
exact formula for the radius of an $\epsilon$–thin tube.

Section 4 describes a family of examples showing the sharpness of the lower
bound of Theorem 1.1. Section 5 proves the upper bound of Theorem 1.1 and
shows that it is sharp.

The lower bound of Theorem 1.1 requires a delicate case analysis, treating
shallow and deep tubes separately. For a maximally sharp estimate, we rely on
a result of Zagier [19], later improved by Cao, Gehring, and Martin [9]. We
obtain a bound on the Euclidean metric on the tube boundary in Section 6, and
prove a lower bound on the depth of an $\epsilon$–tube in Section 7. These
ingredients are combined to give the final proof in Section 8.

Appendix A contains several elementary lemmas in hyperbolic trigonometry that
are useful elsewhere in the paper.

###  1.6. Acknowledgements

Futer was supported in part by NSF grant DMS–1408682. Purcell was supported in
part by the Australian Research Council. All three authors acknowledge support
from NSF grants DMS–1107452, 1107263, 1107367, “RNMS: Geometric Structures and
Representation Varieties” (the GEAR Network), which funded an international
trip to collaborate on this paper.

We thank Ian Biringer and Yair Minsky for a number of enlightening
conversations. We also thank Tarik Aougab, Marc Culler, Priyam Patel, and Sam
Taylor for their helpful suggestions.

##  2\. Tubes and equidistant tori

In this section, we set notation and give definitions for solid tori, tubes,
and injectivity radii that will be used for the remainder of the paper.

To form a nonsingular hyperbolic tube, one starts with a neighborhood of a
geodesic in ${\mathbb{H}}^{3}$ and takes a quotient under a loxodromic
isometry fixing that geodesic. In order for our results to hold for both
nonsingular and singular tubes, we will take a quotient of a more complicated
space, as in the following definition. Fix $0\in{\mathbb{H}}^{3}$ to be an
arbitrary basepoint.

######  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 with
infinite cone angle.

There is a natural action of ${\mathbb{C}}$ (as an additive group) on
${\hat{{\mathbb{H}}}^{3}}$, where $z\in{\mathbb{C}}$ translates $\hat{\sigma}$
by distance $\operatorname{Re}(z)$ and rotates by angle
$\operatorname{Im}(z)$. Since $\hat{\sigma}$ has infinite cone angle, we have
that angles of rotation are 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 reference 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 $ds$,
where

(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 $\Sigma$ is a closed geodesic of complex length
$\lambda+i\tau$, and with a cone singularity of angle $\alpha$ at the core. We
call $N=N_{\alpha,\lambda,\tau}$ a _model solid torus_.

Note that if $\alpha=2\pi$, the quotient of ${\hat{{\mathbb{H}}}^{3}}$ by the
elliptic $\psi_{i\alpha}$ recovers ${\mathbb{H}}^{3}$. In this case, the model
solid torus $N_{2\pi,\lambda,\tau}$ is _nonsingular_ , and can be identified
with ${\mathbb{H}}^{3}/\langle\varphi_{\lambda+i\tau}\rangle$, where
$\varphi_{\lambda+i\tau}$ is a loxodromic isometry of ${\mathbb{H}}^{3}$ with
complex length $\lambda+i\tau$.

######  Definition 2.4.

Let $N=N_{\alpha,\lambda,\tau}$ be a model solid torus, and let $x\in N$. Then
the _injectivity radius_ of $x$, denoted ${\operatorname{injrad}}(x)$, is the
supremal radius $r$ such that an open metric $r$–ball about $x$ is isometric
to a ball $B_{r}(0)\subset{\mathbb{H}}^{3}$. Since we are using open balls,
the supremal radius is attained. If $\alpha\neq 2\pi$ and $x$ lies on the
singular core geodesic, we set ${\operatorname{injrad}}(x)=0$.

For $\epsilon>0$, the _$\epsilon$ –thick part_ of $N$ is

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

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

We emphasize that our definition of the _$\epsilon$ –thick part_ corresponds
to injectivity radius $\epsilon/2$ rather than $\epsilon$. Both choices seem
to be common in the literature on Kleinian groups. Our convention agrees with
that of Minsky [20, 21] and Brock–Canary–Minsky [7], while differing from the
convention of Brock–Bromberg [5] and Namazi–Souto [22].

If $M$ is a non-singular manifold, every essential loop through a point $x\in
M^{\geq\epsilon}$ has length at least $\epsilon$. A similar statement holds
for singular tubes.

######  Lemma 2.5.

Let $N=N_{\alpha,\lambda,\tau}$ be a model solid torus whose core is singular.
That is, assume $N$ is the quotient of ${\hat{{\mathbb{H}}}^{3}}$ by
${\mathbb{Z}}^{2}\cong\langle\psi_{i\alpha},\varphi_{\lambda+i\tau}\rangle$
for $\alpha\neq 2\pi$. Choose a point $x\in N$ and a lift
$\widetilde{x}\in{\hat{{\mathbb{H}}}^{3}}$. Set
$\epsilon=2\,{\operatorname{injrad}}(x)$. Then

(2.6) $\displaystyle\epsilon$
$\displaystyle=\min\left\\{d(\widetilde{x},\eta\widetilde{x}):\eta\in{\mathbb{Z}}^{2}-\\{0\\}\right\\}.$

Similarly, if $N$ is a nonsingular solid torus, the quotient of
${\mathbb{H}}^{3}$ by
${\mathbb{Z}}\cong\langle\varphi_{\lambda+i\tau}\rangle$, and
$\widetilde{x}\in{\mathbb{H}}^{3}$ is a point covering $x$, then

(2.7) $\displaystyle\epsilon$
$\displaystyle=\min\left\\{d(\widetilde{x},\eta\widetilde{x}):\eta=\varphi_{\lambda+i\tau}^{n}\in{\mathbb{Z}}-\\{0\\}\right\\}.$

###### Proof.

We focus on the singular case, as the nonsingular case is well-known. For an
arbitrary basepoint $0\in{\mathbb{H}}^{3}$, there is an isometric embedding
$f\colon B_{\epsilon/2}(0)\to N$, such that $f(0)=x$. Then any non-trivial
element $\eta\in{\mathbb{Z}}^{2}$ must translate
$B_{\epsilon/2}(\widetilde{x})$ by distance at least $\epsilon$, so

$\min\\{d(\widetilde{x},\eta\widetilde{x}):\eta\in{\mathbb{Z}}^{2}-\\{0\\}\\}\geq\epsilon.$

Next, since ${\operatorname{injrad}}(x)=\epsilon/2$, the continuous extension
of $f$ to $\overline{B_{\epsilon/2}(0)}$ either hits the core $\Sigma$ or
fails to be one to one. If it fails to be one to one, then a lifted ball is
tangent to a translate of itself under a nontrivial element
$\eta\in{\mathbb{Z}}^{2}$. On the other hand, if it meets the core $\Sigma$,
then there is a point $z\in f(\overline{B_{\epsilon/2}(0)})\cap\Sigma$. The
preimage $\widetilde{z}$ of $z$ is fixed by an elliptic subgroup
$\langle\psi_{i\alpha}\rangle\subset{\mathbb{Z}}^{2}$. Thus a translate of the
lifted ball by the generator $\eta=\psi_{i\alpha}$ will be tangent to the
ball.

In either case, there are two distinct lifts of the ball, namely
$B_{\epsilon/2}(\widetilde{x})$ and $B_{\epsilon/2}(\eta\widetilde{x})$, that
are tangent in ${\hat{{\mathbb{H}}}^{3}}$. Therefore,
$d(\widetilde{x},\eta\widetilde{x})=\epsilon$, and the minimum over all group
elements must be at most $\epsilon$. The minimum is attained because
${\mathbb{Z}}^{2}$ acts discretely. ∎

######  Definition 2.8.

If $N$ is a nonsingular tube and $\epsilon\geq\lambda$, we define the _power_
for $\epsilon$ to be any $n\in{\mathbb{N}}$ so that the deck transformation
$\eta=\varphi^{n}$ realizes the minimum in Equation (2.7). If $N$ is a
singular tube and $\epsilon>0$, we define the _power_ for $\epsilon$ to be any
$n\in{\mathbb{N}}\cup\\{0\\}$ so that the deck transformation
$\eta=\varphi^{n}\psi^{m}$ realizes the minimum in Equation (2.6), for some
$m\in{\mathbb{Z}}$. The power is uniquely defined for almost every $\epsilon$.

We remark that the power for $\epsilon$ is genuinely a function of $\epsilon$
and $N$, and does not depend on the choice of $x\in N$ such that
${\operatorname{injrad}}(x)=\epsilon/2$. This is because for every
$\epsilon>0$ and every loxodromic
$\eta\in{\operatorname{Isom}}({\hat{{\mathbb{H}}}^{3}})$, the set
$\\{y\in{\hat{{\mathbb{H}}}^{3}}:d(y,\eta y)=\epsilon\\}$ is a Euclidean plane
at fixed radius from the core geodesic $\hat{\sigma}$ (or equal to
$\hat{\sigma}$, or empty). The quotient of this set in $N$ is equidistant from
the core of $N$. This is a distinctly $3$–dimensional phenomenon: already for
$\eta\in{\operatorname{Isom}}({\mathbb{H}}^{4})$, the set of points moved by
distance $\epsilon$ can be far more complicated [26].

##  3\. Tube radii

We now provide a bit of background for tube radii in hyperbolic $3$–manifolds,
well-known to the experts. Let $N=N_{\alpha,\lambda,\tau}$ be a model solid
torus, as in Definition 2.3. Our eventual goal is to bound distances between
the boundaries of the $\epsilon$–thin and $\delta$–thin tubes
$N^{\leq\epsilon}$ and $N^{\leq\delta}$, independently of $\alpha$, $\lambda$,
and $\tau$. In this section, we derive a formula for the radius of one such
tube.

######  Definition 3.1.

Let $\epsilon>0$, and assume the $\epsilon$–thin part $N^{<\epsilon}$ is non-
empty. Then $T^{\epsilon}=\partial N^{\leq\epsilon}$ 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 the core geodesic $\Sigma$. We
denote this radius by

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

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

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

Given $0<\delta<\epsilon$, where $N^{<\delta}\neq\emptyset$, we define the
distance $d(\delta,\epsilon)$ between $T^{\delta}$ and $T^{\epsilon}$ as
follows:

(3.2)
$d(\delta,\epsilon)=d_{\alpha,\lambda,\tau}(\delta,\epsilon)=r_{\alpha,\lambda,\tau}(\epsilon)-r_{\alpha,\lambda,\tau}(\delta).$

Our next goal is to find a formula for
$r(\epsilon)=r_{\alpha,\lambda,\tau}(\epsilon)$ in terms of complex lengths of
isometries stabilizing the singular geodesic of ${\hat{{\mathbb{H}}}^{3}}$.

