2.1 Hyperbolic 3-manifolds

Let M be a connected oriented 3-manifold (without boundary). A hyperbolic metric on M is a complete Riemannian metric of constant sectional curvature $-1$ . A hyperbolic structure on M is a pair $(M,\rho)$ , where $\rho$ is a complete hyperbolic metric. Every hyperbolic structure is isometric to a quotient $\mathbb{H}^3/\Gamma$ of the hyperbolic 3-space $\mathbb{H}^3$ by a discrete torsion free subgroup $\Gamma<{\rm Isom}^+(\mathbb{H}^3)$ . We refer to a connected oriented 3-manifold that admits a hyperbolic structure as a hyperbolic manifold. Hyperbolic structures of finite volume are unique, up to homotopy, by the Mostow–Prasad rigidity theorem. Hyperbolic manifolds with totally geodesic boundary can also be defined and will play a role in Part 2.

The group $\Gamma$ acts on the boundary $\mathbb{CP}^1=\partial\mathbb{H}^3$ by Möbius transformations. The action $\Gamma\curvearrowright\mathbb{CP}^1$ preserves the limit set $\Lambda\subset\mathbb{CP}^1$ , which is the set of accumulation points of an orbit $\Gamma o\subset\mathbb{H}^3\cup\partial\mathbb{H}^3$ , and its complement $\Omega:=\mathbb{CP}^1-\Lambda$ , the domain of discontinuity. The action $\Gamma\curvearrowright\Omega$ is free and properly discontinuous. The quotient $(\mathbb{H}^3\cup\Omega)/\Gamma$ is a 3-manifold with boundary $\Omega/\Gamma$ and the boundary has a natural conformal structure. If $\Gamma$ is finitely generated and non-abelian, then, by Ahlfors finiteness theorem [Reference AhlforsAhl64], the quotient $\Omega/\Gamma$ is a surface of finite type. In particular, $\Omega/\Gamma$ admits a Poincaré metric which is the unique complete finite area hyperbolic metric in the same conformal class.

2.2 Geometrically finite structures

We describe some flexible classes of geometrically finite hyperbolic structures on a pair of fixed topological spaces, namely handlebodies $H_g$ and I-bundles $\Sigma\times[a,b]$ . When dealing with such hyperbolic structures with cusps, the notion of pared (acylindrical) manifold naturally occurs (see Thurston [Reference ThurstonThu86a, Reference ThurstonThu86b]).

1. (1) The fundamental group of each component of P injects in $\pi_1(N)$ .

2. (2) Every $\pi_1$ -injective homotopy $(A,\partial A)\to (N,U)$ is homotopic as a map of pairs to a map that sends A to U.

We say that (N,P) is also acylindrical if furthermore:

1. (3) every $\pi_1$ -injective map $(A,\partial A)\to (N,\partial N-U)$ is homotopic as a map of pairs into $\partial N-U$ .

We refer to Chapter 5 of [Reference Canary and McCulloughCM04] for the topological features of pared (acylindrical) manifolds. We can now give the following definition.

If we have different identifications $f:N-P\to\mathbb{H}^3\cup\Omega/\Gamma$ and $f':N-P\to\mathbb{H}^3\cup\Omega'/\Gamma'$ , the composition $g=f'f^{-1}:\mathbb{H}^3\cup\Omega/\Gamma\to\mathbb{H}^3\cup\Omega'/\Gamma'$ restricts to an isometry $g:\mathbb{H}^3\cup\Omega/\Gamma\to\mathbb{H}^3\cup\Omega'/\Gamma'$ and, hence, the restriction to the boundary $g:\Omega/\Gamma\to\Omega'/\Gamma'$ is conformal. Thus, every geometrically finite hyperbolic metric induces a well-defined Riemann surface structure on $\partial N-P$ , called the conformal boundary, which we can think as a point in the Teichmüller space of $\partial N-P$ .

Two geometrically finite hyperbolic metrics $\rho,\rho'$ on $M={\rm int}(N)$ are said to be equivalent if there is a diffeomorphism isotopic to the identity $\phi:N-P\to N-P$ such that $\rho=\phi^*\rho'$ . The restriction of the diffeomorphism to the boundary $\phi:\partial N-P\to\partial N-P$ is isotopic to the identity and conformal with respect to the Riemann surface structures induced by $\rho$ and $\rho'$ . Therefore, equivalent geometrically finite hyperbolic metrics have equivalent conformal boundaries.

2.3 Maximally cusped structures

Maximally cusped structures play a crucial role in Part 2. Let N be either $H_g$ or $\Sigma\times[a,b]$ .

Definition (Maximally cusped). A hyperbolic structure on $M={\rm int}(N)$ is maximally cusped or maximally cusped at P if it is geometrically finite and has parabolic locus P which is a pants decomposition of $\partial N$ .

Work of Maskit [Reference MaskitMas83], Keen, Maskit and Series [Reference Keen, Maskit and SeriesKMS93], and Ohshika [Reference OhshikaOhs98] gives the following.

Regarding convex cores, work of Keen, Maskit and Series [Reference Keen, Maskit and SeriesKMS93] provides the following description.

2.4 Curve graph and disk set

We now introduce the curve graph and the disk sets and relate them to the topology of Heegaard splittings and handlebodies. We also describe some of their geometric properties.

If $\Sigma=\partial H_g$ is the boundary of a handlebody $H_g$ , we have an associated disk set $\mathcal{D}\subset\mathcal{C}$ consisting of those essential simple closed curves of $\Sigma$ that bound a properly embedded disk in $H_g$ .

