H_{i})\geq\sqrt{3}/2$ for every $i$. Let $s_{i}$ be a Euclidean geodesic on
$\partial H_{i}$ representing the meridian of $\sigma_{i}$. Let
$L_{i}=\operatorname{len}(s_{i})/\sqrt{\operatorname{area}(\partial H_{i})}$
be the normalized length of $s_{i}$, and $L=L({\mathbf{s}})$ the total
normalized length of ${\mathbf{s}}=(s_{1},\ldots,s_{k})$. Thus our hypothesis
$L({\mathbf{s}})^{2}\geq 230.1$ combined with (1.1) implies

(7.1)
$\frac{1}{230.1}\geq\frac{1}{L({\mathbf{s}})^{2}}\geq\frac{1}{L(s_{i})^{2}}=\frac{\operatorname{area}(\partial
H_{i})}{\operatorname{len}(s_{i})^{2}}\geq\frac{\sqrt{3}/2}{\operatorname{len}(s_{i})^{2}}\,,$

hence $\operatorname{len}(s_{i})>14$ for each $i$. Consequently, Theorem 3.1
tells us that $(M,P)$ is a pared manifold that admits a hyperbolic structure
$Y$ with end invariants matching those of $Z$.

Starting from $Z$, we apply strong density, Theorem 2.12, to find a sequence
of geometrically finite, minimally parabolic representations $\tau_{n}\in
AH(M-\mathcal{N}(\Sigma),P\cup\partial\mathcal{N}(\Sigma))$, such that
$\tau_{n}\to\xi$ is a strong limit. The end invariants of the associated
hyperbolic manifolds $Z_{n}$ form a filling sequence, converging to the end
invariants of $Z$. The Kleinian groups $\Delta_{n}$ associated to $Z_{n}$
converge (both geometrically and algebraically) to the Kleinian group $\Delta$
associated to $Z$. In particular, we have $\Delta_{n}\to\Delta$ in the
Chabauty topology. We also choose an open neighborhood system $Q_{n}$ about
$[\xi]$, such that $[\tau_{n}]\in Q_{n}$.

For each $n$, we choose a positive constant $\delta_{n}$ such that
$\lim\delta_{n}=0$ and several more constraints (specified below) are all
satisfied. For each $n$, Brooks’ Theorem 2.15 says that there is a
geometrically finite Kleinian group $\Delta_{n,\delta_{n}}$ representing an
$e^{\delta_{n}}$–quasiconformal deformation of $\Delta_{n}$, and such that the
conformal boundary of $\Delta_{n,\delta_{n}}$ admits a circle packing. By
picking $\delta_{n}$ small enough, we ensure that $d_{\rm
Chaub}(\Delta_{n,\delta_{n}},\Delta_{n})<2^{-n}$. Picking $\delta_{n}$ small
enough also ensures that the associated representation $\xi_{n}$ satisfies
$[\xi_{n}]\in Q_{n}$. These choices imply that $\xi_{n}\to\xi$ is a strong
limit. We let $W_{n}={\mathbb{H}}^{3}/\Delta_{n,\delta_{n}}$.

Next, we fill in the meridians of $\Sigma$ to recover the pared manifold
$(M,P)$. We obtain three sets of hyperbolic structures on $(M,P)$:

  * •

A hyperbolic structure $Y$, whose end invariants agree with those of $Z$,

  * •

A hyperbolic structure $Y_{n}$, whose end invariants agree with to those of
$Z_{n}$,

  * •

A hyperbolic structure $V_{n}$, whose end invariants agree with those of
$W_{n}$.

Since $(M,P)$ is a pared manifold, Theorem 2.6 says there is also a
geometrically finite, minimally parabolic representation $\rho^{\prime}\in
AH(M,P)$. The hyperbolic structures $Y_{n}$ and $V_{n}$, represented by
$\sigma_{n}\in QH(\rho^{\prime})$ and $\rho_{n}\in QH(\rho^{\prime})$,
respectively, exist by Theorem 2.7.

By construction, the conformal boundary of $V_{n}$ is an
$e^{\delta_{n}}$–quasiconformal deformation of the conformal boundary of
$Y_{n}$. Thus the distance in $\mathcal{T}(\partial_{0}M)$ between the
conformal boundaries of those manifolds is at most $\delta_{n}$. By choosing
$\delta_{n}$ small enough, we ensure that $Y_{n}$ and $V_{n}$ are represented
by Kleinian groups $\Gamma_{n}$ and $\Gamma_{n,\delta_{n}}$, respectively,
such that $d_{\rm Chaub}(\Gamma_{n,\delta_{n}},\Gamma_{n})<2^{-n}$. We also
ensure that, for a neighborhood system $\\{O_{n}\\}$ about $[\rho]$, we have
$\rho_{n}\in O_{m(n)}$ whenever $\rho\neq\sigma_{n}\in O_{m(n)}$, and
$\rho_{n}\in O_{n}$ whenever $\sigma_{n}=\rho$. This completes the list of
conditions that $\delta_{n}$ needs to satisfy.

Now, we check the conclusions of the proposition. Conclusion (1) holds by
construction, because we have chosen $\Delta_{n,\delta_{n}}$ so that the
conformal boundary of $W_{n}={\mathbb{H}}^{3}/\Delta_{n,\delta_{n}}$ admits a
circle packing $C_{n}$, and because we have chosen $\delta_{n}$ small enough
to ensure $\xi_{n}\to\xi$ is a strong limit.

For conclusion (2), recall that we have already checked that
$\Delta_{n,\delta_{n}}\to\Delta$ in the Chabauty topology. Thus, for every
basepoint $z\in Z$, there exist basepoints $w_{n}\in W_{n}$ such that
$(W_{n},w_{n})\to(Z,z)$. For any fixed radius, the ball $B_{R}(w_{n})$ will
lie in the scooped manifold $W_{n}^{\circ}$ when $n\gg 0$, hence the sequence
of scooped manifolds $(W_{n}^{\circ},w_{n})$ also converges geometrically to
$(Z,z)$.

We will prove conclusions (3) and (5) together. Observe that since the
conformal boundary of $V_{n}$ agrees with that of $W_{n}$, it also admits the
same circle packing $C_{n}$. Thus we may double-double the scooped manifolds
$V_{n}^{\circ}$ and $W_{n}^{\circ}$ to obtain finite-volume manifolds
$DD(V_{n}^{\circ})$ and $DD(W_{n}^{\circ})$, respectively. The scooped
manifold $W_{n}^{\circ}$ contains a tuple of slopes
${\mathbf{s}}^{n}=(s_{1}^{n},\ldots,s_{k}^{n})$ corresponding to the meridians
of $\Sigma$, with the property that $L(s_{i}^{n})\to L(s_{i})$ in the
geometric limit $(W_{n}^{\circ},w_{n})\to(Z,z)$. Thus, for $n\gg 0$, we have
$L({\mathbf{s}}^{n})^{2}>230.08$. Thus the finite-volume manifold
$DD(W_{n}^{\circ})$ contains a tuple of slopes $DD({\mathbf{s}}^{n})$,
containing four copies of each meridian of $\Sigma$. The definition of
normalized length in (1.1) implies that for all $n\gg 0$, we have

(7.2)
$\frac{1}{L(DD({\mathbf{s}}^{n}))^{2}}=\frac{4}{L({\mathbf{s}}^{n})^{2}}<\frac{4}{230.08}=\frac{1}{57.52},\quad\text{hence}\quad
L(DD({\mathbf{s}}^{n}))>\sqrt{57.52}>7.584.$

Consequently, a theorem of Hodgson and Kerckhoff [24, Theorem 1.2], implies
that Dehn filling $DD(W_{n}^{\circ})$ along the tuple of slopes
$DD({\mathbf{s}}^{n})$ produces a hyperbolic manifold $U_{n}$, where the union
of cores of the Dehn filling solid tori is a geodesic link $\Upsilon_{n}$. By
[24, Corollary 5.13], the total length of $\Upsilon_{n}$ is less than $0.16$.