######  Definition 3.3.

Let $\varphi_{\lambda+i\tau}$ be an isometry of ${\hat{{\mathbb{H}}}^{3}}$,
fixing the singular geodesic $\hat{\sigma}$, with complex length
$\lambda+i\tau$. For $\epsilon\geq\lambda$, define the _translation radius_ ,
denoted ${\operatorname{trad}}_{\lambda,\tau}(\epsilon)$, to be the value of
$r$ such that $\varphi_{\lambda+i\tau}$ translates all points of the form
$(r,\zeta,\theta)\in{\hat{{\mathbb{H}}}^{3}}$ by distance $\epsilon$.

We also need to compute the translation radius of a loxodromic isometry
$\varphi_{\lambda+i\tau}$ acting on ${\mathbb{H}}^{3}$ instead of
${\hat{{\mathbb{H}}}^{3}}$. Since angles in ${\mathbb{H}}^{3}$ are only
defined modulo $2\pi$, this radius coincides with the translation radius of
$\varphi_{\lambda+i(\tau{\\!\\!\\!\mod{}}2\pi)}$ acting on
${\hat{{\mathbb{H}}}^{3}}$, namely
${\operatorname{trad}}_{\lambda,\,\tau{\\!\\!\\!\mod{}}2\pi}(\epsilon)$.

The usage $(\tau{\\!\\!\\!\mod{}}2\pi)$ can be generalized to angles modulo
other numbers.

######  Definition 3.4.

Given $a\in{\mathbb{R}}$ and $b>0$, we define $(a\,{\\!\\!\\!\mod{}}b)$ to be

$a{\\!\\!\\!\mod{}}b=x\in[-b/2,\,b/2)\quad\text{such that}\quad(a-x)\in
b{\mathbb{Z}}.$

In many situations, the translation radius can be computed in closed form.

######  Lemma 3.5.

Let $(r,\zeta_{1},\theta_{1})$ and $(r,\zeta_{2},\theta_{2})$ be points of
${\hat{{\mathbb{H}}}^{3}}$ in cylindrical coordinates, and let $d$ be the
distance between those points. If $|\theta_{1}-\theta_{2}|\leq\pi$, then

$\displaystyle\cosh d$
$\displaystyle=\cosh(\zeta_{1}-\zeta_{2})\cosh^{2}r-\cos(\theta_{1}-\theta_{2})\sinh^{2}r$
(3.6)
$\displaystyle=(\cosh(\zeta_{1}-\zeta_{2})-\cos(\theta_{1}-\theta_{2}))\cosh^{2}r+\cos(\theta_{1}-\theta_{2}).$

If $|\theta_{1}-\theta_{2}|\geq\pi$, then

$\cosh d=(\cosh(\zeta_{1}-\zeta_{2})+1)\cosh^{2}r)-1.$

Consequently, when $\epsilon\geq\lambda$ and $0\leq|\tau|\leq\pi$, we have

(3.7)
$r={\operatorname{trad}}_{\lambda,\tau}(\epsilon)=\operatorname{arccosh}\sqrt{\frac{\cosh\epsilon-\cos(\tau)}{\cosh\lambda-\cos(\tau)}.}$

###### Proof.

Equation (3.6) is proved in Gabai–Meyerhoff–Milley [15, Lemma 2.1]. See
Hodgson–Kerckhoff [17, Lemma 4.2] for the extension to singular tubes in the
case where $|\theta_{1}-\theta_{2}|\leq\pi$. If
$|\theta_{1}-\theta_{2}|\geq\pi$, then the geodesic in
${\hat{{\mathbb{H}}}^{3}}$ between the two points will pass through the
singular axis $\hat{\sigma}$. Thus the distance remains the same if we set
$|\theta_{1}-\theta_{2}|=\pi$, hence $\cos(\theta_{1}-\theta_{2})=-1$.

For Equation (3.7), we substitute $d=\epsilon$ and
$(\zeta_{1},\theta_{1})=(\lambda,\tau)$ and $(\zeta_{2},\theta_{2})=(0,0)$.
Solving for $r$ in (3.6) gives the result. ∎

######  Remark 3.8.

It will be convenient to refer to Definition 3.3 and Lemma 3.5 even in
situations when $\epsilon<\lambda$, hence when there _do not exist_ points of
${\hat{{\mathbb{H}}}^{3}}$ that $\varphi_{\lambda+i\tau}$ translates by
distance $\epsilon$. If $\epsilon<\lambda$, we define
${\operatorname{trad}}_{\lambda,\tau}(\epsilon)=-\infty$. Similarly, when
$x<1$, we set $\operatorname{arccosh}(x)=-\infty$. Under this convention,
(3.7) holds for all pairs $(\lambda,\epsilon)$.

The translation radius ${\operatorname{trad}}_{\lambda,\tau}(\epsilon)$ is
related to the radius of the $\epsilon$–thin part of
$N=N_{\alpha,\lambda,\epsilon}$, but they are not necessarily identical.

$\varphi$$\varphi^{2}$$\varphi^{2}$$\varphi$ Figure 3. Schematic showing how
the tangency pattern of balls $B_{\epsilon}$, $\varphi(B_{\epsilon})$, and
$\varphi^{2}(B_{\epsilon})$ depends on $\epsilon$. In this figure,
$\alpha=2\pi$, $\lambda=0.1$, $\tau=\pi$ are fixed. When $\epsilon$ is small,
the balls $B_{\epsilon}$ and $\varphi(B_{\epsilon})$ are tangent first
(shaded). When $\epsilon$ is larger, $B_{\epsilon}$ and
$\varphi^{2}(B_{\epsilon})$ become tangent first.

######  Example 3.9.

Set $\alpha=2\pi$, $\lambda=0.1$, and $\tau=\pi$. Then
$N=N_{\alpha,\lambda,\tau}$ is a quotient of ${\mathbb{H}}^{3}$. The generator
$\varphi=\varphi_{\lambda,\tau}$ of $\pi_{1}N\cong{\mathbb{Z}}$ will translate
any point $\widetilde{x}\subset{\mathbb{H}}^{3}$ along the invariant geodesic
$\sigma$, and also rotate it by $\pi$ about $\sigma$. When $\epsilon<0.1$, we
have $N^{\leq\epsilon}=\emptyset$, or equivalently
${\operatorname{trad}}_{\lambda,\tau}(\epsilon)=-\infty$. When
$0.1\leq\epsilon\leq 0.2$, the map $\varphi^{n}$ for $n\geq 2$ will translate
any point of ${\mathbb{H}}^{3}$, including points of $\sigma$, by a distance
larger than $\epsilon$. Thus, the radius of $N^{\leq\epsilon}$ is governed by
$\varphi$ alone: that is,
$r(\epsilon)={\operatorname{trad}}_{\lambda,\tau}(\epsilon)$ and the power for
$\epsilon$ is $1$.

Now, consider what happens when $\epsilon$ is ever so slightly larger than
$0.2$, say $\epsilon=0.201$. Because $\varphi^{2}$ has trivial rotational
part, a point $\widetilde{x}$ can lie relatively far from $\sigma$ (at radius
$r=0.1001\ldots$) and still be translated by distance $\epsilon$. In symbols,
for a point $\widetilde{x}=(r,0,0)\in{\mathbb{H}}^{3}$, we have

$d(\widetilde{x},\varphi^{2}\widetilde{x})=\epsilon\qquad\text{and}\qquad{\operatorname{trad}}_{2\lambda,\,2\tau\\!{\\!\\!\\!\mod{}}2\pi}(\epsilon)={\operatorname{trad}}_{2\lambda,\,0}(\epsilon)=r=0.1001\ldots.$

Meanwhile, because $\varphi$ rotates points by angle $\pi$ about $\sigma$, it
will move the same point $\widetilde{x}=(r,0,0)\in{\mathbb{H}}^{3}$ by a
distance much larger than $\epsilon$, whereas the points translated by
distance $\epsilon$ lie closer to $\sigma$:

$d(\widetilde{x},\varphi\widetilde{x})=0.2239\ldots>\epsilon\qquad\text{and}\qquad{\operatorname{trad}}_{\lambda,\tau}(\epsilon)=0.0871\ldots<r.$

Finally, $\varphi^{n}$ for $n>2$ has both translational and rotational part
larger than that of $\varphi^{2}$, so $\varphi^{n}$ will definitely move any
point of ${\mathbb{H}}^{3}$ further than $\varphi^{2}$. In fact, for every
$\epsilon\geq 0.201$, the radius of $N^{\leq\epsilon}$ is governed by
$\varphi^{2}$: that is,
$r(\epsilon)={\operatorname{trad}}_{2\lambda,2\tau}(\epsilon)$ and the power
for $\epsilon$ is $2$. See Figure 3.

The above example is instructive in two ways. First, it illustrates that the
power for $\epsilon$ is locally constant but jumps when $\epsilon$ crosses
certain isolated values. Additional examples of this phenomenon are described
in Section 4. Second, the displayed equations above show that while
$\epsilon=2{\operatorname{injrad}}(x)$ is determined by taking a _minimum_
distance over all nonzero powers of $\varphi$ (see Lemma 2.5), the tube radius
$r(\epsilon)$ is determined by taking a _maximum_ value of
${\operatorname{trad}}_{n\lambda,n\tau}(\epsilon)$ over all nonzero powers.
Proposition 3.10 makes this idea precise, for singular as well as nonsingular
tubes.

Before moving on, we note that the power for $\epsilon$ can only jump upward
as $\epsilon$ increases; see Remark 5.4. When $\epsilon$ is fixed but
$\lambda+i\tau$ varies, the jumping behavior of powers follows the
combinatorics of the Farey graph, as illustrated in Figure 1. The jumping
behavior also explains why the function $r_{\alpha,\lambda,\tau}(\epsilon)$
and the related function $d_{\alpha,\lambda,\tau}(\delta,\epsilon)$ have
points of non-differentiability, as visible in Figure 2.

We are now ready to give a formula for the radius of the $\epsilon$–tube
$N^{\leq\epsilon}$. The following is the main result of this section.

######  Proposition 3.10.

Let $N=N_{\alpha,\lambda,\tau}$ be a model solid torus of cone angle
$\alpha\leq 2\pi$. For any $\epsilon$ such that
$N^{\leq\epsilon}\neq\emptyset$, the radius
$r(\epsilon)=r_{\alpha,\lambda,\tau}(\epsilon)$ of the tube $N^{\leq\epsilon}$
can be computed as follows.