If $f:\Sigma\to\Sigma$ is a gluing map defining the Heegaard splitting $M_f=H_g\cup_fH_g$ , then the Hempel distance of f is the quantity

\[d_{\mathcal{C}}(\mathcal{D},f\mathcal{D}).\]

Splittings with large Hempel distance have strong topological properties:

Combined with the solution of the geometrization conjecture by Perelman, Theorem 2.4 implies that if $d_{\mathcal{C}}(\mathcal{D},f\mathcal{D})\ge 3$ , then the splitting $M_f$ is hyperbolic. In [Reference MaherMah10b], Maher proved that the Hempel distance of a random mapping class grows coarsely linearly and, hence, that random Heegaard splittings are hyperbolic.

In our random 3-manifolds setup, we also exploit the curve graph to control the topology of our building blocks. For example, we use it to check that the blocks satisfy the assumptions of Theorems 2.2 and 4.1: from a curve graph point of view, we have the following useful criterion.

The proof can be found in Appendix A.

From a geometric point of view, Masur and Minsky proved that the curve graph $\mathcal{C}$ is a Gromov hyperbolic space (see [Reference Masur and MinskyMM99]) and the disk set $\mathcal{D}\subset\mathcal{C}$ is a uniformly quasi-convex subspace (see [Reference Masur and MinskyMM04]).

As discovered by Minsky [Reference MinskyMin00, Reference MinskyMin01, Reference MinskyMin10], and Brock, Canary and Minsky [Reference Brock, Canary and MinskyBCM12] in groundbreaking work that led to the solution of the ending lamination conjecture, the geometry of the curve graph is related to the internal geometry of hyperbolic metrics on $\Sigma\times[0,1]$ : the relation is established via end invariants and the devices of subsurface projections and hierarchies of tight geodesics introduced in [Reference Masur and MinskyMM00]. We recall the definition of subsurface projection in the case that is most relevant for us.

If both $\alpha$ and $\beta$ intersect W essentially, we define

\[d_W(\alpha,\beta)={\rm diam}_{\mathcal{C}(W)}(\pi_W(\alpha)\cup\pi_W(\beta)).\]

2.5 Filling and drilling

One of the most important tools needed in our constructions is the universal Dehn surgery theorem by Hodgson and Kerckhoff [Reference Hodgson and KerckhoffHK05]. We briefly recall the basic setup and statements.

Let M be a compact oriented 3-manifold with boundary. For a component of the boundary T that is a torus, a choice $\mu$ of a homotopy class of a non-null homotopic simple closed curve in T is called a slope. Gluing a solid torus $S^1\times D^2$ to M via a gluing homeomorphism $\partial (S^1\times D^2)=S^1\times S^1\to T$ that sends the homotopy class of $\{1\}\times S^1$ to the slope $\mu$ is called Dehn filling of M along the slope $\mu$ and yields an compact oriented 3-manifold that contains M as a submanifold. We call the slope along which we perform a Dehn filling the filling slope.

Let $\eta_3>0$ be a Margulis constant which we fix once and for all.

Let $\mathbb{M}$ be a complete finite-volume hyperbolic 3-manifold with cusps. Each cusp $\gamma$ of $\mathbb{M}$ has a standard $\eta_3$ -Margulis neighborhood $\mathbb{T}_{\eta_3}(\gamma)\subset\mathbb{M}$ isometric to a quotient $\mathcal{O}/\mathbb{Z}^2$ where $\mathcal{O}\subset\mathbb{H}^3$ is a horoball and $\mathbb{Z}^2<{\rm Isom}^+(\mathbb{H}^3)$ consists of parabolic isometries stabilizing $\mathcal{O}$ . The boundary $T_\gamma:=\partial\mathbb{T}_{\eta_3}(\gamma)\subset\mathbb{M}$ is a standard torus horosection of $\gamma$ . The intrinsic metric is of $T_\gamma$ is flat.

Definition (Normalized length). Let $\mu\subset T_\gamma$ be a simple closed geodesic. The normalized length of $\mu$ , is the quantity defined by

\[{\rm nl}(\mu):=\ell_{T_\gamma}(\mu)/\sqrt{{\rm Area}(T_\gamma)}.\]

By Dehn filling $\mathbb{M}$ along a slope $\mu$ of $T_\gamma$ , we mean Dehn filling the underlying manifold of $\mathbb{M}\setminus{\rm int}(\mathbb{T}_{\eta_3}(\gamma))$ along a slope $\mu$ of the boundary component $T_\gamma$ of M. Hodgson and Kerckhoff proved the following.

If we have a geodesic link $\Gamma\subset M$ in a complete finite-volume hyperbolic 3-manifold M and the complement $M-\Gamma$ admits a complete finite-volume hyperbolic metric as well, then, by the techniques from [Reference Hodgson and KerckhoffHK05, Reference Hodgson and KerckhoffHK08], there is a relation between the length of the link $\Gamma\subset M$ and the normalized length of the standard meridians of $\Gamma$ in $M-\Gamma$ made fully explicit in [Reference Futer, Purcell and SchleimerFPS19].

Theorem 2.7 [Reference Futer, Purcell and SchleimerFPS19, Corollary 6.13]. Let $\Gamma\subset M$ be a geodesic link in a complete finite-volume hyperbolic 3-manifold, and denote by $\ell$ the total length of $\Gamma$ . Then $M-\Gamma$ has a complete finite-volume hyperbolic metric, and we denote by L the total normalized length L of the meridians of $\Gamma$ . Moreover, suppose that either $L\ge 7.823$ , or that each component of $\Gamma$ has length at most 0.0996 and $\ell\le 0.1396$ . Then

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

Brock and Bromberg studied the change in the geometry of a family $M_t$ of hyperbolic cone manifold structures on a topological model M in a very general setup and proved that away from the singular locus $\Gamma$ one can get uniform bilipschitz control only depending on the length of $\Gamma$ . This is the content of the following drilling theorem.