Now, recall that each $V_{n}$ is homeomorphic to $M$, and contains a smoothly
embedded copy of $\Sigma$, such that $W_{n}$ is homeomorphic to
$V_{n}-\Sigma$. After an isotopy, we may assume that the embedded copy of
$\Sigma$ lies in $V_{n}^{\circ}$, hence $DD(V_{n}^{\circ})$ contains the
double-doubled link $DD(\Sigma)$. Since $\partial V_{n}^{\circ}$ and $\partial
W_{n}^{\circ}$ contain exactly the same pattern of red and blue faces, the
double-doubling construction shows that $DD(W_{n}^{\circ})$ is homeomorphic to
$DD(V_{n}^{\circ})-DD(\Sigma)$ by a homeomorphism that carries
$DD({\mathbf{s}}^{n})$ to the tuple of meridians of $DD(\Sigma)$. The above
homeomorphism extends to a homeomorphism of pairs
$(U_{n},\Upsilon_{n})\to(DD(V_{n}^{\circ}),DD(\Sigma))$. Since
$DD(V_{n}^{\circ})$ is Haken, Waldhausen’s topological rigidity theorem [39,
Theorem 7.1] implies that $DD(\Sigma)$ is isotopic to the geodesic link
$\Upsilon_{n}$. The components of $\Upsilon_{n}$ that lie in the original copy
of $V_{n}^{\circ}$ form a geodesic link $\Sigma_{n}$, with the property that

$DD(V_{n}^{\circ})-DD(\Sigma_{n})\cong DD(V_{n}^{\circ}-\Sigma_{n})\cong
DD(W_{n}^{\circ}),$

proving (3) and (5). We note that
$\operatorname{len}(\Sigma_{n})=\operatorname{len}(\Upsilon_{n})/4<0.04$.

Conclusion (4) is established exactly as in Theorem 6.2. By construction, the
end invariants of $Y_{n}$ agree with those of $Z_{n}$. Thus those end
invariants form a filling sequence that limits to the end invariants of $Y$.
Thus, by the approximation Theorem 2.13, the representations $\sigma_{n}$
associated to $Y_{n}$ converge strongly (after a subsequence) to a
representation of a hyperbolic manifold homeomorphic to $Y$ and having the
same end invariants as $Y$. By the ending lamination Theorem 2.5, we have a
strong limit $\sigma_{n}\to\rho$ for a representation $\rho$ associated to
$Y$. In particular, $\Gamma_{n}\to\Gamma$ in the Chabauty topology and
$[\sigma_{n}]\in O_{m(n)}$ for a sequence $m(n)\to\infty$.

By our choice of $\delta_{n}$, we have a Chabauty limit
$\Gamma_{n,\delta_{n}}\to\Gamma$ and $[\rho_{n}]\in O_{m(n)}$ for the same
sequence $m(n)$, hence $\rho_{n}\to\rho$ is a strong limit. As above, we can
fix a basepoint $y\in Y$ and then find basepoints $v_{n}\in V_{n}$, which lie
in $V_{n}^{\circ}$ for $n\gg 0$, such that $(V_{n}^{\circ},v_{n})$ converges
geometrically to $(Y,y)$. Since $\rho_{n}\to\rho$ is a strong limit, and
$\operatorname{len}(\Sigma_{n})<0.04$ for all $n\gg 0$, Proposition 4.1 says
that $\Sigma_{n}\to\Sigma_{\infty}\subset Y$, a geodesic link isotopic to
$\Sigma$.

Finally, conclusion (6) is a corollary of (2) and (4). ∎

######  Remark 7.3.

If the end invariants of $Y$ (equivalently, the end invariants of $Z$) are
geometrically finite, the preceding proof becomes considerably more
lightweight. In this case, one can take constant sequences $Y_{n}=N$ and
$Z_{n}=Z$. Thus, as in Remark 6.3, the hyperbolization, strong density, ending
lamination, and approximation theorems become unnecessary. Thus, in the
geometrically finite case, the only tools required in the proof are Theorem
2.7 and Brooks’ Theorem 2.15.

Next, we record a finite-volume version of Theorem 1.3, with some additional
information. The following result is [20, Corollary 9.34]. As in Remark 6.6,
the hypotheses of Theorem 7.4 differ from those of Theorem 1.3 by a factor of
4.

######  Theorem 7.4 (Effective filling in finite volume, [20]).

Fix any $0<\epsilon\leq\log 3$ and any $J>1$. Let $M$ be a 3-manifold with
empty or toroidal boundary, and $\Sigma$ a link in $M$. Suppose that
$M-\Sigma$ admits a complete, finite-volume hyperbolic metric $W$, in which
the total normalized length of the meridians of $\Sigma$ satisfies

$L^{2}\geq\max\left\\{\frac{2\pi\cdot
6771\cosh^{5}(0.6\epsilon+0.1475)}{\epsilon^{5}}+11.7,\>\frac{2\pi\cdot
11.35}{\epsilon^{5/2}\log(J)}+11.7\right\\}.$

Then $M$ has a hyperbolic metric $V$, in which $\Sigma$ a geodesic link.
Furthermore, there are canonical $J$–bilipschitz inclusions

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

which are equivariant with respect to the symmetry group of the pair
$(M,\Sigma)$.

We can now use Theorems 7.2 and 7.4 to prove Theorem 1.3. The proof involves
chasing the left half of the diagram in Figure 2. Starting from the hyperbolic
structure $Z$ on the complement of $\Sigma$, we will consider a circle-packed
approximating manifold $W_{n}$, the finite-volume manifold
$DD(W_{n}^{\circ})$, its filling $DD(V_{n}^{\circ})$, and the scooped
submanifold $V_{n}^{\circ}$ that approximates $Y$.

###### Proof of Theorem 1.3.

Let $M$ be a tame 3-manifold and $\Sigma\subset M$ a link such that $M-\Sigma$
admits a hyperbolic structure $Z$. If $\operatorname{vol}(Z)<\infty$, then the
desired conclusion is covered by Theorem 7.4, substituting $V=Y$ and $W=Z$.
For the rest of the proof, we assume $\operatorname{vol}(Z)=\infty$.