When the tube is nonsingular (that is, $\alpha=2\pi$)

$r_{2\pi,\lambda,\tau}(\epsilon)=\max_{n\in{\mathbb{N}}}\big{\\{}{\operatorname{trad}}_{n\lambda,\,n\tau\\!{\\!\\!\\!\mod{}}2\pi}(\epsilon)\big{\\}}=\max_{n\in{\mathbb{N}}}\left\\{\operatorname{arccosh}\sqrt{\frac{\cosh\epsilon-\cos(n\tau)}{\cosh(n\lambda)-\cos(n\tau)}}\right\\}.$

When the tube is singular (that is, $0<\alpha<2\pi$)

$r_{\alpha,\lambda,\tau}(\epsilon)=\max\left\\{{\operatorname{trad}}_{0,\alpha}(\epsilon),\,\max_{n\in{\mathbb{N}}}\left\\{{\operatorname{trad}}_{n\lambda,\,n\tau\\!{\\!\\!\\!\mod{}}\alpha}(\epsilon)\right\\}\right\\}.$

Furthermore, if $\pi\leq\alpha<2\pi$, then
${\operatorname{trad}}_{0,\alpha}(\epsilon)=\epsilon/2.$

######  Remark 3.11.

In each case of Proposition 3.10, it suffices to take a maximum over a finite
set. Indeed, when $n\lambda>\epsilon$, the translation length of $\varphi^{n}$
is sure to be larger than $\epsilon$. Thus, by Remark 3.8, all values of $n$
larger than $\epsilon/\lambda$ will contribute $-\infty$ to the set over which
we are taking a maximum, and can therefore be ignored.

The integer $n\in{\mathbb{N}}\cup\\{0\\}$ that realizes the maximum is the
same as the power for $\epsilon$, defined in Definition 2.8. This shows that
$r(\epsilon)$ is actually a maximum rather than a supremum.

###### Proof of Proposition 3.10.

First, assume that $N$ is nonsingular. In this case, the solid torus $N$ can
be described as
$N_{2\pi,\lambda,\tau}={\mathbb{H}}^{3}/\langle\varphi_{\lambda+i\tau}\rangle\cong{\mathbb{H}}^{3}/{\mathbb{Z}}$.
Let $x\in N$ be a point such that $2{\operatorname{injrad}}(x)=\epsilon$, and
let $\widetilde{x}$ be a preimage in ${\mathbb{H}}^{3}$. By Lemma 2.5, we have

$\epsilon=\min\left\\{d(\widetilde{x},\varphi^{n}\widetilde{x}):n\in{\mathbb{Z}}-\\{0\\}\right\\}.$

Let $m\in{\mathbb{Z}}-\\{0\\}$ be a power realizing the minimum. Without loss
of generality, we may assume $m>0$. Then, for every $n\in{\mathbb{N}}$, we
have $d(\widetilde{x},\varphi^{n}\widetilde{x})\geq
d(\widetilde{x},\varphi^{m}\widetilde{x})$. In other words, a point
$\widetilde{y}\in{\mathbb{H}}^{3}$ that $\varphi^{n}$ moves by distance
$\epsilon$ would have to be _closer_ to the core geodesic than $\widetilde{x}$
is, hence
${\operatorname{trad}}_{m\lambda,\,m\tau{\\!\\!\\!\mod{}}2\pi}(\epsilon)\geq{\operatorname{trad}}_{n\lambda,\,n\tau{\\!\\!\\!\mod{}}2\pi}(\epsilon)$.
(This includes the possibility that no such point $\widetilde{y}$ exists: that
is,
${\operatorname{trad}}_{n\lambda,n\tau{\\!\\!\\!\mod{}}2\pi}(\epsilon)=-\infty$;
see Remark 3.8.) We conclude that

$r_{2\pi,\lambda,\tau}(\epsilon)={\operatorname{trad}}_{m\lambda,\,m\tau{\\!\\!\\!\mod{}}2\pi}(\epsilon)=\max_{n\in{\mathbb{N}}}\big{\\{}{\operatorname{trad}}_{n\lambda,\,n\tau{\\!\\!\\!\mod{}}2\pi}(\epsilon)\big{\\}}.$

For every $n\in{\mathbb{N}}$, we may compute
${\operatorname{trad}}_{n\lambda,\,n\tau{\\!\\!\\!\mod{}}2\pi}(\epsilon)$ via
Equation (3.7):

${\operatorname{trad}}_{n\lambda,\,n\tau\\!{\\!\\!\\!\mod{}}2\pi}(\epsilon)=\operatorname{arccosh}\sqrt{\frac{\cosh\epsilon-\cos(n\tau)}{\cosh(n\lambda)-\cos(n\tau)}},$

where both sides might be $-\infty$ as in Remark 3.8.

Now, assume that $N$ is a singular tube of cone angle $\alpha<2\pi$. In this
case, $N$ can be described as
$N_{\alpha,\lambda,\tau}={\hat{{\mathbb{H}}}^{3}}/{\mathbb{Z}}^{2}$, where
${\mathbb{Z}}^{2}=\langle\varphi,\psi\rangle=\langle\varphi_{\lambda+i\tau},\psi_{i\alpha}\rangle$.
Let $x\in N$ be a point such that $2{\operatorname{injrad}}(x)=\epsilon$, and
let $\widetilde{x}$ be a preimage in ${\hat{{\mathbb{H}}}^{3}}$. As above,
Lemma 2.5 describes $\epsilon$ as a minimum of
$d(\widetilde{x},\eta\widetilde{x})$ over all choices of
$\eta\in{\mathbb{Z}}^{2}-\\{0\\}$. We begin by restricting the values of
$\eta$ that need to be considered.

Fix $n>0$, and consider all isometries of the form $\eta=\varphi^{n}\psi^{m}$
as $m$ varies over ${\mathbb{Z}}$. All of these isometries translate the
singular geodesic $\sigma\subset{\hat{{\mathbb{H}}}^{3}}$ by the same
distance, so $d(\widetilde{x},\eta\widetilde{x})$ will be smallest when the
rotational angle is smallest (in absolute value). Equivalently, the
translation radius will be largest when the rotational angle is smallest. Thus

$\max_{m\in{\mathbb{Z}}}\big{\\{}{\operatorname{trad}}_{n\lambda,n\tau+m\alpha}(\epsilon)\big{\\}}={\operatorname{trad}}_{n\lambda,n\tau{\\!\\!\\!\mod{}}\alpha}(\epsilon).$

Similarly, if $\eta=\varphi^{0}\psi^{m}$ for $m\neq 0$, then
$d(\widetilde{x},\eta\widetilde{x})$ will be smallest for $\eta=\psi$. We
conclude that

$r_{\alpha,\lambda,\tau}(\epsilon)=\max\left\\{{\operatorname{trad}}_{0,\alpha}(\epsilon),\,\max_{n\in{\mathbb{N}}}\left\\{{\operatorname{trad}}_{n\lambda,\,n\tau{\\!\\!\\!\mod{}}\alpha}(\epsilon)\right\\}\right\\}.$

It remains to interpret the quantity
${\operatorname{trad}}_{0,\alpha}(\epsilon)$ for various values of $\alpha$.
If $0<\alpha<\pi$, a ball $B_{\epsilon/2}(\widetilde{x})$ will be tangent to
its image under $\psi$ without meeting the axis of ${\hat{{\mathbb{H}}}^{3}}$.
In this case, ${\operatorname{trad}}_{0,\alpha}(\epsilon)$ can be computed as
in Equation (3.7). If $\pi\leq\alpha<2\pi$, the ball
$B_{\epsilon/2}(\widetilde{x})$ will bump directly into the core geodesic
$\sigma\subset{\hat{{\mathbb{H}}}^{3}}$, hence
${\operatorname{trad}}_{0,\alpha}(\epsilon)=\epsilon/2$. ∎

##  4\. Examples demonstrating sharpness

In Section 7 below, we will prove lower bounds on the radius of an
$\epsilon$–tube in terms of $\epsilon$ and the core length $\lambda$.
Quantitatively, the lower bound on $\cosh r(\epsilon)$ will be on the order of
$\epsilon/\sqrt{\lambda}$. The following family of examples, suggested by Ian
Biringer, shows that an $O(\epsilon/\sqrt{\lambda})$ bound is in fact optimal.
As a consequence, we show in Theorem 4.6 that the lower bound of Theorem 1.1
is sharp up to additive error.

######  Proposition 4.1.

For any $n\geq 4$, let $N_{n}=N_{2\pi,\lambda,\tau}$ be a nonsingular model
solid torus whose core geodesic has complex length

$\lambda+i\tau=\frac{1}{n^{2}}+i\frac{2\pi}{n}.$

Then, for every $\epsilon$ in the range $1.016/n\leq\epsilon\leq 0.3$, the
tube radius in $N_{n}$ satisfies

$\cosh
r(\epsilon)=\sqrt{\frac{\cosh\epsilon-1}{\cosh\sqrt{\lambda}-1}}\quad\in\left(\frac{\epsilon}{\sqrt{\lambda}},\>1.004\frac{\epsilon}{\sqrt{\lambda}}\right).$

Plugging the minimal and maximal values of $\epsilon$ into the above estimate
shows that the radii appearing in Proposition 4.1 range from
$\operatorname{arccosh}(1.016)=0.1767\ldots$ to
$\operatorname{arccosh}(0.3n)$, which goes to $\infty$ with $n$. We suspect
that similar behavior cannot hold for very small radii.

The proof of Proposition 4.1 needs the following easy monotonicity result,
which will be used repeatedly below.

######  Lemma 4.2.

Let $a,b\in{\mathbb{R}}$ be constants, and consider the function

$g(x)=\frac{a-x}{b-x}=\frac{x-a}{x-b}.$

Then $g(x)$ is strictly increasing in $x$ if $b<a$ and strictly decreasing in
$x$ if $a<b$.