Theorem 2.8 [Reference Brock and BrombergBB04, Theorem 6.2]. For every $K\in(1,2)$ there exists $\eta_{{\rm drill}}\in(0,\eta_3)$ such that the following holds. Let M be a complete finite-volume hyperbolic 3-manifold. Let $\Gamma\subset M$ be a geodesic link and let M’ be a complete hyperbolic structure on $M-\Gamma$ . Suppose that $\ell_M(\Gamma)<\eta_{{\rm drill}}$ . Then, there exists a K-bilipschitz diffeomorphism of pairs

\[\bigg(M'-\bigsqcup_{\alpha\subset\Gamma}{\mathbb{T}_{\eta_3}(\alpha)},\bigsqcup_{\alpha\subset\Gamma}{T_\alpha}\bigg)\to\bigg(M-\bigsqcup_{\alpha\subset\Gamma}{\mathbb{T}_{\eta_3}(\alpha)},\bigsqcup_{\alpha\subset\Gamma}{T_\alpha}\bigg),\]

where $\mathbb{T}_{\eta_3}(\alpha)$ denotes a standard $\eta_3$ -Margulis neighborhood for $\alpha$ and $T_\alpha=\partial\mathbb{T}_{\eta_3}(\alpha)$ its boundary.

3.1 Outline

We now outline the strategy to produce a hyperbolic metric on $M_f$ for which $\gamma\subset\Sigma$ is a short geodesic.

We start by associating to $\gamma$ the 3-manifold

\[M_f-\gamma=(H_g-\gamma)\cup_f(H_g-f(\gamma)).\]

The Heegaard splitting $M_f$ is obtained from $M_f-\gamma$ by Dehn filling along a filling slope which is completely determined by the topology.

According to Thurston’s hyperbolization theorem, the manifold $M_f-\gamma$ admits a complete hyperbolic metric provided that it is irreducible, atoroidal and Haken; see, for example, Theorem 1.42 in [Reference KapovichKap09]. We take [Reference KapovichKap09] as our general reference for basic 3-manifold topology. Note, however, that, since it is the interior of a compact manifold with non-empty boundary, by basic 3-manifold topology, if $M_f-\gamma$ is irreducible, then it will also be automatically Haken (see Corollary 1.24 in [Reference KapovichKap09]).

Both irreducibility and atoroidality will follow from our assumption that both $(H_g,\gamma)$ and $(H_g,f(\gamma))$ are so-called pared acylindrical handlebodies. This is condition (a) of Theorem 6 and is borrowed from the theory of Jaco, Shalen and Johannson (see Chapter 5 of [Reference Canary and McCulloughCM04]).

Once we have a hyperbolic metric on $M_f-\gamma$ we use it to find a hyperbolic metric on $M_f$ . The tool for such an operation is Hodgson and Kerckhoff effective version of the hyperbolic Dehn surgery theorem (see Theorem 2.6). We need to certify that the canonical filling slope has large normalized length on a standard torus horosection $T:=\partial\mathbb{T}_{\eta_3}(\gamma)$ of the cusp of $M_f-\gamma$ .

Actually, we check something stronger, namely that, if condition (b) of Theorem 6 holds, then the flat torus T itself will be long and skinny so that every filling slope different from the one corresponding to $(\Sigma-\gamma)\cap T$ will have large normalized length.

The central point of the proof is to show that T is long and skinny provided that the subsurface projection $d_{\Sigma-\gamma}(\mathcal{D},f\mathcal{D})$ is large. The main ingredient for the argument is the model manifold technology by Minsky [Reference MinskyMin10] and Brock, Canary and Minsky [Reference Brock, Canary and MinskyBCM12].

At this point, before going on, we need a further consequence of the condition (a): if $(H_g,\gamma)$ is pared acylindrical, then the inclusion $\Sigma-\gamma\subset H_g$ is doubly incompressible. Double incompressibility allows us to use Thurston’s uniform injectivity for pleated surfaces [Reference ThurstonThu86a] and its consequences. In particular, we show that there are two simple closed curves $\alpha\subset\Sigma-\gamma$ and $\beta\subset\Sigma-f(\gamma)$ which are represented by geodesics in $M_f-\gamma$ with moderate length and which are combinatorially close to the disk set projections as follows: $d_{\Sigma-\gamma}(\mathcal{D},\alpha)\le 2$ and $d_{\Sigma-\gamma}(f\mathcal{D},\beta)\le2$ . Note that, since $d_{\Sigma-\gamma}(\mathcal{D},f\mathcal{D})$ is large, $d_{\Sigma-\gamma}(\alpha,\beta)$ will also be large. In order to produce the moderate-length curves $\alpha$ and $\beta$ it turns out to be convenient to work with the coverings of $M_f-\gamma$ corresponding to $\pi_1(H_g-\gamma)$ and $\pi_1(H_g-f(\gamma))$ instead of $M_f-\gamma$ itself.

As a last step, incompressibility of $\Sigma-\gamma\subset M_f-\gamma$ and the fact that the curves $\alpha,\beta\subset\Sigma-\gamma$ are represented by moderate-length curves in $M_f-\gamma$ and are combinatorially far apart in the curve graph $\mathcal{C}(\Sigma-\gamma)$ implies, via the model manifold technology, that $T=\partial\mathbb{T}_{\eta_3}(\gamma)$ is long and skinny.

This concludes the outline. The remainder of this section is dedicated to the topological part of the proof. We establish the topological properties that imply that $M_f-\gamma$ is hyperbolic and check that $\Sigma-\gamma$ is doubly incompressible. The geometric part of the argument is discussed in the next section.

3.2 Hyperbolizable drilled Heegaard splittings

Gluing together two pared acylindrical handlebodies produces an irreducible and atoroidal 3-manifold.

In particular, by Thurston’s hyperbolization theorem for Haken manifolds, the drilled splitting $M_f-\gamma$ admits a hyperbolic structure, which is also unique by Mostow–Prasad rigidity.

We split the proof of Proposition 3.1 into small steps, Lemmas 3.2, 3.3 and 3.4. First we observe that $\Sigma-\gamma\subset M_f-\gamma$ is incompressible.