Let ${\mathbf{s}}=(s_{1},\dots,s_{k})$ be a tuple of slopes on the cusps of
$Z$, with one slope for the meridian of each component of $\Sigma$. We may
think of each $s_{i}$ as a slope on a torus of $\mathcal{N}(\Sigma)$. Recall
our hypothesis on the total normalized length of the meridians:

(7.3) $L({\mathbf{s}})^{2}>4\,\max\left\\{\frac{2\pi\cdot
6771\cosh^{5}(0.6\epsilon+0.1475)}{\epsilon^{5}}+11.7,\>\frac{2\pi\cdot
11.35}{\epsilon^{5/2}\log(J)}+11.7\right\\}.$

Since $\epsilon\leq\log 3$, and since $\cosh^{5}(x)\geq 1$ for any $x$,
hypothesis (7.3) is considerably stronger than the normalized length
hypothesis of Theorem 7.2. Thus $Y=Z({\mathbf{s}})$ is a hyperbolic structure
on $M$ with the same end invariants as those of $Z$.

Fix a basepoint $z\in Z$ so that $\epsilon=2{\operatorname{injrad}}(z)$, and
furthermore $z$ lies on an embedded, $\epsilon$–thick horotorus
$T^{\eta}(\sigma)$ about the first component $\sigma\subset\Sigma$. Such a
choice of $z\in Z^{\geq\epsilon}$ is possible because $\epsilon$ is a Margulis
number for $Z$ by Theorem 2.2; hence the thick part $Z^{>\epsilon}$ is non-
empty.

Let $V_{n}$ and $W_{n}$ be the sequences of geometrically finite manifolds
constructed in Theorem 7.2. In particular, the conformal boundaries of each
$V_{n}$ and each $W_{n}$ admit the same circle packing $C_{n}$. Let
$V_{n}^{\circ}\subset V_{n}$ and $W_{n}^{\circ}\subset W_{n}$ be the scooped
submanifolds defined by $C_{n}$, as in Definition 2.16. We also have strong
limits $\rho_{n}\to\rho$ and $\xi_{n}\to\xi$, as described in Theorem 7.2.

By Theorem 7.2, each approximating manifold $W_{n}$ has pared manifold
$(M-\mathcal{N}(\Sigma),P\cup\partial\mathcal{N}(\Sigma))$. Thus each $W_{n}$
has a rank-2 horocusp $H_{i}^{n}$ corresponding to each component
$\sigma_{i}\subset\Sigma$, with meridian slope $s_{i}^{n}$. We may assume
$H_{i}^{n}\subset W_{n}^{\circ}$, after shrinking the horocusp as needed. For
each $i$, the normalized length $L(s_{i}^{n})$ converges to the normalized
length $L(s_{i})$ measured in $Z$. Write
${\mathbf{s}}^{n}=(s_{1}^{n},\ldots,s_{k}^{n})$ for the tuple of slopes in
$W_{n}$ representing the meridians of $\Sigma$. Since $L(s_{i}^{n})\to
L(s_{i})$ as $n\to\infty$, it follows that for $n\gg 0$, the total normalized
length $L({\mathbf{s}}^{n})$ must also satisfy the lower bound (7.3).

After doubling $W_{n}^{\circ}$ twice, as in Definition 2.16, we obtain a
finite-volume hyperbolic manifold $DD(W_{n}^{\circ})$. This manifold contains
the disjoint union of four isometric copies of $H_{i}^{n}$ for each $i$. Then
the tuple of slopes ${\mathbf{s}}^{n}$ is also double-doubled to become a
tuple of slopes $DD({\mathbf{s}}^{n})$ on the cusps of $DD(W_{n}^{\circ})$.
Because each slope $s_{i}^{n}$ appears four times in $DD({\mathbf{s}}^{n})$,
the total normalized length of the meridians of $DD(H^{n})$ satisfies

$L^{2}>\max\left\\{\frac{2\pi\cdot
6771\cosh^{5}(0.6\epsilon+0.1475)}{\epsilon^{5}}+11.7,\>\frac{2\pi\cdot
11.35}{\epsilon^{5/2}\log(J)}+11.7\right\\}.$

Thus, for $n\gg 0$, Theorem 7.4 enables us to fill $DD(W_{n}^{\circ})$ along
the tuple of meridians of ${\mathbf{s}}^{n}$ and obtain a hyperbolic
3-manifold $U_{n}$, in which the union of cores of the filled solid tori is a
geodesic link $\Upsilon_{n}$. By Theorem 7.2, we have
$U_{n}=DD(V_{n}^{\circ})$ and $\Upsilon_{n}=DD(\Sigma_{n})$ for a geodesic
link $\Sigma_{n}\subset V_{n}^{\circ}$.

Now, Theorem 7.4 says that there are $J$–bilipschitz inclusions

$\varphi_{n}\colon U_{n}^{\geq\epsilon}\to
DD(W_{n}^{\circ})^{\geq\epsilon/1.2},\qquad\psi_{n}\colon
DD(W_{n}^{\circ})^{\geq\epsilon}\to U_{n}^{\geq\epsilon/1.2},$

which are equivariant with respect to the
${\mathbb{Z}}_{2}\times{\mathbb{Z}}_{2}$ group of symmetries of the pair
$(U_{n},\Upsilon_{n})$. Since $(V_{n}^{\circ},\Sigma_{n})$ is a fundamental
domain for this group action, $\varphi_{n}$ and $\psi_{n}$ restrict to
$J$–bilipschitz inclusions

$\varphi_{n}\colon(V_{n}^{\circ})^{\geq\epsilon}\to(W_{n}^{\circ})^{\geq\epsilon/1.2},\qquad\psi_{n}\colon
DD(W_{n}^{\circ})^{\geq\epsilon}\to(V_{n}^{\circ})^{\geq\epsilon/1.2}.$

By Theorem 7.2, the geodesic links $\Sigma_{n}\subset V_{n}^{\circ}$ converge
to a geodesic link $\Sigma_{\infty}\subset Y$, isotopic to $\Sigma$. After
performing this isotopy, we may suppose that $\Sigma\subset Y$ is a geodesic
link.

In preparation for Theorem 5.1, we choose appropriate basepoints for our
geometric limits. Recall that we have picked a component
$\sigma\subset\Sigma$, and chosen a basepoint $z\in Z$ so that
$2{\operatorname{injrad}}(z)=\epsilon$, and furthermore $z$ lies on an
embedded $\epsilon$–thick horotorus $T^{\epsilon}(\sigma)$. In a similar
fashion, we choose a basepoint $y\in Y$ so that
$2{\operatorname{injrad}}(y)=\eta\in(\epsilon,2\epsilon)$, and furthermore $y$
lies on an embedded, $\eta$–thick equidistant torus $T^{\eta}(\sigma)$ about
the same component $\sigma\subset\Sigma$. Such a choice of $y\in
Y^{>\epsilon}$ is possible because $\epsilon$ is a Margulis number for $N$;
hence the thick part $N^{>\epsilon}$ is non-empty.