###### Proof.

Compute the derivative. Alternately, consider $g$ as a linear fractional
transformation of ${\mathbb{R}}{\mathbb{P}}^{1}$, which preserves orientation
if and only if $\det\left[\begin{smallmatrix}1&-a\\\
1&-b\end{smallmatrix}\right]>0$. ∎

###### Proof of Proposition 4.1.

Fix $\epsilon\geq\sqrt{\lambda}=1/n$. Let
$r(\epsilon)=r_{2\pi,\lambda,\tau}(\epsilon)$ be the radius of the
$\epsilon$–tube, as in Definition 3.1. We begin by proving a lower bound on
$r(\epsilon)$, using Proposition 3.10:

$\displaystyle r_{2\pi,\lambda,\tau}(\epsilon)$
$\displaystyle=\max_{m\in{\mathbb{N}}}\left\\{{\operatorname{trad}}_{m\lambda,\,m\tau\\!{\\!\\!\\!\mod{}}2\pi}(\epsilon)\right\\}$
$\displaystyle\geq{\operatorname{trad}}_{n\lambda,\,n\tau\\!{\\!\\!\\!\mod{}}2\pi}(\epsilon)$
$\displaystyle={\operatorname{trad}}_{\sqrt{\lambda},\,0}(\epsilon)$ (4.3)
$\displaystyle=\operatorname{arccosh}\sqrt{\frac{\cosh\epsilon-1}{\cosh\sqrt{\lambda}-1}}\,.$

We will eventually show that, for $\epsilon\geq 1.016/n$, the above inequality
is actually equality. This amounts to showing that $n$ is indeed the power for
$\epsilon$ (see Definition 2.8). This requires a few estimates.

First, we get a tight two-sided estimate on the quantity in (4.3). Assume that
$1/n=\sqrt{\lambda}<\epsilon\leq 0.3$. The monotonicity of the function
$h(x,y)$ in Lemma A.3 implies that

$1\><\>\frac{\cosh\epsilon-1}{\epsilon^{2}}\cdot\frac{\lambda}{\cosh\sqrt{\lambda}-1}\>\leq\>\frac{\cosh(0.3)-1}{(0.3)^{2}}\cdot\frac{1}{1/2}\>=\>(1.00375\ldots)^{2}.$

Multiplying all of this by $\epsilon^{2}/\lambda$ and taking square roots
gives

(4.4)
$\frac{\epsilon}{\sqrt{\lambda}}<\sqrt{\frac{\cosh\epsilon-1}{\cosh\sqrt{\lambda}-1}}<1.004\frac{\epsilon}{\sqrt{\lambda}},$

as claimed in the statement of the Proposition.

Now, assume that $\epsilon\geq 1.016/n$. Then (4.3) and (4.4) combine to give

$\cosh
r(\epsilon)\geq\sqrt{\frac{\cosh\epsilon-1}{\cosh\sqrt{\lambda}-1}}>\frac{\epsilon}{\sqrt{\lambda}}=\epsilon
n\geq 1.016,$

which implies

(4.5) $\tanh^{2}r(\epsilon)>0.0312.$

Next, we claim that (for any $\epsilon\geq\lambda$) the only possible powers
for $\epsilon$ are either $1$ or $n$. This can be seen from Equation (3.7),
substituting $\tau=2\pi/n$:

$\cosh^{2}{\operatorname{trad}}_{k\lambda,\,k\tau\\!{\\!\\!\\!\mod{}}2\pi}(\epsilon)=\frac{\cosh\epsilon-\cos(2\pi
k/n)}{\cosh(k\lambda)-\cos(2\pi k/n)}.$

When $k\in n{\mathbb{Z}}$, the subtracted term is $\cos(2\pi k/n)=1$, hence
constant, and the denominator is smallest for $k=n$. When $k\notin
n{\mathbb{Z}}$, we have $\cos(2\pi k/n)\leq\cos(2\pi/n)$, hence Lemma 4.2
gives

$\cosh^{2}{\operatorname{trad}}_{k\lambda,\,k\tau\\!{\\!\\!\\!\mod{}}2\pi}(\epsilon)\leq\frac{\cosh\epsilon-\cos(2\pi
k/n)}{\cosh(\lambda)-\cos(2\pi
k/n)}\leq\frac{\cosh\epsilon-\cos(2\pi/n)}{\cosh(\lambda)-\cos(2\pi/n)}.$

Thus only $k=1$ or $k=n$ can give a maximal value of ${\operatorname{trad}}$.

Finally, we claim that $n$ is indeed the power for $\epsilon$. Let
$\varphi=\varphi_{\lambda,\tau}$ be the loxodromic isometry of
${\mathbb{H}}^{3}$ that generates the deck group of $N_{n}$. Fix
$r=r(\epsilon)$, and let $\widetilde{T}_{r}$ be the lift to ${\mathbb{H}}^{3}$
of the equidistant torus $T_{r}\subset N_{n}$. For
$\widetilde{x}\in\widetilde{T}_{r}$, let
$d_{1}=d(\widetilde{x},\varphi\widetilde{x})$ and
$d_{n}=d(\widetilde{x},\varphi^{n}\widetilde{x})$. By Lemma 3.5,

$\displaystyle\cosh d_{1}$
$\displaystyle=\cosh\lambda\cosh^{2}r-\cos\tau\sinh^{2}r\qquad=\cosh\left(\tfrac{1}{n^{2}}\right)\cosh^{2}r-\cos\left(\tfrac{2\pi}{n}\right)\sinh^{2}r$
$\displaystyle\cosh d_{n}$
$\displaystyle=\cosh(n\lambda)\cosh^{2}r-\cos(n\tau)\sinh^{2}r=\cosh\left(\tfrac{1}{n}\right)\cosh^{2}r-\cos(0)\sinh^{2}r$

hence

$\cosh d_{n}-\cosh
d_{1}=\left[\cosh\left(\tfrac{1}{n}\right)-\cosh\left(\tfrac{1}{n^{2}}\right)\right]\cosh^{2}r-\left[1-\cos\left(\tfrac{2\pi}{n}\right)\right]\sinh^{2}r.$

Using calculus, we check that when $n\geq 4$,

$\frac{\cosh\left(\tfrac{1}{n}\right)-\cosh\left(\tfrac{1}{n^{2}}\right)}{1-\cos\left(\tfrac{2\pi}{n}\right)}\leq
0.02945<\tanh^{2}r,$

where the last inequality is by (4.5). Multiplying both sides by $\cosh^{2}r$
gives

$\left[\cosh\left(\tfrac{1}{n}\right)-\cosh\left(\tfrac{1}{n^{2}}\right)\right]\cosh^{2}r<\left[1-\cos\left(\tfrac{2\pi}{n}\right)\right]\sinh^{2}r,$

hence $\cosh d_{n}<\cosh d_{1}$. This means that $\varphi^{n}$ translates
points of $\widetilde{T}_{r}$ by _less_ than $\varphi$ for all radii
satisfying (4.5), hence $n$ is the power for $\epsilon$.

We conclude that the inequality (4.3) is equality, as desired. ∎

We use Proposition 4.1 to prove Theorem 4.6, below. Recall from Section 1.3
that the upper bound of Theorem 4.6 differs from the lower bound of Theorem
1.1 by an additive error of less than 2.2.

######  Theorem 4.6.

Fix a pair $(\delta,\epsilon)$ such that $0<\sqrt{7.256\delta}\leq\epsilon\leq
0.3$. Given the pair $(\delta,\epsilon)$, there is a model solid torus
$N=N_{2\pi,\lambda,\tau}$ with $\lambda\leq\delta$, such that

$d_{2\pi,\lambda,\tau}(\delta,\epsilon)\leq\operatorname{arccosh}\left(1.116\frac{\epsilon}{\sqrt{\delta}}\right).$

###### Proof.

Given $\delta$ as above, let $n$ be the unique integer satisfying

$\frac{1}{n}\leq\sqrt{\delta}<\frac{1}{n-1}.$

Now, set $\lambda+i\tau=1/n^{2}+2\pi i/n$, and let
$N=N_{n}=N_{2\pi,\lambda,\tau}$ be a model solid torus as in Proposition 4.1.
Since $\sqrt{\delta}\leq 0.3/\sqrt{7.256}<0.1114$, we have $n\geq 9$. In
addition, $\epsilon>\sqrt{7\delta}\geq\sqrt{7}/n$, verifying that the
hypotheses of Proposition 4.1 hold in our setting.

We will use Proposition 4.1 to get an upper bound on
$d_{2\pi,\lambda,\tau}(\delta,\epsilon)$. To do so, we need an upper bound on
$\sqrt{\delta}/\sqrt{\lambda}$. Observe that

$\frac{\sqrt{\delta}}{\sqrt{\lambda}}=n\sqrt{\delta}<\frac{n}{n-1}.$

Thus, if $n\geq 10$, we have $\sqrt{\delta/\lambda}\leq 10/9$. Meanwhile, if
$n=9$, recall that $\sqrt{\delta}<0.1114$, which implies
$\sqrt{\delta/\lambda}=9\sqrt{\delta}<1.0026<10/9$ as well. Now, we compute:

$\displaystyle d_{2\pi,\lambda,\tau}(\delta,\epsilon)$
$\displaystyle=r_{2\pi,\lambda,\tau}(\epsilon)-r_{2\pi,\lambda,\tau}(\delta)$
$\displaystyle\leq r_{2\pi,\lambda,\tau}(\epsilon)$
$\displaystyle\leq\operatorname{arccosh}\left(1.004\,\frac{\epsilon}{\sqrt{\lambda}}\right)$
$\displaystyle\leq\operatorname{arccosh}\left(1.004\cdot\frac{10}{9}\,\frac{\epsilon}{\sqrt{\delta}}\right).\qed$

##  5\. Distance between tubes: upper bound

In this section we prove the upper bound of Theorem 1.1. This result requires
some lemmas, the first of which is also used in the lower bound.

######  Lemma 5.1.

Let $N=N_{\alpha,\lambda,\tau}$ be a model solid torus. Let $x,y$ be points of
$N$ such that $2\,{\operatorname{injrad}}(x)=\delta>0$ and
$2\,{\operatorname{injrad}}(y)=\epsilon>0$. Then

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

###### Proof.

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 2.4, 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

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

The following lemma controls the type of isometry that realizes injectivity
radius in singular tubes. Recall that $T^{\epsilon}=\partial
N^{\leq\epsilon}\subset N$ denotes the equidistant torus consisting of points
whose injectivity radius is exactly $\epsilon/2$.

######  Lemma 5.2.

Consider a singular solid torus $N=N_{\alpha,\lambda,\tau}$ of cone angle
$\alpha<2\pi$, and let $0<\delta<\epsilon$. Suppose that, for $q\in
T^{\epsilon}\subset N$, the injectivity radius ${\operatorname{injrad}}(q)$ is
realized by an elliptic isometry $\psi_{i\alpha}$ (compare Lemma 2.5). Then,
for every $p\in T^{\delta}$, the injectivity radius
${\operatorname{injrad}}(p)$ is also realized by the same elliptic isometry
$\psi_{i\alpha}$.

###### Proof.

There are two cases: $\alpha<\pi$ and $\alpha\geq\pi$. We consider the latter
case first.