We now use transversality arguments together with acylindricity of the handlebodies to establish irreducibility and atoroidality of $M_f-\gamma$ .

Consider now some $\alpha_i$ , and let H be the handlebody containing the annulus $A\subseteq T$ bounded by $\alpha_i$ and $\alpha_{i+1}$ (note that if there is one $\alpha_i$ , then in fact there are at least two because $\Sigma-\gamma$ separates $M_f-\gamma$ , so that $(\Sigma-\gamma)\cap T$ needs to have at least two connected components).

Let $I\subset A$ be a properly embedded interval connecting the two boundary components of A. In addition, let $I'\subset A'$ be an arc that is homotopic rel boundary to I in H. An innermost circle argument gives that $I'\cup I$ is also null-homotopic in N, hence by Dehn’s lemma there exists a disk $D\subset N$ with boundary $I'\cup I$ . Let S be the sphere obtained as the union of the disk given by boundary compression of A along D and the disk in $\Sigma$ with the same boundary. This sphere S bounds a ball in H, and thus N, by irreducibility of handlebodies. Thus N compresses to a ball; hence, N is a solid torus as desired.

Let A’ be the annulus in $\Sigma$ that union A forms a torus that bounds a solid torus in H. If A’ does not contain $\gamma$ , then we can reduce the number of components of $(\Sigma-\gamma)\cap T$ by isotoping A in $M_f-\gamma$ to the other handlebody. Otherwise, A’ is (up to isotopy in $\Sigma$ ) $N(\gamma)$ , and we can apply an isotopy in $M_f$ to move A inside a regular neighborhood N of $\gamma$ in $M_f$ . Note that there is at least one $\alpha_j$ as otherwise T would be contained in one of the handlebodies; however, handlebodies do not contain incompressible tori since incompressible tori are $\pi_1$ -injective and the fundamental groups of handlebodies do not contain $\mathbb Z^2$ subgroups. In particular, T is a union of annuli as above and, hence, after applying finitely many isotopies, we reduce to the case that every annulus of T is entirely contained in N (we can assume that the isotopies we found above move N inside itself). Hence, T can be thought of as an incompressible surface in $N-\gamma$ . There is a classification of incompressible surfaces in $S^1$ -bundles; see [Reference WaldhausenWal67, Satz 2.8], which in our case implies that T is boundary parallel, as required.

Hence, $M_f-\gamma$ is irreducible and atoroidal. As mentioned before, irreducibility is already enough to ensure that it is also Haken. Incidentally, we observe that $\Sigma-\gamma\subset M_f-\gamma$ is a Haken surface.

Combining with Lemma 2.5, we get the following.

3.3 Double incompressibility

The second crucial topological property of a pared acylindrical handlebody $(H_g,\gamma)$ that we need is the fact that the inclusion of the boundary $\Sigma-\gamma\subset H_g$ is doubly incompressible.

Double incompressibility was defined by Thurston in [Reference ThurstonThu86a] for maps from hyperbolic surfaces to hyperbolic 3-manifolds. In the setting we are interested in, we can rephrase Thurston’s definition purely topologically as in the following definition.

1. (a) The inclusion $\Sigma-\gamma\subset H_g$ is $\pi_1$ -injective.

2. (b) Essential relative homotopy classes of maps $(I,\partial I)\to(\Sigma-N(\gamma),\partial N(\gamma))$ are mapped injectively to relative homotopy classes of maps $(I,\partial I)\to (H_g-U(\gamma),\partial U(\gamma))$ .

3. (c) There are no essential cylinders in $\Sigma-N(\gamma)$ : this means that every essential map $f:(A,\partial A)\to (H_g,\Sigma-N(\gamma))$ is either homotopic into $U(\gamma)$ or the restriction of f to $\partial A$ extends to a map into $\Sigma-N(\gamma)$ .

4. (d) Each maximal abelian subgroup of $\pi_1(\Sigma-\gamma)$ is mapped to a maximal abelian subgroup of $\pi_1(H_g)$ .

Since maximal abelian subgroups of a free group are infinite cyclic groups, the last condition is equivalent to the following.

1. (e) Each maximal cyclic subgroup of $\pi_1(\Sigma-\gamma)$ is mapped to a maximal cyclic subgroup of $\pi_1(H_g)$ .

As Thurston essentially observes in § 7 of [Reference ThurstonThu86a], we have the following.

Proposition 3.7 is well-known to experts and follows from JSJ decomposition theory. However, it might not be easy to extract from the literature. For this reason, and for the sake of being more self-contained, we include a proof in Appendix B.

4.1 Handlebody covering

In order to find the curves $\alpha$ and $\beta$ , we work with handlebody coverings, which we now describe.

Consider first the pared handlebody $(H_g,\gamma)$ . The fundamental group of $H_g-\gamma\subset M_f-\gamma$ injects into $\pi_1(M_f-\gamma)$ (see Lemma 3.2) and determines a covering N of $M_f-\gamma$ to which $H_g-\gamma$ lifts homeomorphically. By slight abuse of notation we do not distinguish between $H_g-\gamma$ and its homeomorphic lift to N. Recall, from the outline, that we want to find a simple closed curve $\alpha\subset\Sigma-\gamma$ that has moderate-length representative in N and satisfies $d_W(\alpha,\mathcal{D})\le 2$ .

As both $H_g-\gamma$ and N are aspherical, the inclusion $H_g-\gamma\subset N$ is a homotopy equivalence. Since the pair $(N,H_g-\gamma)$ has also the homotopy extension property (e.g. since it can be give the structure of a CW-pair), it follows that the manifold N deformation retracts to $H_g-\gamma$ (this is a general fact; see, for example, [Reference HatcherHat02, Corollary 0.20]). Therefore, according to Proposition 3.7 together with Remark 3.6 to relate the topological and hyperbolic versions of double incompressibility, the inclusion $W=\Sigma-\gamma\subset N$ is doubly incompressible in the sense of [Reference ThurstonThu86a].