Theorem 7.2 says that for $n\gg 0$, there exist choices of basepoints
$v_{n}\in V_{n}^{\circ}$ and $w_{n}\in W_{n}^{\circ}$, such that
$(V_{n}^{\circ},v_{n})\to(Y,y)$ and $(W_{n}^{\circ},w_{n})\to(Z,z)$. The
convergence of injectivity radii in a geometric limit implies that
$2{\operatorname{injrad}}(v_{n})\in(\epsilon,2\epsilon)$ for large $n$.
Similarly, $2{\operatorname{injrad}}(w_{n})\to\epsilon$ as $n\to\infty$.

We are now ready to construct the $J$–bilipschitz inclusion $\varphi\colon
Y^{\geq\epsilon}\hookrightarrow Z^{\geq\epsilon/1.2}$, using Theorem 5.1. We
have geometrically convergent sequences $(V_{n}^{\circ},v_{n})\to(Y,y)$ and
$(W_{n},w_{n})\to(Z,z)$. We have $y\in Y^{>\epsilon}$ and
$v_{n}\in(V_{n}^{\circ})^{\geq\epsilon}$ for large $n$, as required. For large
$n$, we have a $J$–bilipschitz inclusion
$\varphi_{n}\colon(V_{n}^{\circ})^{\geq\epsilon}\to(W_{n}^{\circ})^{\geq\epsilon/1.2}$.
Furthermore, $d(\varphi_{n}(v_{n}),w_{n})$ is uniformly bounded, by exactly
the same argument as in the end of the proof of Theorem 1.2. (Essentially,
this follows because injectivity radii are well-behaved under both geometric
limits and bilipschitz maps.) Thus Theorem 5.1 gives a $J$–bilipschitz
inclusion $\varphi\colon Y^{\geq\epsilon}\hookrightarrow
Z^{\geq\epsilon/1.2}$.

The reverse inclusion $\psi\colon Z^{\geq\epsilon}\hookrightarrow
Y^{\geq\epsilon/1.2}$ is constructed in exactly the same way, tracing points
backwards to check the hypotheses of Theorem 5.1. ∎

##  8\. Short geodesics in infinite-volume manifolds

The main results of this section are Theorem 8.5 and Theorem 8.7, which bound
the change in complex length of a short geodesic under drilling and filling,
respectively. Corollaries of those results include Theorem 1.5 and Corollary
8.9, where the functions that estimate the change in complex length are
replaced by constants.

As in Section 6, the proof of Theorem 8.5 combines an approximation result
(Theorem 6.2) with a previously proved theorem that works in finite volume
(Theorem 8.6). Similarly, the proof of Theorem 8.7 combines an approximation
result (Theorem 7.2) with a finite-volume theorem (Theorem 8.8).

To set up our results, we need to define the functions that will estimate the
change in length.

######  Definition 8.1.

Let $z_{0}=\sqrt{\sqrt{5}-2}=0.5306\ldots$. For $z\in[z_{0},1]$, define a
function

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

By a derivative computation, the function ${\operatorname{haze}}(z)$ is
decreasing and invertible in this domain. Using Cardano’s Formula, the inverse
function ${\operatorname{haze}}^{-1}$ can be expressed as follows:

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

Note that ${\operatorname{haze}}^{-1}$ is defined and monotonically decreasing
on $[0,1.0196]$. Compare [20, Remark 4.23].

Here is the geometric meaning of ${\operatorname{haze}}$ and
${\operatorname{haze}}^{-1}$. If $(N,\Sigma,g_{t})$ is a hyperbolic cone
manifold whose singular locus $\Sigma$ has angle $\alpha\in[0,2\pi]$, the
_visual area of $\Sigma$_ is defined to be
$\mathcal{A}(\Sigma)=\alpha\operatorname{len}(\Sigma)$. Hodgson and Kerckhoff
showed that under appropriate hypotheses, there is an embedded tube about
$\Sigma$ of radius
$R\geq\operatorname{arctanh}({\operatorname{haze}}^{-1}(\mathcal{A}(\Sigma)))$.
See [23, Theorem 5.6] and [20, Corollary 4.25] for details. In turn, the
radius of this tube is used to control a number of geometric quantities
through the cone deformation [20, Sections 5–7]. One of those quantities is
the complex length of a non-singular closed geodesic, which we seek to control
here.

######  Definition 8.2.

Let $\gamma$ be a closed geodesic in a hyperbolic 3-manifold $N$. Then
$\gamma$ 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 $\operatorname{len}(\gamma)$ and rotates by
angle $\tau(\gamma)$. We define the complex length
$\mathcal{L}_{N}(\gamma)=\mathcal{L}(\gamma)=\operatorname{len}(\gamma)+i\tau(\gamma)$.
Observe that $i\mathcal{L}(\gamma)$ lies in the upper half-plane of
${\mathbb{C}}$, which we identify with the hyperbolic plane
${\mathbb{H}}^{2}$.

Given two complex lengths $\mathcal{L}_{Y}(\gamma)$,
$\mathcal{L}_{Z}(\delta)$, we define the _hyperbolic distance_ between them to
be

${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\mathcal{L}_{Z}(\delta))=d_{{\mathbb{H}}^{2}}(i\mathcal{L}_{Y}(\gamma),\,i\mathcal{L}_{Z}(\delta)).$

This is closely related to distance in the Teichmüller space of the torus,
which is isometric to ${\mathbb{H}}^{2}$. See Minsky [32, Section 6.2] for
details.

The hyperbolic distance between lengths can be translated into a bound on the
real and imaginary parts of length. The following elementary lemma is [20,
Lemma 7.14].

######  Lemma 8.3.

Let $\mathcal{L}_{Y}(\gamma)$ and $\mathcal{L}_{Z}(\delta)$ be complex
lengths. Suppose that
${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\,\mathcal{L}_{Z}(\delta))\leq K$
for some $K>0$. Then the real and imaginary parts of $\mathcal{L}_{Y}(\gamma)$
and $\mathcal{L}_{Z}(\delta)$ are bounded as follows:

$e^{-K}\>\leq\>\frac{\operatorname{len}_{Z}(\delta)}{\operatorname{len}_{Y}(\gamma)}\>\leq\>e^{K},\qquad|\tau_{Z}(\delta)-\tau_{Y}(\gamma)|\>\leq\>\sinh(K)\cdot\min\\{\operatorname{len}_{Y}(\gamma),\,\operatorname{len}_{z}(\delta)\\}.$

Finally, given a filled manifold $Y$ and a drilled manifold $Z$, we control
the hyperbolic distance
${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\,\mathcal{L}_{Z}(\gamma))$ via
the following function.

######  Definition 8.4.

For $z\in[z_{0},1]$ and $\ell\in(0,0.5085]$, define a function

$F(z,\ell)=\frac{(1+z^{2})}{z^{3}(3-z^{2})}\cdot\frac{\ell}{10.667-20.977\ell}\,.$

Note that $F$ is positive everywhere on its domain, decreasing in $z$, and
increasing in $\ell$. Compare [20, Definition 7.2].