Suppose that $\pi\leq\alpha<2\pi$, and let $q\in T^{\epsilon}$. By Proposition
3.10, we must have $r(\epsilon)=\epsilon/2$, hence there is a point of
intersection $z\in\Sigma\cap\overline{B_{\epsilon/2}(q)}$, where $\Sigma$ is
the singular core of $N$. Let $\alpha$ be the geodesic segment of length
$\epsilon/2$ from $q$ to $z$.

Let $p\in T^{\delta}$. Since the symmetry group of $N$ acts transitively on
$T^{\delta}$, we may assume without loss of generality that $p\in\alpha\cap
T^{\delta}$. By Lemma 5.1, we have

$r(\delta)\>=\>d(z,p)\>=\>d(z,q)-d(p,q)\>\leq\>\frac{\epsilon}{2}-\frac{\epsilon-\delta}{2}\>=\>\frac{\delta}{2}.$

On the other hand, Proposition 3.10 gives $r(\delta)\geq\delta/2$. Thus
$r(\delta)=\delta/2$, hence the injectivity radius of $T^{\delta}$ is realized
by the elliptic generator $\psi_{i\alpha}$.

Next, suppose that $0<\alpha<\pi$. According to Proposition 3.10, the
injectivity radius of $q\in T^{\epsilon}$ is realized by an elliptic isometry
$\psi_{i\alpha}$ precisely when
${\operatorname{trad}}_{0,\alpha}(\epsilon)\leq{\operatorname{trad}}_{n\lambda,n\tau{\\!\\!\\!\mod{}}\alpha}(\epsilon)$
for all $n\in{\mathbb{N}}$. Unwinding the definition of
${\operatorname{trad}}(\epsilon)$ via Lemma 3.5, we see that for $n\geq 1$,

(5.3)
$\frac{\cosh\epsilon-\cos(n\tau{\\!\\!\\!\mod{}}\alpha)}{\cosh(n\lambda)-\cos(n\tau{\\!\\!\\!\mod{}}\alpha)}\leq\frac{\cosh\epsilon-\cos\alpha}{1-\cos\alpha}.$

For simplicity of the following calculations, set

$x=\cosh\epsilon,\quad a=\cos\alpha,\quad b=1,\quad
a^{\prime}=\cos(n\tau{\\!\\!\\!\mod{}}\alpha),\quad
b^{\prime}=\cosh(n\lambda).$

We consider $a,b,a^{\prime},b^{\prime}$ to be constants in the following
argument, because the integer $n$ will stay fixed. Note that $b<b^{\prime}$
and $a<a^{\prime}$. The inequality (5.3) can be rewritten as

$k(x)\cdot\frac{x-a^{\prime}}{b^{\prime}-a^{\prime}}=\frac{x-a}{b-a},\mbox{
where }k(x)\geq 1.$

This, in turn, can be rewritten as

$k(x)\cdot\frac{b-a}{b^{\prime}-a^{\prime}}=\frac{x-a}{x-a^{\prime}}.$

Since $a<a^{\prime}$, Lemma 4.2 implies that $g(x)=\frac{x-a}{x-a^{\prime}}$
is a strictly decreasing function of $x$. Now, for any $0<\delta<\epsilon$,
set $y=\cosh\delta$. We have

$k(y)\cdot\frac{b-a}{b^{\prime}-a^{\prime}}=\frac{y-a}{y-a^{\prime}}>\frac{x-a}{x-a^{\prime}}=k(x)\cdot\frac{b-a}{b^{\prime}-a^{\prime}}.$

We conclude that

$k(y)>k(x)\geq 1,\quad\mbox{that
is}\quad\frac{y-a}{b-a}>\frac{y-a^{\prime}}{b^{\prime}-a^{\prime}}.$

This means that for every $n\in{\mathbb{N}}$, we have that
${\operatorname{trad}}_{n\lambda,n\tau{\\!\\!\\!\mod{}}\alpha}(\delta)$ is
strictly smaller than ${\operatorname{trad}}_{0,\alpha}(\delta)$. Thus
$r(\delta)={\operatorname{trad}}_{0,\alpha}(\delta)$, as desired. ∎

######  Remark 5.4.

The above proof can be modified to establish the following “monotonicity of
powers” statement. Suppose that $0<\delta<\epsilon$, and the power for
$\delta$ is $n\in{\mathbb{N}}$. That is, for $p\in T^{\delta}$, the
injectivity radius ${\operatorname{injrad}}(p)$ is realized by a loxodromic
isometry $\varphi_{\lambda+i\tau}^{n}$. Then the power for $\epsilon$ is at
least $n$. Note that by Lemma 5.2, the realizing isometry must be loxodromic.
The proof starts with the following analogue of (5.3): for all $1\leq m<n$, we
have

(5.5)
$\frac{\cosh\delta-\cos(m\tau{\\!\\!\\!\mod{}}\alpha)}{\cosh(m\lambda)-\cos(m\tau{\\!\\!\\!\mod{}}\alpha)}\leq\frac{\cosh\delta-\cos(n\tau{\\!\\!\\!\mod{}}\alpha)}{\cosh(n\lambda)-\cos(n\tau{\\!\\!\\!\mod{}}\alpha)}.$

Now, (5.5) combined with the monotonicity of the function $g(x)$ of Lemma 4.2
(in the opposite direction compared to the last proof) will imply that (5.5)
also holds with $\epsilon$ instead of $\delta$. Since we do not need this
statement, we omit the details.

We can now prove the following statement, which will quickly imply the upper
bound of Theorem 1.1.

######  Proposition 5.6.

Suppose $0<\delta<\epsilon$ and $0<\alpha\leq 2\pi$. Then, in any model solid
torus $N=N_{\alpha,\lambda,\tau}$ such that $N^{\leq\delta}\neq\emptyset$, we
have

$\frac{\cosh r_{\alpha,\lambda,\tau}(\epsilon)}{\cosh
r_{\alpha,\lambda,\tau}(\delta)}\leq\sqrt{\frac{\cosh\epsilon-1}{\cosh\delta-1}}.$

Equality holds if and only if the injectivity radii of $T^{\delta}$ and
$T^{\epsilon}$ are realized by the same loxodromic isometry, whose rotational
part is trivial. In particular, if $\alpha=2\pi$ and $\tau=0$, then equality
holds.

###### Proof.

We consider three cases, depending on the value of $\alpha$ and the power for
$\epsilon$.

First, suppose that the power for $\epsilon$ is $m\in{\mathbb{N}}$, hence the
injectivity radius of $T^{\epsilon}$ is realized by a loxodromic isometry
$\varphi_{\lambda+i\tau}^{m}$. In the following computation, using Proposition
3.10, the ellipsis $(\cdots)$ denotes any elliptic terms that arise when
$\alpha<2\pi$.

$\displaystyle\frac{\cosh r_{\alpha,\lambda,\tau}(\epsilon)}{\cosh
r_{\alpha,\lambda,\tau}(\delta)}$
$\displaystyle=\frac{\max\left\\{\max_{n\in{\mathbb{N}}}\left\\{\cosh{\operatorname{trad}}_{n\lambda,n\tau{\\!\\!\\!\mod{}}\alpha}(\epsilon)\right\\},\cdots\right\\}}{\max\left\\{\max_{n\in{\mathbb{N}}}\left\\{\cosh{\operatorname{trad}}_{n\lambda,n\tau{\\!\\!\\!\mod{}}\alpha}(\delta)\right\\},\cdots\right\\}}\quad\text{by
Proposition~{}\ref{Prop:TubeRadFormula}}$
$\displaystyle=\frac{\cosh{\operatorname{trad}}_{m\lambda,m\tau{\\!\\!\\!\mod{}}\alpha}(\epsilon)}{\max\left\\{\max_{n\in{\mathbb{N}}}\left\\{\cosh{\operatorname{trad}}_{n\lambda,n\tau{\\!\\!\\!\mod{}}\alpha}(\delta)\right\\},\cdots\right\\}}\quad\text{by
definition of $m$}$
$\displaystyle\leq\frac{\cosh{\operatorname{trad}}_{m\lambda,m\tau{\\!\\!\\!\mod{}}\alpha}(\epsilon)}{\cosh{\operatorname{trad}}_{m\lambda,m\tau{\\!\\!\\!\mod{}}\alpha}(\delta)}$
$\displaystyle=\sqrt{\frac{\cosh\epsilon-\cos(m\tau{\\!\\!\\!\mod{}}\alpha)}{\cosh
m\lambda-\cos(m\tau{\\!\\!\\!\mod{}}\alpha)}\cdot\frac{\cosh
m\lambda-\cos(m\tau{\\!\\!\\!\mod{}}\alpha)}{\cosh\delta-\cos(m\tau{\\!\\!\\!\mod{}}\alpha)}}\quad\text{
by~{}\eqref{Eqn:TradDistance}}$
$\displaystyle=\sqrt{\frac{\cosh\epsilon-\cos(m\tau{\\!\\!\\!\mod{}}\alpha)}{\cosh\delta-\cos(m\tau{\\!\\!\\!\mod{}}\alpha)}}$
$\displaystyle\leq\sqrt{\frac{\cosh\epsilon-1}{\cosh\delta-1}}\quad\text{by
Lemma~{}\ref{Lem:EasyMonotonicity}}.$

Observe that the first inequality is equality precisely when $m$ is also the
power for $\delta$. The second inequality is equality precisely when
$(m\tau{\\!\\!\\!\mod{}}\alpha)=0$: that is, when the realizing isometry has
trivial rotational part.

Next, suppose that the injectivity radius of $T^{\epsilon}$ is realized by an
elliptic isometry $\psi_{i\alpha}$, and furthermore $0<\alpha<\pi$. Then Lemma
5.2 says that the injectivity radius of $T^{\delta}$ is realized by the same
elliptic. Thus we may compute as above:

$\displaystyle\frac{\cosh r_{\alpha,\lambda,\tau}(\epsilon)}{\cosh
r_{\alpha,\lambda,\tau}(\delta)}$
$\displaystyle=\frac{\cosh{\operatorname{trad}}_{0,\alpha}(\epsilon)}{\cosh{\operatorname{trad}}_{0,\alpha}(\delta)}$
$\displaystyle=\sqrt{\frac{\cosh\epsilon-\cos(\alpha)}{1-\cos(\alpha)}\cdot\frac{1-\cos(\alpha)}{\cosh\delta-\cos(\alpha)}}\quad\text{
by~{}\eqref{Eqn:TradDistance}}$
$\displaystyle=\sqrt{\frac{\cosh\epsilon-\cos(\alpha)}{\cosh\delta-\cos(\alpha)}}$
$\displaystyle<\sqrt{\frac{\cosh\epsilon-1}{\cosh\delta-1}}.$

The inequality is strict because $0<\alpha<\pi$, hence $\cos\alpha<1$.