Even though we will not need it, for the sake of clarity, we provide a more complete description of the covering. The hyperbolic structure on N is isometric to a geometrically finite structure on $\textrm{int}(H_g)$ with a rank-one cusp at $\gamma$ and the submanifold $H_g-\gamma\subset N$ is a so-called relative Scott core for N (see [Reference ScottSco73, Reference McCulloughMcC86, Reference Kulkarni and ShalenKS89]). The fact that N is homeomorphic to the interior of $H_g$ follows from Bonahon’s tameness theorem [Reference BonahonBon86]. Geometric finiteness is, instead, a consequence of Canary and Thurston’s covering theorem [Reference CanaryCan96].

We now come back to double incompressibility. Crucially, it allows us to use Thurston’s uniform injectivity for pleated surfaces [Reference ThurstonThu86a].

Theorem 4.1 (Uniform injectivity [Reference ThurstonThu86a, Theorem 5.7]). Fix $\eta>0$ , a Margulis constant. For every ${\epsilon}>0$ there exists $\delta>0$ such that for any type-preserving doubly incompressible pleated surface $g:W\to N$ with pleating locus $\lambda$ and induced metric $\sigma$ , if $x,y\in\lambda$ lie in the $\eta$ -thick part of $(W,\sigma)$ , then

\[d_{{\mathbf P}(N)}({\mathbf p}_g(x),{\mathbf p}_g(y))\le\delta\Longrightarrow d_\sigma(x,y)\le{\epsilon}.\]

Here ${\mathbf p}_g:\lambda\to {\mathbf P}(N)$ denotes the map induced by g from the lamination $\lambda$ to the projective unit tangent bundle of N.

We use Theorem 4.1 to prove the following.

Finally, the moderate-length segments provided by Proposition 4.2 can be promoted to moderate-length curves $\alpha$ and $\beta$ in a simple way.

The proof of Proposition 4.2 is where we fully exploit the assumption that W is the complement of a non-separating simple closed curve. The main ingredients of the proof are Thurston’s uniform injectivity and a technical lemma about quasi-geodesic concatenations in $\mathbb{H}^3$ . We also use the following general property of pleated surfaces observed by Thurston in [Reference ThurstonThu86a].

4.2 Quasi-geodesic concatenations

In addition to the use of uniform injectivity, our proof of Proposition 4.2 is elementary and rests on the following fact about piecewise broken geodesics in $\mathbb{H}^3$ , which we state without proof. In the statement, the breaking angle of a break point I(t) of a piecewise geodesic $\gamma\colon I\to \mathbb{H}^3$ is the angle formed by the positively oriented left and right tangent at I(t).

In particular, if the length of the geodesic segments is large enough, then $\gamma$ cannot be a loop. Lemma 4.5 follows from the fact that, in a hyperbolic space, local quasi-geodesics, such as $\gamma$ (as is not difficult to check), are global quasi-geodesics.

Using Lemma 4.5, we find the following.

• • $\gamma_{2j+1}$ is a geodesic of length at least ${\epsilon}$ on the horosphere $\mathcal{H}_j=\partial\mathcal{O}_j$ ;

• • $\gamma_{2j}$ is the orthogonal segment connecting $\mathcal{H}_{j-1}$ and $\mathcal{H}_j$ ; we require that $\gamma_{2j}$ has length at least $L_0$ .

Then $\gamma$ is not a loop.

Proof. We use the following fact that can be easily checked in the upper half-space model of $\mathbb{H}^3$ . If x,y lie on the same horosphere $\mathcal{H}$ , then we have

\[\sinh(d_{\mathbb{H}^3}(x,y))=d_{\mathcal{H}}(x,y).\]

Denote by $\mathcal{H}_j=\partial\mathcal{O}_j$ the boundary horosphere of the horoball $\mathcal{O}_j$ . Observe that, by assumption, the flat geodesic $\gamma_{2j}\subset\mathcal{H}_j$ has length $\ell(\gamma_{2j})\ge{\epsilon}$ .

If we expand $\mathcal{H}_j$ radially from its center at infinity, the intrinsic geometry of the horosphere expands exponentially with exponent equal to the increase in the radius. Hence, if we inflate all the horoballs $\mathcal{O}_j$ by r, then each $\gamma_{2j+1}$ is shortened to an arc $\gamma_{2j+1}^r$ of length

\[\ell(\gamma_{2j+1}^r)=\ell(\gamma_{2j+1})-2r\ge L-2r,\]

while the length of the inflated $\gamma_{2j}$ , denoted by $\gamma_{2j}^r$ , becomes

\[\ell(\gamma_{2j}^r)=e^r\ell(\gamma_{2j}).\]

We now straighten all the $\gamma_{2j}^r$ relative to the endpoints and obtain geodesic arcs $\alpha_{2j}^r$ of length $\ell(\alpha_{2j}^r)=\sinh^{-1}(e^r\ell(\gamma_{2j}))\ge\sinh^{-1}(e^r{\epsilon})$ .

Let us now consider the angles between the segments $\gamma_{2j-1}$ and $\alpha_{2j}$ . Observe that, by assumption, $\gamma_{2j-1}^r$ is orthogonal to $\mathcal{H}_j^r$ , the inflated horosphere. Therefore, as, by convexity of horoballs, $\alpha_{2j}$ is contained in $\mathcal{O}_j^r$ , the angle $\angle\gamma_{2j-1}^r\alpha_{2j}^r$ between the two geodesics $\gamma_{2j-1}$ and $\alpha_{2j}$ is in $(0,\pi/2)$ .

In conclusion, if the length of each $\gamma_{2j+1}$ is much larger than $L+2r$ and $\sinh^{-1}(e^r{\epsilon})\ge L$ , where L is the constant of Lemma 4.5, then the broken piecewise geodesic $\gamma=\gamma_1^r*\alpha_2^r*\cdots*\gamma_{2n-1}^r*\alpha_{2n}^r$ is a A-quasi-geodesic. If it is also sufficiently long compared with A, which can again be achieved by assuming that each $\gamma_{2j+1}$ is long enough, it cannot be a loop.