###  8.1. Short geodesics under drilling

The first main result of this section controls the complex length of a short
geodesic $\gamma$ under the drilling of a geodesic link $\Sigma$. In the next
theorem, $\ell=\operatorname{len}_{Y}(\Sigma)$ is the length of the geodesic
link that we wish to drill, $m=\operatorname{len}_{Y}(\gamma)$ is the length
of the geodesic that we wish to control, and $z={z_{\min}}=\tanh{R_{\min}}$,
where ${R_{\min}}$ is the minimum radius of an embedded tube about
$\Sigma^{+}=\Sigma\cup\gamma$. Then the function $F(z,\ell)$ controls the
change in the complex length $\mathcal{L}(\gamma)$. We will not compute
${R_{\min}}$ or ${z_{\min}}$ directly; we will merely estimate ${z_{\min}}$ as
a function of $\ell$ and $m$.

######  Theorem 8.5.

Let $Y$ be a tame hyperbolic 3-manifold. Let $\Sigma$ be a geodesic link in
$Y$, and $\gamma$ a closed geodesic disjoint from $\Sigma$. Let
$\ell=\operatorname{len}_{Y}(\Sigma)$ and $m=\operatorname{len}_{Y}(\gamma)$
be the lengths of $\Sigma$ and $\gamma$ in the complete metric on $Y$. Suppose
$\ell<0.018375$ and $m<0.0996-1.408\cdot\ell$. Let

${z_{\min}}={\operatorname{haze}}^{-1}(2\pi(4\ell+m+{10^{-5}})).$

Then $Y-\Sigma$ also admits a complete hyperbolic metric $Z$, with the same
end invariants as those of $Y$. The closed curve $\gamma$ is isotopic to a
geodesic in this metric. Furthermore, the complex lengths of $\gamma$ in $Y$
and $Z$ are related as follows:

${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\,\mathcal{L}_{Z}(\gamma))\>\leq\>4\pi^{2}\,F({z_{\min}},4\ell).$

The proof of Theorem 8.5 relies on the following finite-volume analogue [20,
Theorem 7.19].

######  Theorem 8.6 (Short geodesics under drilling, [20]).

Let $V$ be a complete, finite volume hyperbolic 3-manifold. Let $\Sigma$ be a
geodesic link in $V$, and $\gamma$ a closed geodesic disjoint from $\Sigma$.
Let $\ell=\operatorname{len}_{V}(\Sigma)$ and
$m=\operatorname{len}_{V}(\gamma)$ be the lengths of $\Sigma$ and $\gamma$ in
the complete metric on $V$. Suppose $\ell\leq 0.0735$ and $m\leq
0.0996-0.352\cdot\ell$. Let

$z^{\prime}_{\min}={\operatorname{haze}}^{-1}(2\pi(\ell+m+{10^{-5}}))>0.6288.$

Then $V-\Sigma$ also admits a complete hyperbolic metric $W$, in which
$\gamma$ is again isotopic to a geodesic. Furthermore, the complex lengths of
$\gamma$ in $V$ and $W$ are related as follows:

${d_{\mathrm{hyp}}}(\mathcal{L}_{V}(\gamma),\,\mathcal{L}_{W}(\gamma))\>\leq\>4\pi^{2}\,F(z^{\prime}_{\min},\ell).$

###### Proof of Theorem 8.5.

If $\operatorname{vol}(Y)<\infty$, the desired result already follows from
Theorem 8.6. (Although the definition of $z^{\prime}_{\min}$ in Theorem 8.6
differs from the definition of ${z_{\min}}$ in Theorem 8.5, the monotonicity
of ${\operatorname{haze}}^{-1}$ implies that $z^{\prime}_{\min}>{z_{\min}}$.
Then, the monotonicity of $F$ ensures that the conclusion of Theorem 8.6 still
applies with ${z_{\min}}$ in place of $z^{\prime}_{\min}$ and $4\ell$ in place
of $\ell$.) For the rest of the proof, we assume that
$\operatorname{vol}(Y)=\infty$.

We will apply Theorem 6.2. Let $V_{n}$ and $W_{n}$ be the sequences of
geometrically finite manifolds constructed in that theorem. Let $Z$ be the
hyperbolic manifold homeomorphic to $Y-\Sigma$, with the same end invariants
as those of $Y$. Then, by Theorem 6.2, the conformal boundaries of each
$V_{n}$ and each $W_{n}$ admit the same circle packing $C_{n}$. That theorem
also guarantees a strong limit $\rho_{n}\to\rho$ (where $\rho_{n}$ is the
representation corresponding to $W_{n}$ and $\rho$ corresponds to $Y$) and a
strong limit $\xi_{i}\to\xi$ (where $\xi_{n}$ is the representation
corresponding to $W_{n}$ and $\xi$ corresponds to $Z$).

Now, let $\gamma\subset Y$ be a closed geodesic satisfying the length bound of
the theorem. By Meyerhoff’s theorem [30, Section 7], we have
$\gamma\cap\Sigma=\emptyset$, hence $\Sigma^{+}=\Sigma\cup\gamma$ is a
geodesic link where each component is shorter than $0.1$. Then Proposition 4.1
implies that for $n\gg 0$, the approximating manifold $V_{n}$ contains a
geodesic link $\Sigma_{n}^{+}=\Sigma_{n}\cup\gamma_{n}$, where the sequence
$\\{\Sigma_{n}^{+}\\}$ converges to $\Sigma^{+}$ as $n\to\infty$. In
particular, setting $\ell_{n}=\operatorname{len}_{V_{n}}(\Sigma_{n})$ and
$m_{n}=\operatorname{len}_{V_{n}}(\gamma_{n})$, we have $\ell_{n}\to\ell$ and
$m_{n}\to m$. Consequently, for all $n\gg 0$, we have $\ell_{n}\leq 0.018375$
and $m_{n}\leq 0.0996-1.408\cdot\ell_{n}$.

For every $n$ where $\Sigma_{n}^{+}$ is defined, we have
$\Sigma_{n}^{+}\subset CC(V_{n})\subset V_{n}^{\circ}$. Thus, for all $n\gg
0$, the double-double $DD(V_{n}^{\circ})$ contains the double-double
$DD(\Sigma_{n})$, a geodesic link of total length $4\ell_{n}\leq 0.0735$.
Furthermore, by construction, we have

$m_{n}\leq 0.0996-1.408\cdot\ell_{n}=0.0996-0.352\cdot 4\ell_{n}.$

Thus $DD(V_{n}^{\circ})$ satisfies the hypotheses of Theorem 8.6. Combining
that result with Theorem 6.2, we may drill the link $DD(\Sigma_{n})$ and
obtain a cusped hyperbolic 3-manifold
$DD(W_{n}^{\circ})=DD(V_{n}^{\circ})-DD(\Sigma_{n})$ containing a closed
geodesic isotopic to $\gamma_{n}$. Furthermore, Theorem 8.6 gives