Finally, suppose that the injectivity radius of $T^{\epsilon}$ is realized by
an elliptic isometry $\psi_{i\alpha}$, and furthermore $\pi\leq\alpha<2\pi$.
Then Lemma 5.2 says that the injectivity radius of $T^{\delta}$ is realized by
the same elliptic. Furthermore, $r(\epsilon)=\epsilon/2$ and
$r(\delta)=\delta/2$. Thus

$\displaystyle\frac{\cosh^{2}(r_{\alpha,\lambda,\tau}(\epsilon))}{\cosh^{2}(r_{\alpha,\lambda,\tau}(\delta))}$
$\displaystyle=\frac{\cosh^{2}(\epsilon/2)}{\cosh^{2}(\delta/2)}$
$\displaystyle<\frac{\cosh^{2}(\epsilon/2)-1}{\cosh^{2}(\delta/2)-1},\quad\text{by
Lemma~{}\ref{Lem:EasyMonotonicity} }$
$\displaystyle=\frac{(2\cosh^{2}(\epsilon/2)-1)-1}{(2\cosh^{2}(\delta/2)-1)-1}$
$\displaystyle=\frac{\cosh(\epsilon)-1}{\cosh(\delta)-1}.$

Again, the inequality is strict in this case. ∎

We can now prove the upper bound of Theorem 1.1, including its sharpness.

######  Proposition 5.7.

Suppose $0<\delta<\epsilon$ and $0<\alpha\leq 2\pi$. Then, in any model solid
torus $N_{\alpha,\lambda,\tau}$ with $N^{\leq\delta}$ non-empty, we have

$d_{\alpha,\lambda,\tau}(\delta,\epsilon)\leq\operatorname{arccosh}\sqrt{\frac{\cosh\epsilon-1}{\cosh\delta-1}}.$

Furthermore, equality holds if and only if
$(\alpha,\lambda,\tau)=(2\pi,\delta,0)$.

###### Proof.

Combining Lemma A.1 and Proposition 5.6 gives

(5.8) $\cosh
d_{\alpha,\lambda,\tau}(\delta,\epsilon)=\cosh\big{(}r_{\alpha,\lambda,\tau}(\epsilon)-r_{\alpha,\lambda,\tau}(\delta)\big{)}\leq\frac{\cosh
r_{\alpha,\lambda,\tau}(\epsilon)}{\cosh
r_{\alpha,\lambda,\tau}(\delta)}\leq\sqrt{\frac{\cosh\epsilon-1}{\cosh\delta-1}}.$

Next, let us analyze when equality can hold. Assuming $\delta<\epsilon$, hence
$r(\delta)<r(\epsilon)$, Lemma A.1 implies that the first inequality of (5.8)
will be equality if and only if $r(\delta)=0$. But $r(\delta)\geq\delta/2>0$
when $N$ is singular, hence equality cannot hold for singular tori.

From now on, assume that $\alpha=2\pi$, hence $N$ is nonsingular. By Lemma
A.1, the first inequality of (5.8) will be equality precisely when
$r(\delta)=0$, hence when $\lambda=\delta$ and the realizing isometry is a
generator of $\pi_{1}(N)={\mathbb{Z}}$. By Proposition 5.6, the second
inequality of (5.8) is equality precisely when this realizing isometry has
rotational part $\tau=0$. ∎

##  6\. Euclidean bounds

We now begin working toward the lower bound of Theorem 1.1. In that argument,
we will need to estimate distances on the torus $T^{\epsilon}$ that forms the
boundary of an $\epsilon$–tube. The torus $T^{\epsilon}\subset N$ inherits a
Euclidean metric, as does its preimage
$\widetilde{T}^{\epsilon}\subset{\hat{{\mathbb{H}}}^{3}}$, and we may use that
metric to measure the distance between points. In this section, we consider
how the Euclidean distance between points on $\widetilde{T}^{\epsilon}$
relates to the actual hyperbolic distance between these points in
${\hat{{\mathbb{H}}}^{3}}$.

For points $p$ and $q$ in an equidistant torus
$\widetilde{T}^{\epsilon}=\widetilde{T}_{r}\subset{\hat{{\mathbb{H}}}^{3}}$,
define $d_{E}(p,q)$ to be the distance between them in the Euclidean metric on
$\widetilde{T}_{r}$. Note that if $p=(r,0,0)$ and $q=(r,\zeta,\theta)$ in
cylindrical coordinates, then (2.2) gives

(6.1) $d_{E}(p,q)^{2}=\zeta^{2}\cosh^{2}r+\theta^{2}\sinh^{2}r.$

######  Lemma 6.2.

Let $\widetilde{T}_{r}\subset{\hat{{\mathbb{H}}}^{3}}$ be a plane at fixed
distance $r>0$ from the singular geodesic $\hat{\sigma}$. 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.

Suppose without loss of generality that the cylindrical coordinates of $p$ are
$(r,0,0)$ and the coordinates of of $q$ are $(r,\zeta,\theta)$, where
$|\theta|\leq A$. We can now compute using Lemma 3.5:

$\displaystyle\cosh d(p,q)-1$
$\displaystyle=\cosh\zeta\cosh^{2}r-\cos\theta\sinh^{2}r-1$
$\displaystyle=(\cosh\zeta-1)\cosh^{2}r+(\cosh^{2}r-1)-\cos\theta\sinh^{2}r$
$\displaystyle=(\cosh\zeta-1)\cosh^{2}r+(1-\cos\theta)\sinh^{2}r$ (6.3)
$\displaystyle=\frac{\cosh\zeta-1}{\zeta^{2}}(\zeta\cosh
r)^{2}+\frac{1-\cos\theta}{\theta^{2}}(\theta\sinh r)^{2}.$

If either $\zeta=0$ or $\theta=0$, equation (6.3) should be interpreted by
substituting the limiting value of $1/2$:

(6.4) $\lim_{\zeta\to
0}\frac{\cosh\zeta-1}{\zeta^{2}}=\frac{1}{2}=\lim_{\theta\to
0}\frac{1-\cos\theta}{\theta^{2}}.$

The monotonicity of the functions in (6.4) leads to the following chain of
inequalities:

$\frac{1-\cos
A}{A^{2}}\leq\frac{1-\cos\theta}{\theta^{2}}\leq\frac{1}{2}\leq\frac{\cosh\zeta-1}{\zeta^{2}}\leq\frac{\cosh
B-1}{B^{2}}.$

To get the lower bound of the lemma, we return to (6.3), and substitute the
lower bound $(1-\cos A)/(A^{2})$ for both $(1-\cos\theta)/(\theta^{2})$ and
$(\cosh\zeta-1)/(\zeta^{2})$. The upper bound follows similarly. ∎

The last result should be interpreted in light of Lemma A.3: so long as
$d(p,q)$ is bounded above, $(\cosh d(p,q)-1)$ is not too different from
$\frac{1}{2}d(p,q)^{2}$. Thus Lemma 6.2 can be interpreted as saying that the
Euclidean and hyperbolic distances are quite similar. The next lemma makes
this idea precise, in a special case.

######  Lemma 6.5.

Let $\widetilde{T}_{r}\subset{\hat{{\mathbb{H}}}^{3}}$ be a plane at fixed
distance $r>0$ from the singular geodesic $\hat{\sigma}$. Let
$p,q\in\widetilde{T}_{r}$ be points whose $\theta$–coordinates differ by at
most $A\leq\pi$. Suppose that $d(p,q)\leq 0.3$. Then

$0.6342\,d_{E}(p,q)<d(p,q)<d_{E}(p,q).$

###### Proof.

The upper bound $d(p,q)<d_{E}(p,q)$ is immediate, because tubes are strictly
convex. For the lower bound, we use the monotonicity of the function
$f(x)=(\cosh x-1)/(x^{2})$. Lemma A.3 implies that on the interval
$[0,\,0.3]$, we have

(6.6) $\frac{\cosh d(p,q)-1}{d(p,q)^{2}}\leq f(0.3)=0.50376\ldots.$

Similarly, $g(x)=(1-\cos x)/(x^{2})$ is strictly decreasing on $[0,\pi]$,
hence (setting $A$ as in Lemma 6.2) we get

(6.7) $\frac{1-\cos A}{A^{2}}\geq g(\pi)=\frac{2}{\pi^{2}}.$

Plugging (6.6) and (6.7) into Lemma 6.2 gives

$0.50377\,d(p,q)^{2}\geq\cosh d(p,q)-1\geq\frac{1-\cos
A}{A^{2}}d_{E}(p,q)^{2}\geq\frac{2}{\pi^{2}}d_{E}(p,q)^{2},$

which simplifies to the desired result. ∎

##  7\. The depth of a tube

The next step in the proof of Theorem 1.1 is Proposition 7.1, which provides a
lower bound on the radius of an $\epsilon$–tube. Note that by Proposition 4.1,
the estimate of Proposition 7.1 is sharp up to multiplicative error in $\cosh
r(\epsilon)$, hence up to additive error in $r(\epsilon)$.

######  Proposition 7.1.

Let $N=N_{\alpha,\lambda,\tau}$ be a model solid torus whose core curve has
cone angle $0<\alpha\leq 2\pi$ and length $0<\lambda\leq 2.97$. Suppose that
$N^{\leq\epsilon}\neq\emptyset$. Then the tube radius $r(\epsilon)$ satisfies

$\cosh
r_{\alpha,\lambda,\tau}(\epsilon)\>\geq\>\frac{\epsilon}{\sqrt{(4\pi/\sqrt{3})\,\lambda}}\>\geq\>\frac{\epsilon}{\sqrt{7.256\,\lambda}}\,.$

The proof breaks into separate cases, depending on whether $N$ is singular or
nonsingular. We will handle singular tubes first.

######  Lemma 7.2.

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.