4.3 Moderate-length surgeries

We now prove Proposition 4.2.

Now, after collapsing parallel components to a single one, $\lambda$ can be realized as a sublamination of the pleating locus of some pleated surface $g:(\Sigma-\gamma,\sigma)\to N$ in the proper (relative to cusps) homotopy class of the inclusion $\Sigma-\gamma\subset N$ (see, for example, Theorems I.5.3.6 and I.5.3.9 in [Reference Canary, Epstein and GreenCEG06]).

When regarding $\delta$ as a simple closed curve on $\Sigma$ , we can think of it as a concatenation $\delta=\beta_1*\cdots*\beta_{2n}$ of arcs $\beta_{2j}$ in W and arcs $\beta_{2j+1}$ crossing a regular annular neighborhood of $\gamma$ . Here we are using the fact that W is the complement of a non-separating simple closed curve.

Using the proper homotopy between the inclusion $W=\Sigma-\gamma\subset N$ and g, we simultaneously straighten all the $\beta_{2j}$ to subarcs $\alpha_{2j}$ of the geodesic leaves of $g(\lambda)$ that start and end in the standard horosection $T=\partial\mathbb{T}_{\eta_3}(\gamma)$ . Then, again, using the proper homotopy between $\Sigma-\gamma\subset N$ and the deformation retraction of N to $N-\mathbb{T}_{\eta_3}(\gamma)$ , we replace the arcs $\beta_{2j+1}$ with arcs $\alpha_{2j+1}$ on the horosection T that are geodesic with respect to the intrinsic flat metric and join the endpoints of the previously obtained $\alpha_{2j}$ and $\alpha_{2(j+1)}$ .

We have that $\delta\subset\Sigma$ is homotopic in N to a concatenation $\alpha_1*\cdots*\alpha_{2n}$ of closed arcs $\alpha_j$ , where each $\alpha_{2j+1}$ is a geodesic in the boundary of the standard horosection T, while each $\alpha_{2j}$ is a proper geodesic arc in $N-\mathbb{T}_{\eta_3}(\gamma)$ contained in the pleating locus $g(\lambda)$ .

Note that, by Lemma 4.4, if $\eta_3$ is sufficiently small, then, when we look at $\alpha_{2j}$ in the intrinsic metric of the pleated surface, it will join two points of the $\eta_0$ -cuspidal part for some universal $\eta_0>0$ .

Now, by Theorem 4.1, which applies to g because it is doubly incompressible, there is a uniform ${\epsilon}>0$ such that the length of each $\alpha_{2j+1}$ , is at least ${\epsilon}$ . Let $L_0$ be as in Lemma 4.6, for the given ${\epsilon}$ . Note that this constant only depends on $\Sigma$ . Being homotopically trivial in N, the loop $\alpha_1*\cdots*\alpha_{2n}$ lifts to a closed loop in $\mathbb{H}^3$ . Hence, by Lemma 4.6, some $\alpha_{2i}$ must have length less than $L_0$ .

4.4 Size of the standard horosection

Consider $T=\partial\mathbb{T}_{\eta_3}$ , the boundary of the standard horosection. We show that T is long and skinny, meaning that the length of any slope $\mu\subset T$ different from that coming from $(\Sigma-\gamma)\cap T$ is very long, provided that $d_W(\alpha,\beta)$ is sufficiently large.

Proposition 4.7. There exists $c=c(\Sigma)>0$ such that, for $\alpha$ and $\beta$ any pair of curves as in Corollary 4.3, we have

\[ d_W(\alpha,\beta)\le c\ell_T(\mu)+c,\]

for every slope $\mu$ in T that is not homotopic to a component of $(\Sigma-\gamma)\cap T$ .

Observe now that the intrinsic diameter of T satisfies

\[{\rm diam}(T)\le D:=\Big(\inf_{\mu}\ell_T(\mu)+D'\Big)\Big/2,\]

where the $\inf$ runs over all $\mu$ as in the statement. Hence, any two points on T can be joined by a flat geodesic of length at most D.

Consider the covering $p:Q\to M_f-\gamma$ corresponding to $\pi_1(W=\Sigma-\gamma)<\pi_1(M_f-\gamma)$ . By Bonahon’s tameness [Reference BonahonBon86] Q is geometrically and topologically tame. Moreover, since $\Sigma-\gamma$ is not a virtual fiber (for example, because it separates $M_f-\gamma$ ), the covering Q is a geometrically finite hyperbolic metric on $W\times\mathbb{R}$ (without accidental parabolics) by Thurston and Canary’s covering theorem [Reference CanaryCan96]. Denote by $X\sqcup Y=\partial\mathcal{CC}(Q)$ the boundary of the convex core.

By the work of Minsky [Reference MinskyMin10] (see also Theorem 2.1.3 of Bowditch [Reference BowditchBow11]), there exists a uniform quasi-geodesic $l:=\alpha_0,\alpha_1,\ldots,\alpha_n$ in $\mathcal{C}(W)$ , that is,

\[\frac{1}{C}|i-j|-C\le d_W(\alpha_i,\alpha_j)\le C|i-j|+C\]

for some uniform constant $C>0$ , with the following properties.

1. (i) Every $\alpha_j$ has a geodesic representative $\alpha_j^*$ in Q of moderate length, that is, $\ell(\alpha_j^*)\le L$ for some uniform $L>0$ .

2. (ii) The initial and terminal curves have moderate length on the X and Y boundary components, that is, $\ell_X(\alpha_0),\ell_Y(\alpha_n)\le L$ , with L as before.

3. (iii) Every curve $\beta\in\mathcal{C}(W)$ such that $\ell_Q(\beta)\le L$ lies uniformly close to the quasi-geodesic l, that is, $d_W(\beta,l)\le R$ for some uniform $R>0$ .