${d_{\mathrm{hyp}}}(\mathcal{L}_{DD(V_{n}^{\circ})}(\gamma_{n}),\,\mathcal{L}_{DD(V_{n}^{\circ})}(\gamma_{n}))\leq
4\pi^{2}\,F(z_{\min}^{n},4\ell_{n}),$

where

$z_{\min}^{n}={\operatorname{haze}}^{-1}(2\pi(4\ell_{n}+m_{n}+{10^{-5}})).$

Observe that the isotopy class of $\gamma_{n}$ in $V_{n}^{\circ}-\Sigma_{n}$
contains a representative disjoint from the scooped boundary (the red and blue
faces). Thus the closed geodesic $\gamma_{n}$ in the hyperbolic metric on
$DD(W_{n}^{\circ})=DD(V_{n}^{\circ})-DD(\Sigma_{n})$ must be disjoint from the
red and blue totally geodesic surfaces that partition the four copies of the
fundamental domains $W_{n}^{\circ}$. In short, we may take $\gamma_{n}$ to be
a closed geodesic in $W_{n}^{\circ}$. Thus, by the above displayed equation,
we also have

(8.1)
${d_{\mathrm{hyp}}}(\mathcal{L}_{V_{n}}(\gamma_{n}),\,\mathcal{L}_{W_{n}}(\gamma_{n}))\leq
4\pi^{2}\,F(z_{\min}^{n},4\ell_{n}),$

Observe that

$\lim_{n\to\infty}z_{\min}^{n}=\lim_{n\to\infty}{\operatorname{haze}}^{-1}(2\pi(4\ell_{n}+m_{n}+{10^{-5}}))={\operatorname{haze}}^{-1}(2\pi(4\ell+m+{10^{-5}}))={z_{\min}}.$

Since $z_{\min}^{n}\geq 0.6288$ by Theorem 8.6, we can substitute
$F(z_{\min}^{n},4\ell_{n})\leq 0.0174$ in (8.1), hence Lemma 8.3 implies
$\operatorname{len}_{W_{n}}(\gamma_{n})<2m_{n}<0.2$.

Now, recall the strong limit $\xi_{n}\to\xi$. The closed geodesics
$\gamma_{n}\subset W_{n}$ have length universally bounded by a constant less
than $\log 3$, hence Proposition 4.1, says that $\gamma_{n}\subset W_{n}$
converge to a closed geodesic $\gamma\subset Z$ in the geometric limit. In
particular, $\mathcal{L}_{W_{n}}(\gamma_{n})\to\mathcal{L}_{Z}(\gamma)$.
Taking limits of the bound in (8.1) as $n\to\infty$ gives

${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\,\mathcal{L}_{Z}(\gamma))\leq
4\pi^{2}\,F({z_{\min}},4\ell),$

as desired. ∎

We can now derive Theorem 1.5, which was stated in the introduction.

###### Proof of Theorem 1.5.

Let $\ell=\operatorname{len}_{Y}(\Sigma)$ and
$m=\operatorname{len}_{Y}(\gamma)$, and assume $\max\\{4\ell,m\\}<0.0735$.
This hypothesis implies $\ell<0.018375$ and thus $m+1.408\ell<0.0996$, hence
the hypotheses of Theorem 8.5 are satisfied. By Lemma 6.1, $Y-\Sigma$ admits a
hyperbolic metric $Z$, with the same end invariants. In addition, the above
hypothesis on $\ell$ and $m$, combined with the monotonicity of
${\operatorname{haze}}^{-1}$, implies

${z_{\min}}={\operatorname{haze}}^{-1}(2\pi(4\ell+m+{10^{-5}}))\geq 0.6299.$

Now, Theorem 8.5 gives

${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\,\mathcal{L}_{Z}(\gamma))\leq
4\pi^{2}F({z_{\min}},4\ell)\leq 4\pi^{2}F(0.6299,0.0735)\leq 0.6827,$

where the second inequality uses the monotonicity of $F$ and the third
inequality comes from evaluating Definition 8.4. Finally, Lemma 8.3 gives

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

###  8.2. Short geodesics under filling

Next, we turn our attention to bounding the length of a short geodesic under
filling rather than drilling. The following result is the filling analogue of
Theorem 8.5.

######  Theorem 8.7.

Let $V$ be a tame, hyperbolic 3-manifold and $\Sigma$ a geodesic link in $V$.
Suppose that $V-\Sigma$ admits a hyperbolic structure $W$ with the same end
invariants as those of $V$, such that the total normalized length of the
meridians of $\Sigma$ in $W$ satisfies $L^{2}>512$. Let $\gamma\subset W$ be a
closed geodesic of length $m=\operatorname{len}_{W}(\gamma)<0.056$. Define

${z_{\min}}={\operatorname{haze}}^{-1}\left(\frac{(2\pi)^{2}}{(L/2)^{2}-14.7}+2\pi\cdot
1.656\,m\right).$

Then $\gamma$ is isotopic to a closed geodesic in $V$. Furthermore, the
complex lengths of $\gamma$ in $V$ and $W$ are related as follows:

${d_{\mathrm{hyp}}}(\mathcal{L}_{V}(\gamma),\,\mathcal{L}_{W}(\gamma))\>\leq\>4\pi^{2}\,F\big{(}{z_{\min}},\tfrac{2\pi}{(L/2)^{2}-14.7}\big{)}.$

Just as with the drilling argument, the proof of Theorem 8.7 relies on the
following finite-volume analogue [20, Theorem 7.21].

######  Theorem 8.8 (Short geodesics under filling, [20]).

Let $Y$ be a complete, finite-volume hyperbolic 3-manifold and $\Sigma$ a
geodesic link in $Y$. Suppose that $Y-\Sigma$ admits a hyperbolic structure
$Z$, such that the total normalized length of the meridians of $\Sigma$ in $Z$
satisfies $L^{2}\geq 128$. Let $\gamma\subset Z$ be a closed geodesic of
length $m=\operatorname{len}_{Z}(\gamma)\leq 0.056$. Define

$z_{\min}^{\prime}={\operatorname{haze}}^{-1}\left(\frac{(2\pi)^{2}}{L^{2}-14.7}+2\pi\cdot
1.656\,m\right)>0.624.$

Then $\gamma$ is isotopic to a closed geodesic in $Y$. Furthermore, the
complex lengths of $\gamma$ in $Y$ and $Z$ are related as follows:

${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\,\mathcal{L}_{Z}(\gamma))\>\leq\>4\pi^{2}\,F\big{(}z_{\min}^{\prime},\tfrac{2\pi}{L^{2}-14.7}\big{)}.$

###### Proof of Theorem 8.7.

If $\operatorname{vol}(Y)<\infty$, the desired result already follows from
Theorem 8.8. (Although the definition of $z^{\prime}_{\min}$ in Theorem 8.8
differs from the definition of ${z_{\min}}$ in Theorem 8.7, the monotonicity
of ${\operatorname{haze}}^{-1}$ and $F$ ensures that the conclusion of Theorem
8.6 still applies with $(L/2)^{2}$ in place of $L^{2}$.) For the rest of the
proof, we assume that $\operatorname{vol}(Y)=\infty$.

Let $V_{n}$ and $W_{n}$ be the sequences of geometrically finite manifolds
constructed in Theorem 6.2. By that theorem, the conformal boundaries of each
$V_{n}$ and each $W_{n}$ admit the same circle packing $C_{n}$. Furthermore,
there is a strong limit $\rho_{n}\to\rho$ (where $\rho_{n}$ is the
representation corresponding to $W_{n}$ and $\rho$ corresponds to $Y$) and a
strong limit $\xi_{i}\to\xi$ (where $\xi_{n}$ is the representation
corresponding to $W_{n}$ and $\xi$ corresponds to $Z$).