Let $T_{r}\subset N$ be the torus at radius $r$ from the core, and let
$\widetilde{T}_{r}$ be is preimage in ${\hat{{\mathbb{H}}}^{3}}$. By Equation
(2.2), the area of $T_{r}$ is

(7.3) $\operatorname{area}(T_{r})=\alpha\lambda\sinh r\cosh
r=\frac{\alpha\lambda}{2}\sinh 2r.$

Now, suppose that an (arbitrary) point $x\in T_{r}$ has
${\operatorname{injrad}}(x)=\epsilon/2$. Let
$\widetilde{x}\in\widetilde{T}_{r}$ be a lift of $x$. By Lemma 2.5, every non-
trivial deck transformation $\eta\in{\mathbb{Z}}^{2}-\\{0\\}$ satisfies

$d(\widetilde{x},\eta\widetilde{x})\geq\epsilon.$

This includes elliptic isometries of ${\hat{{\mathbb{H}}}^{3}}$, because $N$
is singular. Observe that Euclidean distance along $\widetilde{T}_{r}$ is
greater than hyperbolic distance, because tubes are strictly convex (compare
Lemma 6.5). Thus

$d_{E}(\widetilde{x},\eta\widetilde{x})>d(\widetilde{x},\eta\widetilde{x})\geq\epsilon.$

This means $\widetilde{x}$ is the center of a Euclidean disk of radius
$\epsilon/2$, disjoint from all of its translates by the deck group.
Projecting down to $T_{r}$ gives an embedded Euclidean disk of radius
$\epsilon/2$. Since a packing of ${\mathbb{R}}^{2}$ by isometric disks has
density at most $\pi/(2\sqrt{3})$, we have

(7.4)
$\operatorname{area}(T^{\epsilon})\>\geq\>\frac{\pi\epsilon^{2}}{4}\cdot\frac{2\sqrt{3}}{\pi}\>=\>\frac{\sqrt{3}}{2}\epsilon^{2}.$

Combining this result with Equation (7.3), we obtain

$\operatorname{area}(T^{\epsilon})=\frac{\alpha\lambda}{2}\sinh
2r(\epsilon)\geq\frac{\sqrt{3}\,\epsilon^{2}}{2}.\qed$

The nonsingular case of Proposition 7.1 uses the following result by Cao,
Gehring, and Martin [9, Lemma 3.4], sharpening an earlier lemma by Zagier [19,
Page 1045]. We remark that more elementary forms of Lemma 7.5, such as a
pigeonhole principle argument due to Meyerhoff [19, Page 1048], suffice to
show a version of Proposition 7.1, but with a larger constant in front of
$\sqrt{\lambda}$.

######  Lemma 7.5.

Consider a nonsingular tube whose core geodesic has complex length
$\lambda+i\tau$, where $0<\lambda\leq 2.97$. Then there is an integer $m\geq
1$ such that

$\cosh(m\lambda)-\cos(m\tau)\leq\frac{2\pi\lambda}{\sqrt{3}}\leq
3.628\lambda.$

###### Proof.

This is a restatement of [9, Lemma 3.4]. To convert their result to the form
stated above, one needs the identity

$\left|2\sinh^{2}\left(\frac{m(\lambda+i\tau)}{2}\right)\right|=\left|\cosh\big{(}m(\lambda+i\tau)\big{)}-1\right|=\cosh(m\lambda)-\cos(m\tau),$

which is readily verified. See [9, Equation (3.10)]. ∎

Using this, we can prove the nonsingular case of Proposition 7.1.

######  Lemma 7.6.

Let $r_{2\pi,\lambda,\tau}(\epsilon)$ be the radius of a nonsingular tube that
has injectivity radius $\epsilon$. Suppose the core curve has length
$\lambda\leq 2.97$. Then, for every $\epsilon\geq\lambda$,

$\cosh^{2}(r_{2\pi,\lambda,\tau}(\epsilon))\geq\frac{\sqrt{3}\,\epsilon^{2}}{4\pi\lambda}.$

###### Proof.

Let $m\geq 1$ be the integer guaranteed by Lemma 7.5. Then Proposition 3.10
gives

$\displaystyle\cosh^{2}(r_{2\pi,\lambda,\tau}(\epsilon))$
$\displaystyle=\max_{n\in{\mathbb{N}}}\,\\{\cosh^{2}({\operatorname{trad}}_{n\lambda,\,n\tau{\\!\\!\\!\mod{}}\alpha}(\epsilon))\\}$
$\displaystyle\geq\cosh^{2}({\operatorname{trad}}_{m\lambda,\,m\tau{\\!\\!\\!\mod{}}\alpha}(\epsilon))$
$\displaystyle=\frac{\cosh\epsilon-\cos(m\tau)}{\cosh(m\lambda)-\cos(m\tau)}$
$\displaystyle\geq(\cosh\epsilon-1)\cdot\frac{\sqrt{3}}{2\pi\lambda}$
$\displaystyle\geq\frac{\epsilon^{2}}{2}\cdot\frac{\sqrt{3}}{2\pi\lambda}.\qed$

###### Proof of Proposition 7.1.

If $N$ is a nonsingular tube whose core has length $\lambda\leq 2.97$, the
proposition holds by Lemma 7.6. Meanwhile, if $N$ is a singular tube whose
core curve has cone angle $\alpha<2\pi$, Lemma 7.2 implies

$\cosh^{2}r(\epsilon)>\sinh r(\epsilon)\cosh r(\epsilon)=\frac{1}{2}\sinh
2r(\epsilon)\geq\frac{\sqrt{3}\,\epsilon^{2}}{4\pi\lambda}.\qed$

##  8\. Distance between tubes: lower bound

In this section, we prove the lower bound of Theorem 1.1. The proof breaks
into two cases: shallow and deep. A tube is said to be _shallow_ if its radius
is bounded above by some constant denoted ${r_{\max}}$. Similarly, a tube is
said to be _deep_ if its radius is bounded below by some constant denoted
${r_{\min}}$. The optimal values of ${r_{\min}}$ and ${r_{\max}}$ will be
determined later.

The following lemma gives a bound for shallow tubes.

######  Lemma 8.1.

Suppose that $0<\delta<\epsilon$. Let $N=N_{\alpha,\lambda,\tau}$ be a model
solid torus, where $0<\alpha\leq 2\pi$ and $0<\lambda\leq\min(\delta,\,2.97)$.
Suppose as well that $r_{\alpha,\lambda,\tau}(\delta)\leq{r_{\max}}$, for some
${r_{\max}}>0$. Then

$d_{\alpha,\lambda,\tau}(\delta,\epsilon)\geq\operatorname{arccosh}\frac{\epsilon}{\sqrt{7.256\,\delta}}-{r_{\max}}.$

###### Proof.

An immediate consequence of Proposition 3.10 is that the tube radius
$r_{\alpha,\lambda,\tau}(\epsilon)$ is decreasing in $\lambda$. Combining this
fact with Proposition 7.1 gives

$\displaystyle d_{\alpha,\lambda,\tau}(\delta,\epsilon)$
$\displaystyle=r_{\alpha,\lambda,\tau}(\epsilon)-r_{\alpha,\lambda,\tau}(\delta)$
$\displaystyle\geq
r_{\alpha,\delta,\tau}(\epsilon)-r_{\alpha,\lambda,\tau}(\delta)$
$\displaystyle\geq\operatorname{arccosh}\frac{\epsilon}{\sqrt{7.256\,\delta}}-{r_{\max}}.\qed$

The corresponding statement for deep tubes is:

######  Lemma 8.2.

Suppose that $0<\delta<\epsilon$, where $\delta\leq 0.3$. Let $0<\alpha\leq
2\pi$, and let $N=N_{\alpha,\lambda,\tau}$ be a model solid torus such that
$N^{\leq\delta}\neq\emptyset$. Suppose as well that
$r_{\alpha,\lambda,\tau}(\delta)\geq{r_{\min}}>0$. Then

$d_{\alpha,\lambda,\tau}(\delta,\epsilon)\>\geq\>\log\left(\frac{\epsilon}{\delta}\cdot
1.268\sinh{r_{\min}}\right)-{r_{\min}}.$

To prove Lemma 8.2, we need to compute how fast the Euclidean injectivity
radius changes.

######  Lemma 8.3.

Let $0<r<R$, and consider equidistant tori $T_{r},T_{R}\subset
N_{\alpha,\lambda,\tau}$. Let $c_{r}\subset T_{r}$ be a rectifiable curve, and
let $c_{R}\subset T_{R}$ be the cylindrical projection of $c_{r}$ to $T_{R}$.
Then

$\frac{\cosh R}{\cosh r}\leq\frac{\ell(c_{R})}{\ell(c_{r})}\leq\frac{\sinh
R}{\sinh r}.$

###### Proof.

Let $(r,\zeta(t),\theta(t))$ be a parametrization of $c_{r}$, where
$t\in[0,1]$. By (2.2), the distance element on $T_{r}$ satisfies

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

Thus we may compute that

$\ell(c_{r})=\int_{0}^{1}\,ds=\int_{0}^{1}\sqrt{\cosh^{2}r\left(\frac{d\zeta}{dt}\right)^{2}+\sinh^{2}r\left(\frac{d\theta}{dt}\right)^{2}}dt.$

Similarly,

$\displaystyle\ell(c_{R})$
$\displaystyle=\int_{0}^{1}\sqrt{\cosh^{2}R\left(\frac{d\zeta}{dt}\right)^{2}+\sinh^{2}R\left(\frac{d\theta}{dt}\right)^{2}}dt$
$\displaystyle=\int_{0}^{1}\sqrt{\frac{\cosh^{2}R}{\cosh^{2}r}\cdot\cosh^{2}r\left(\frac{d\zeta}{dt}\right)^{2}+\frac{\sinh^{2}R}{\sinh^{2}r}\cdot\sinh^{2}r\left(\frac{d\theta}{dt}\right)^{2}}dt$
$\displaystyle\geq\int_{0}^{1}\frac{\cosh R}{\cosh
r}\,\sqrt{\cosh^{2}r\left(\frac{d\zeta}{dt}\right)^{2}+\sinh^{2}r\left(\frac{d\theta}{dt}\right)^{2}}dt$
$\displaystyle=\frac{\cosh R}{\cosh r}\,\ell(c_{r}),$

where the inequality in the next-to-last line is Lemma A.2.

A similar computation proves the other inequality in the lemma. ∎

###### Proof of Lemma 8.2.

Let $x\in T^{\delta}\subset N$. Then, by Lemma 2.5, there is a lift
$\widetilde{x}\in{\hat{{\mathbb{H}}}^{3}}$ and a deck transformation $\eta$
such that $d(\widetilde{x},\eta\widetilde{x})=\delta$. Furthermore, $\eta$ is
a loxodromic if $N$ is nonsingular. The points
$\widetilde{x},\eta(\widetilde{x})$ are connected by a Euclidean geodesic arc
$\widetilde{c}^{\delta}\subset\widetilde{T}^{\delta}$. Projecting back down to
$N$, we have a Euclidean geodesic $c^{\delta}$ whose length satisfies

(8.4)
$0.634\,\ell(c^{\delta})=0.634\,d_{E}(\widetilde{x},\eta\widetilde{x})<\delta,$

where the inequality is by Lemma 6.5.