We note that

\[ n\ge c_0d_W(\alpha,\beta)-c_0\]

for some uniform $c_0>0$ . In fact, on one hand, by property (iii), we have $d_W(\alpha,l), d_W(\beta,l)\le R$ . On the other hand, the uniform quasi-geodesic property of $l:=\alpha_0,\ldots,\alpha_n$ and hyperbolicity of the curve graph $\mathcal{C}(W)$ gives $d_W(\alpha_0,\alpha_n)\ge d_W(\alpha,\beta)-R_1$ for some uniform $R_1>0$ . Combined together the two properties give the estimate above.

The moderate-length geodesics $\alpha_j^*$ are well-spaced in Q. This is a consequence of the model manifold technology, which we use in the form of the following result of Bowditch [Reference BowditchBow11] and Brock and Bromberg [Reference Brock and BrombergBB11].

Theorem 4.8 (See Theorem 2.1.4 of [Reference BowditchBow11] and Theorem 7.16 of [Reference Brock and BrombergBB11]). For every $L>0$ there exists $A>1$ such that the following holds. Let Q a hyperbolic structure on $W\times\mathbb{R}$ for which the boundary $\partial W$ is parabolic. Let $Q_0=Q-\mathbb{T}_{\eta_3}(\partial W)$ be the complement of the standard cusp neighborhoods. Suppose that $\alpha,\beta\in\mathcal{C}(W)$ are simple closed curves represented in Q by closed geodesics of length at most L. Then

\[\frac{1}{A}d_{\mathcal{C}(W)}(\alpha,\beta)-A\le\rho_{Q_0}\big(\mathbb{T}_{\eta_3}(\alpha),\mathbb{T}_{\eta_3}(\beta)\big)\le Ad_{\mathcal{C}(W)}(\alpha,\beta)+A,\]

where $\rho_{Q_0}$ denotes the $\eta_3$ -electric distance in $Q_0$ .

The $\eta_3$ -electric distance $\rho_{Q_0}(x,y)$ between two points $x,y\in Q_0$ is defined to be the infimum of the electric lengths of all paths joining them in $Q_0$ . The electric length of a path $\delta$ in $Q_0$ is the length of the portion of $\delta$ that lies in the $\eta_3$ -thick part of $Q_0$ . Observe that, by definition, the electric distance satisfies $\rho_{Q_0}\le d_{Q_0}$ .

The electric distance is also defined on hyperbolic surfaces $(W,\sigma)$ , and it is a fact (bounded diameter lemma; see Lemma 1.10 of [Reference BonahonBon86]) that for any fixed Margulis constant $\eta>0$ , if $(W,\sigma)$ has finite area, then the $\eta$ -electric diameter of $(W,\sigma)$ is uniformly bounded only in terms of $\eta$ and $\chi(W)$ . This fact applies in our setting: the $\eta_3$ -electric diameter of any pleated surface is uniformly bounded.

It follows from Theorem 4.8 and the fact that the sequence $\alpha_0,\ldots,\alpha_n$ is a uniform quasi-geodesic that

\[\frac{1}{B}|i-j|-B\le\rho_{Q_0}(\alpha_i^*,\alpha_j^*)\le B|i-j|+B\]

for some uniform $B>0$ . Also note that X and Y are uniformly close to $\alpha_0^*$ and $\alpha_n^*$ respectively, meaning that

\[\rho_{Q_0}(\alpha_0^*,X),\rho_{Q_0}(\alpha_n^*,Y)\le\log(2L/\eta_3).\]

This follows from the following standard fact, which we state without proof.

Lemma 4.9 (See, e.g., [Reference Biringer and SoutoBS17, Lemma 4.2]). Let $\alpha\subset Q$ be a closed curve homotopic to a geodesic $\alpha^*$ . Then

\[ d_Q(\alpha,\mathbb{T}_{\eta_3}(\alpha^*))\le\log(2\ell(\alpha)/\eta_3).\]

For simplicity, from now on, we assume $n=2m$ since it does not affect the argument and simplifies the notation. Consider a pleated surface $G:(W,\sigma)\to Q$ realizing the middle curve $\alpha_{m=n/2}$ in Q. Under the covering projection G descends to the pleated surface $g:=pG:(W,\sigma)\to M_f-\gamma$ realizing $\alpha_m$ in $M_f-\gamma$ .

The manifold Q has two cusps that cover $\mathbb{T}_{\eta_3}(\gamma)$ ; we fix one of those and denote it by $\mathbb{T}^1_Q$ , with boundary $T^1_Q=\partial\mathbb{T}^1_Q$ . Then, we choose points $x\in T^1_Q\cap X$ and $w\in W$ such that $G(w)\in T^1_Q$ . Observe that ${\mathrm{inj}}_x(X),{\rm inj}_w(W,\sigma)\ge\eta_3$ . As a consequence, since the electric diameter of pleated surfaces is uniformly bounded, we have

\[\rho_{Q_0}(G(w),\alpha_m^*),\rho_{Q_0}(x,\alpha_0^*)\le K\]

for some uniform $K>0$ . Now connect p(x) to g(w) via a shortest flat geodesic $\xi$ between them on T. Denote by $\delta$ the lift of $\xi$ to Q with basepoint G(w). We have $\ell(\delta)\le D=(\ell_T(\mu)+D')/2$ .

In order to conclude the proof of Proposition 4.7 it remains to show the following.

We divide the proof of this claim into two cases.

In this case, we have

\[\ell(\delta)\ge\rho_{Q_0}(x,G(w))\ge\rho_{Q_0}(\alpha_0^*,\alpha_m^*)-2K\ge\frac{m}{B}-B-2K,\]

as desired.

In this case, let $\tau$ be a non peripheral loop on X based at x that has moderate length, say $\ell_X(\tau)\le L_1$ for some uniform $L_1>0$ only depending on W.