Let $\gamma\subset Z$ be a closed geodesic satisfying the length bound of the
theorem. Then Proposition 4.1 implies that for $n\gg 0$, the approximating
manifold $W_{n}$ contains a closed geodesic $\gamma_{n}$, where the sequence
$\\{\gamma_{n}\\}$ converges to $\gamma$ as $n\to\infty$. Consequently,
$\mathcal{L}_{W_{n}}(\gamma_{n})\to\mathcal{L}_{Z}(\gamma)$. In particular,
for all $n\gg 0$, we have $m_{n}=\operatorname{len}_{W_{n}}(\gamma_{n})\leq
0.056$.

Let ${\mathbf{s}}$ be the tuple of slopes in $Z$ corresponding to the
meridians of $\Sigma$, and let ${\mathbf{s}}^{n}$ be the tuple of slopes in
$W_{n}$ corresponding to the meridians of $\Sigma$. Then, as in the proof of
Theorem 7.2, we have $L({\mathbf{s}}^{n})\to L({\mathbf{s}})>512$ as
$n\to\infty$. In the double-doubled manifold $DD(W_{n}^{\circ})$, we obtain a
tuple of slopes $DD({\mathbf{s}}^{n})$, where each coordinate of
${\mathbf{s}}^{n}$ appears four times, once per copy of $W_{n}^{\circ}$. Thus,
just as in (7.2), we get

$\frac{1}{L(DD({\mathbf{s}}^{n}))^{2}}=\frac{4}{L({\mathbf{s}}^{n})^{2}}\leq\frac{4}{512}=\frac{1}{128},$

where the inequality holds for $n\gg 0$. Thus $DD(W_{n}^{\circ})$ satisfies
the hypotheses of Theorem 8.8. By Theorem 7.2, filling $DD(W_{n}^{\circ})$
along the tuple of slopes $DD({\mathbf{s}}^{n})$ produces the finite-volume
hyperbolic manifold $DD(V_{n}^{\circ})$.

By Theorem 8.8, the closed geodesic $\gamma_{n}\subset DD(W_{n}^{\circ})$ is
isotopic to a geodesic in the filled manifold $DD(V_{n}^{\circ})$.
Furthermore, since $\gamma_{n}$ can be isotoped to be disjoint from the red
and blue totally geodesic surfaces that partition the copies of
$V_{n}^{\circ}$, the geodesic representative of $\gamma_{n}$ must be entirely
contained in one copy of $V_{n}^{\circ}$. Applying Theorem 8.8 to
$DD(W_{n}^{\circ})$ and $DD(V_{n}^{\circ})$, we obtain

${d_{\mathrm{hyp}}}(\mathcal{L}_{V_{n}^{\circ}}(\gamma_{n}),\,\mathcal{L}_{W_{n}^{\circ}}(\gamma_{n}))\leq
4\pi^{2}\,F\big{(}z_{\min}^{n},\tfrac{2\pi}{(L({\mathbf{s}}^{n})/2)^{2}-14.7}\big{)},$

where

$z_{\min}^{n}={\operatorname{haze}}^{-1}\left(\tfrac{(2\pi)^{2}}{L(DD({\mathbf{s}}^{n}))^{2}-14.7}+2\pi\cdot
1.656\,m_{n}\right)={\operatorname{haze}}^{-1}\left(\tfrac{(2\pi)^{2}}{(L({\mathbf{s}}^{n})/2)^{2}-14.7}+2\pi\cdot
1.656\,m_{n}\right).$

Since we are using Theorem 8.8 with $z_{\min}^{n}\geq 0.624$ and
$(L({\mathbf{s}}^{n})/2)^{2}-14.7\geq 113.3$, we can substitute
$F(0.642,\frac{2\pi}{113.3})\leq 0.128$ in the above bound on complex length.
Thus Lemma 8.3 implies $\operatorname{len}_{V_{n}}(\gamma_{n})<1.66m_{n}<0.1$,
enabling us to apply Proposition 4.1.

Now, recall the strong limit $\rho_{n}\to\rho$. By Proposition 4.1, the
geodesics $\gamma_{n}\subset V_{n}$ converge to a geodesic $\gamma\subset Y$.
In particular, $\mathcal{L}_{V_{n}}(\gamma_{n})\to\mathcal{L}_{Y}(\gamma)$.
Taking limits of the above bound on
${d_{\mathrm{hyp}}}(\mathcal{L}_{V_{n}^{\circ}}(\gamma_{n}),\,\mathcal{L}_{W_{n}^{\circ}}(\gamma_{n}))$

${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\,\mathcal{L}_{Z}(\gamma))\leq
4\pi^{2}\,F\big{(}{z_{\min}},\tfrac{2\pi}{(L({\mathbf{s}}^{n})/2)^{2}-14.7}\big{)},$

as desired. ∎

As a corollary of Theorem 8.7, we obtain

######  Corollary 8.9.

Let $Y$ be a tame hyperbolic 3-manifold and $\Sigma$ a geodesic link in $Y$.
Suppose that $Y-\Sigma$ admits a hyperbolic structure $Z$ with the same end
invariants as those of $Y$, and such that the total normalized length of the
meridians of $\Sigma$ satisfies $L^{2}>512$. Let $\gamma\subset Z$ be a closed
geodesic of complex length $\operatorname{len}_{Z}(\gamma)+i\tau_{Z}(\gamma)$,
with $\operatorname{len}_{Z}(\gamma)<0.056$. Then $\gamma$ is isotopic to a
closed geodesic in $Y$, and furthermore

$1.657^{-1}\leq\frac{\operatorname{len}_{Z}(\gamma)}{\operatorname{len}_{Y}(\gamma)}\leq
1.657\qquad\text{and}\qquad|\tau_{Z}(\gamma)-\tau_{Y}(\gamma)|\leq 0.0295.$

###### Proof.

The hypotheses of this corollary match those of Theorem 8.7. The assumption
$L^{2}>512$ is equivalent to $(L/2)^{2}-14.7\geq 113.3$, hence

${z_{\min}}={\operatorname{haze}}^{-1}\left(\frac{(2\pi)^{2}}{(L/2)^{2}-14.7}+2\pi\cdot
1.656\,m\right)\geq{\operatorname{haze}}^{-1}\left(\frac{(2\pi)^{2}}{113.3}+2\pi\cdot
1.656\cdot 0.056\right)\geq 0.624.$

Plugging $z={z_{\min}}\geq 0.624$ and
$\ell=\tfrac{2\pi}{(L/2)^{2}-14.7}\leq\frac{2\pi}{113.3}$ into Theorem 8.7, we
obtain