Let $c^{\epsilon}\subset T^{\epsilon}$ be the cylindrical projection of
$c^{\delta}$ to $T^{\epsilon}$. Since every point of $T^{\epsilon}$ has
injectivity radius $\epsilon/2$, we know that
$\ell(c^{\epsilon})\geq\epsilon$. Therefore,

$\frac{\epsilon}{\delta}<\>\frac{\ell(c^{\epsilon})}{0.634\,\ell(c^{\delta})}\leq\>\frac{\sinh
r(\epsilon)}{0.634\,\sinh r(\delta)}<\>\frac{e^{r(\epsilon)}}{2\cdot
0.634}\cdot\frac{e^{-r(\delta)}\cdot e^{r(\delta)}}{\sinh
r(\delta)}\leq\>\frac{e^{r(\epsilon)-r(\delta)}}{1.268}\cdot\frac{e^{{r_{\min}}}}{\sinh{r_{\min}}}.$

Here, the first inequality uses (8.4), the second inequality is Lemma 8.3, the
third inequality uses the definition of $\sinh r$, and the fourth inequality
uses the monotonicity of $e^{-r}\sinh r$. Taking logarithms completes the
proof. ∎

To complete the proof of Theorem 1.1, we need one more elementary lemma.

######  Lemma 8.5.

Consider the function

$j(\delta,\epsilon)=\frac{1}{1.268}\left(\sqrt{\frac{\delta}{7.256}}+\sqrt{\frac{\delta}{7.256}-\frac{\delta^{2}}{\epsilon^{2}}}\right).$

On the domain $Q=\\{(\delta,\epsilon):0\leq\epsilon\leq
0.3,\,0\leq\delta\leq\epsilon^{2}/7.256\\}$, the function satisfies
$j(\delta,\epsilon)\leq 0.0424$.

###### Proof.

It is clear from the definition that $j(\delta,\epsilon)$ is increasing in
$\epsilon$. Thus the maximum over $Q$ occurs on the arc of $\partial Q$ where
$\epsilon=0.3$. On this arc, we compute
$\frac{\partial}{\partial\delta}j(\delta,\epsilon)$ and find that
$j(\delta,\epsilon)$ has a single interior critical point at
$\delta=0.0093026\ldots$, with maximal value
$j(\delta,\epsilon)=0.042357\ldots$. ∎

We can now complete the proof of the main theorem.

###### Proof of Theorem 1.1.

The upper bound of the theorem is proved in Proposition 5.7.

For the lower bound, suppose that $0<\delta<\epsilon\leq 0.3$, and set
${r_{\max}}={r_{\min}}=0.0424$. Suppose that $N=N_{\alpha,\lambda,\tau}$ is a
model solid torus such that $\lambda\leq\delta$. By Lemma 5.1, we have
$d_{\alpha,\lambda,\tau}(\delta,\epsilon)\geq(\epsilon-\delta)/2$. Thus it
remains to show that

(8.6)
$d_{\alpha,\lambda,\tau}(\delta,\epsilon)\geq\operatorname{arccosh}\frac{\epsilon}{\sqrt{7.256\,\delta}}-{r_{\min}}.$

If $\sqrt{7.256\,\delta}>\epsilon$, our convention (see Remark 3.8) is that
$\operatorname{arccosh}(\epsilon/\sqrt{7.256\,\delta})=-\infty$, hence (8.6)
holds trivially. If $r(\delta)\leq{r_{\min}}$, the desired lower bound of
(8.6) holds by Lemma 8.1. From now on, we assume that
$r(\delta)\geq{r_{\min}}=0.0424$ and $\delta\leq\epsilon^{2}/7.256\leq
0.3^{2}/7.256$. This means the hypotheses of both Lemma 8.2 and Lemma 8.5 are
satisfied.

We will use Lemma 8.5 to show that the lower bound of Lemma 8.2 is stronger
than (8.6). We compute as follows, starting from Lemma 8.5:

$\displaystyle 0.0424={r_{\min}}$
$\displaystyle\geq\frac{1}{1.268}\left(\sqrt{\frac{\delta}{7.256}}+\sqrt{\frac{\delta}{7.256}-\frac{\delta^{2}}{\epsilon^{2}}}\right)$
$\displaystyle\sinh{r_{\min}}$
$\displaystyle\geq\frac{1}{1.268}\cdot\frac{\delta}{\epsilon}\left(\sqrt{\frac{\epsilon^{2}}{7.256\delta}}+\sqrt{\frac{\epsilon^{2}}{7.256\delta}-1}\right)$
$\displaystyle\frac{\epsilon}{\delta}\cdot 1.268\sinh{r_{\min}}$
$\displaystyle\geq\sqrt{\frac{\epsilon^{2}}{7.256\delta}}+\sqrt{\frac{\epsilon^{2}}{7.256\delta}-1}$
$\displaystyle\log\left(\frac{\epsilon}{\delta}\cdot
1.268\sinh{r_{\min}}\right)-{r_{\min}}$
$\displaystyle\geq\log\left(\sqrt{\frac{\epsilon^{2}}{7.256\delta}}+\sqrt{\frac{\epsilon^{2}}{7.256\delta}-1}\right)-{r_{\min}}$
$\displaystyle\log\left(\frac{\epsilon}{\delta}\cdot
1.268\sinh{r_{\min}}\right)-{r_{\min}}$
$\displaystyle\geq\operatorname{arccosh}\frac{\epsilon}{\sqrt{7.256\,\delta}}-{r_{\min}}.$

Thus, by Lemma 8.2, the desired lower bound (8.6) holds. ∎

######  Remark 8.7.

In the above proof, the constant ${r_{\min}}=0.0424$ is (a slight overestimate
for) the maximum of the function $j(\delta,\epsilon)$ from Lemma 8.5. The
maximum of $j(\delta,\epsilon)$ on its domain is attained when $\epsilon$
takes the maximal value $0.3$, and is the primary reason why the additive
constant $-0.0424$ in the statement of Theorem 1.1 depends on the upper bound
for $\epsilon$. Meanwhile, the multiplicative constant $7.256$ in the
statement of Theorem 1.1 is a slight overestimate for $\sqrt{4\pi/\sqrt{3}}$.
This multiplicative constant comes from Proposition 7.1 and Lemma 8.1, where
$\delta\leq 2.97$ suffices. In particular, it carries no hypotheses on
$\epsilon$.

Consequently, there is a generalization of Theorem 1.1 that holds for larger
values of $\epsilon$.

######  Theorem 8.8.

Fix a positive constant $\epsilon_{\max}\leq 1.475$, and suppose that
$0<\delta<\epsilon\leq\epsilon_{\max}$. Let $j(\delta,\epsilon)$ be the
function of Lemma 8.5, and let $j_{\max}$ be the maximal value of
$j(\delta,\epsilon_{\max})$ on the interval
$0\leq\delta\leq\epsilon_{\max}^{2}/7.256$. Let
${r_{\min}}=\operatorname{arcsinh}(j_{\max})$.

Then, for every hyperbolic solid torus $N=N_{\alpha,\lambda,\tau}$, where
$0<\alpha\leq 2\pi$ and $\lambda\leq\delta$, we have

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

###### Proof.

The proof is nearly identical to the above proof of Theorem 1.1. We substitute
$\epsilon_{\max}$ in place of $0.3$ and the new definition
${r_{\min}}=\operatorname{arcsinh}(j_{\max})$ in place of $0.0424$. In the
non-trivial case where $\delta\leq\epsilon^{2}/7.256$, the hypothesis
$\epsilon\leq 1.475$ implies $\delta\leq 0.3$, which means Lemma 8.2 still
applies. The final computation goes through verbatim. ∎

Finally, we observe that the hypotheses of Theorem 8.8 can be loosened even
further, to $\epsilon_{\max}\leq\sqrt{7.256\cdot 2.97}=4.642\ldots,$ at the
cost of a more complicated adjustment to the value of ${r_{\min}}$. Given
$\delta_{\max}=\epsilon_{\max}^{2}/7.256$, we have to adjust the statement of
Lemma 8.2 to work for all $\delta$ in the range $0<\delta\leq\delta_{\max}$.
This will cause the multiplicative constant $1.268$ to change, which will
cause the function $j(\delta,\epsilon)$ of Lemma 8.5 to change as well. The
maximum of the adjusted function will then determine the needed value of
${r_{\min}}$. We expect the constants of Theorem 8.8 to suffice for our
applications.

##  Appendix A Hyperbolic trigonometry

This appendix records several easy lemmas involving hyperbolic sines and
cosines.

######  Lemma A.1.

Let $0\leq r\leq s$. Then

$\cosh(s-r)\leq\frac{\cosh s}{\cosh r},$

with equality if and only if $r=0$ or $r=s$.

###### Proof.

Let $h=s-r$. Then

$\displaystyle\cosh(s)$ $\displaystyle=\cosh(r+h)$ $\displaystyle=\cosh r\cosh
h+\sinh r\sinh h$ $\displaystyle\geq\cosh r\cosh h.$

Note that we will have equality if and only if $r=0$ or $h=0$, as desired. ∎

######  Lemma A.2.

Let $0<r\leq s$. Then

$\frac{\cosh s}{\cosh r}\leq e^{s-r}\leq\frac{\sinh s}{\sinh r},$

with equality if and only if $r=s$.

###### Proof.

Again, let $h=s-r$. Then, as above,

$\displaystyle\cosh(s)$ $\displaystyle=\cosh r\cosh h+\sinh r\sinh h$
$\displaystyle\leq\cosh r\cosh h+\cosh r\sinh h$ $\displaystyle=\cosh r\cdot
e^{h},$

proving the first inequality. Observe that equality holds if and only if
$h=0$. The second inequality is proved similarly, using the sum formula for
$\sinh(r+h)$. ∎

######  Lemma A.3.

Suppose $x,y\geq 0$. Define

$f(x)=\begin{cases}(\cosh x-1)/x^{2},&x>0\\\ 1/2,&x=0\\\
\end{cases}\qquad\text{and}\qquad h(x,y)=\frac{f(y)}{f(x)}.$

Then $f(x)$ strictly increasing in $x$, while $h(x,y)$ is increasing in $y$
and decreasing in $x$.

###### Proof.

Expanding the Taylor series

$\cosh x=1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\frac{x^{6}}{6!}+\ldots$

gives

$f(x)=\frac{x^{0}}{2!}+\frac{x^{2}}{4!}+\frac{x^{4}}{6!}+\ldots,$

which is clearly increasing in $x$. The monotonicity of $h$ is now immediate.
∎

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