${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\,\mathcal{L}_{Z}(\gamma))\leq
4\pi^{2}\,F(z,\ell)\leq 4\pi^{2}\,F(0.624,\tfrac{2\pi}{113.3})\leq 0.5045,$

where the second inequality uses the monotonicity of $F$ and the third
inequality comes from evaluating Definition 8.4. Finally, Lemma 8.3 converts
the bound on
${d_{\mathrm{hyp}}}(\mathcal{L}_{Y}(\gamma),\,\mathcal{L}_{Z}(\gamma))$ into
the desired upper bounds on the distance between the real and imaginary parts
of $\mathcal{L}_{Y}(\gamma)$ and $\mathcal{L}_{Z}(\gamma)$. ∎

## References

  * [1] Ian Agol, _Bounds on exceptional Dehn filling_ , Geom. Topol. 4 (2000), 431–449 (electronic).
  * [2] by same author, _Volume change under drilling_ , Geom. Topol. 6 (2002), 905–916.
  * [3] by same author, _Tameness of hyperbolic 3-manifolds_ , arXiv:math/0405568, 2004.
  * [4] James W. Anderson and Richard D. Canary, _Cores of hyperbolic $3$-manifolds and limits of Kleinian groups. II_, J. London Math. Soc. (2) 61 (2000), no. 2, 489–505.
  * [5] Alan F. Beardon and Kenneth Stephenson, _The uniformization theorem for circle packings_ , Indiana Univ. Math. J. 39 (1990), no. 4, 1383–1425.
  * [6] Francis Bonahon, _The geometry of Teichmüller space via geodesic currents_ , Invent. Math. 92 (1988), no. 1, 139–162.
  * [7] Károly Böröczky, _Packing of spheres in spaces of constant curvature_ , Acta Math. Acad. Sci. Hungar. 32 (1978), no. 3-4, 243–261.
  * [8] Brian H. Bowditch, _The ending lamination theorem_ , http://homepages.warwick.ac.uk/~masgak/papers/elt.pdf.
  * [9] Jeffrey F. Brock and Kenneth W. Bromberg, _On the density of geometrically finite Kleinian groups_ , Acta Math. 192 (2004), no. 1, 33–93.
  * [10] Jeffrey F. Brock, Richard D. Canary, and Yair N. Minsky, _The classification of Kleinian surface groups, II: The ending lamination conjecture_ , Ann. of Math. (2) 176 (2012), no. 1, 1–149.
  * [11] Kenneth W. Bromberg, _Hyperbolic cone-manifolds, short geodesics, and Schwarzian derivatives_ , J. Amer. Math. Soc. 17 (2004), no. 4, 783–826.
  * [12] Robert Brooks, _Circle packings and co-compact extensions of Kleinian groups_ , Invent. Math. 86 (1986), no. 3, 461–469.
  * [13] Danny Calegari and David Gabai, _Shrinkwrapping and the taming of hyperbolic 3-manifolds_ , J. Amer. Math. Soc. 19 (2006), no. 2, 385–446.
  * [14] Richard D. Canary, _Ends of hyperbolic $3$-manifolds_, J. Amer. Math. Soc. 6 (1993), no. 1, 1–35.
  * [15] by same author, _Marden’s tameness conjecture: history and applications_ , Geometry, analysis and topology of discrete groups, Adv. Lect. Math. (ALM), vol. 6, Int. Press, Somerville, MA, 2008, pp. 137–162.
  * [16] Richard D. Canary, David B. A. Epstein, and Paul Green, _Notes on notes of Thurston_ , Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 3–92.
  * [17] Richard D. Canary and Darryl McCullough, _Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups_ , Mem. Amer. Math. Soc. 172 (2004), no. 812, xii+218.
  * [18] Marc Culler and Peter B. Shalen, _Paradoxical decompositions, $2$-generator Kleinian groups, and volumes of hyperbolic $3$-manifolds_, J. Amer. Math. Soc. 5 (1992), no. 2, 231–288.
  * [19] David Futer, Jessica S. Purcell, and Saul Schleimer, _Effective distance between nested Margulis tubes_ , Trans. Amer. Math. Soc. 372 (2019), no. 6, 4211–4237.
  * [20] by same author, _Effective bilipschitz bounds on drilling and filling_ , Geom. Topol. 26 (2022), no. 3, 1077–1188.
  * [21] David Futer and Saul Schleimer, _Cusp geometry of fibered 3-manifolds_ , Amer. J. Math. 136 (2014), no. 2, 309–356.
  * [22] Craig D. Hodgson and Steven P. Kerckhoff, _Rigidity of hyperbolic cone-manifolds and hyperbolic Dehn surgery_ , J. Differential Geom. 48 (1998), no. 1, 1–59.
  * [23] by same author, _Universal bounds for hyperbolic Dehn surgery_ , Ann. of Math. (2) 162 (2005), no. 1, 367–421.
  * [24] by same author, _The shape of hyperbolic Dehn surgery space_ , Geom. Topol. 12 (2008), no. 2, 1033–1090.
  * [25] Neil R. Hoffman and Jessica S. Purcell, _Geometry of planar surfaces and exceptional fillings_ , Bull. Lond. Math. Soc. 49 (2017), no. 2, 185–201.
  * [26] William Jaco, _Lectures on three-manifold topology_ , CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980.
  * [27] Troels Jørgensen and Albert Marden, _Algebraic and geometric convergence of Kleinian groups_ , Math. Scand. 66 (1990), no. 1, 47–72.
  * [28] Marc Lackenby, _Word hyperbolic Dehn surgery_ , Invent. Math. 140 (2000), no. 2, 243–282.
  * [29] Albert Marden, _Hyperbolic manifolds_ , Cambridge University Press, Cambridge, 2016, An introduction in 2 and 3 dimensions.
  * [30] Robert Meyerhoff, _A lower bound for the volume of hyperbolic $3$-manifolds_, Canad. J. Math. 39 (1987), no. 5, 1038–1056.
  * [31] Yair Minsky, _The classification of Kleinian surface groups. I. Models and bounds_ , Ann. of Math. (2) 171 (2010), no. 1, 1–107.
  * [32] Yair N. Minsky, _The classification of punctured-torus groups_ , Ann. of Math. (2) 149 (1999), no. 2, 559–626.
  * [33] Hossein Namazi and Juan Souto, _Non-realizability and ending laminations: proof of the density conjecture_ , Acta Math. 209 (2012), no. 2, 323–395.
  * [34] Ken’ichi Ohshika, _Realising end invariants by limits of minimally parabolic, geometrically finite groups_ , Geom. Topol. 15 (2011), no. 2, 827–890.
  * [35] Peter Scott, _A new proof of the annulus and torus theorems_ , Amer. J. Math. 102 (1980), no. 2, 241–277.
  * [36] Peter B. Shalen, _Small optimal Margulis numbers force upper volume bounds_ , Trans. Amer. Math. Soc. 365 (2013), no. 2, 973–999.
  * [37] William P. Thurston, _Three-dimensional manifolds, Kleinian groups and hyperbolic geometry_ , Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381.
  * [38] by same author, _On the geometry and dynamics of diffeomorphisms of surfaces_ , Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431.
  * [39] Friedhelm Waldhausen, _On irreducible $3$-manifolds which are sufficiently large_, Ann. of Math. (2) 87 (1968), 56–88.