We now observe that $p(\tau)$ does not lift to a loop based at z in Q. In fact, we claim that $p(\tau)$ admits a unique lift which is a loop based at a point on the chosen cusp of Q and such a lift is $\tau$ , which is based at $x\neq z$ . In general, lifts of $p(\tau)$ based at a point on $p^{-1}(p(x))\cap T^1_Q$ correspond to elements $\kappa\in\pi_1(T,p(x))$ such that $\kappa p(\tau)\kappa^{-1}\in p_*\pi_1(X,x)$ .

Lemma 4.10. If $c\in\pi_1(\Sigma-\gamma)$ is not peripheral and $\kappa\in\pi_1(M_f-\gamma)-\pi_1(\Sigma-\gamma)$ , then

\[\kappa c\kappa^{-1}\not\in\pi_1(\Sigma-\gamma).\]

Proof. Recall that $\pi_1(M_f-\gamma)$ is a free product with amalgamation

\[\pi_1(M_f-\gamma)=\pi_1(H_g-\gamma)*_{\pi_1(\Sigma-\gamma)}\pi_1(H_g-f(\gamma)).\]

Write $\kappa$ as a reduced word $\kappa=a_1b_1\cdots a_nb_n$ with $a_j\in\pi_1(H_g-\gamma)-\pi_1(\Sigma-f(\gamma))$ for $j>1$ and $b_j\in\pi_1(H_g-\gamma)-\pi_1(\Sigma-\gamma)$ for $j<n$ . Since $\kappa\notin\pi_1(\Sigma-\gamma)$ , either $b_n\not\in\pi_1(\Sigma-\gamma)$ , or we can take $b_n$ to be the identity and $a_n\not\in\pi_1(\Sigma-\gamma)$ .

The two cases can be dealt with in the same way, so we only consider the first case, that is, we assume that $b_n\not\in\pi_1(\Sigma-\gamma)$ . We have

\[\kappa c\kappa^{-1}=a_1b_1\cdots a_nb_ncb_n^{-1}a_n^{-1}\cdots b_1^{-1}a_1^{-1}.\]

We claim that $b_ncb_n^{-1}\in\pi_1(H_g-\gamma)$ is not in $\pi_1(\Sigma-\gamma)$ provided that c is not a peripheral element: this follows from the fact that $(H_g,\gamma)$ is pared acylindrical. In fact, suppose that $b_ncb_n^{-1}\in\pi_1(\Sigma-\gamma)$ and consider the homotopy between c and $b_ncb_n^{-1}$ which takes place in $H_g-\gamma$ . We have the following possibilities: if the homotopy is deformable into the cusp, then c would be peripheral, which is ruled out by our initial assumption. As $(H_g,\gamma)$ is pared acylindrical, if the homotopy is not deformable to the cusp, then it is deformable to the boundary $\Sigma-\gamma$ . In this case $b_n$ would be contained in $\pi_1(\Sigma-\gamma)$ , which is again a contradiction. Therefore, if c is not peripheral, the word $\kappa c\kappa^{-1}=a_1b_1\cdots a_n(b_ncb_n^{-1})a_n^{-1}\cdots b_1^{-1}a_1^{-1}$ is still reduced, and it contains a term not in $\pi_1(\Sigma-\gamma)$ . Hence, it cannot represent an element in $\pi_1(\Sigma-\gamma)$ .

We now return to the main argument for Proposition 4.7. By Lemma 4.10, the loop $g(\tau)$ lifts to an arc $\eta$ with basepoint z on the preferred cusp and another endpoint u on a different component of $p^{-1}(\mathbb{T}_{\eta_3}(\gamma))$ . We now observe that, if $\eta_3$ has been chosen sufficiently short in the beginning, no component of $p^{-1}(\mathbb{T}_{\eta_3}(\gamma))$ different from the cusps of Q intersects the convex core $\mathcal{CC}(Q)$ .

Lemma 4.11. If $\eta_3$ is sufficiently small, only depending on the topological type of W, then

\[p^{-1}\mathbb{T}_{\eta_3}(\gamma)\cap\mathcal{CC}(Q)={\rm cusp}(Q).\]

Proof. Let $p^{-1}\mathbb{T}_{\eta_3}(\gamma)=\bigsqcup_{j\in I}\mathcal{O}_j$ be the full preimage of the cusp under the covering projection $p:Q\to M_f-\gamma$ . Suppose that a component $X\subset\partial\mathcal{CC}(Q)$ of the boundary of the convex core intersects one of the components $\mathcal{O}_j$ of the lift of the Margulis tube. Note that $p:X\to M_f-\gamma$ is a type-preserving pleated surface in the homotopy class of the inclusion $\Sigma-\gamma\subset M_f-\gamma$ and that $p(\mathcal{O}_j)=\mathbb{T}_{\eta_3}(\gamma)$ . By Lemma 4.4, the pleated surface p(X) can only intersect $\mathbb{T}_{\eta_3}(\gamma)$ in its $\eta_0$ -cuspidal part, for some uniform $\eta_0$ , if $\eta_3$ has been chosen sufficiently small in the beginning. This means that $\mathcal{O}_j$ intersects the cuspidal part on X and, hence, $\mathcal{O}_j$ is one of the cusps of Q.

We are now able to conclude: by Lemmas 4.10 and 4.11, the arc $\delta*\eta$ has an endpoint $G(w)\in T^1_Q\cap\mathcal{CC}(Q)$ and another one $u\in p^{-1}(\mathbb{T}_{\eta_3}(\gamma))-\mathcal{CC}(Q)$ outside the convex core. Therefore, it must intersect $\partial\mathcal{CC}(Q)=X\sqcup Y$ . Say it intersects X. In particular $\rho_{Q_0}(G(w),X)\le d_{Q_0}(G(w),X)\le\ell(\delta*\eta)$ , which, combined with the previously established inequalities gives us

\begin{align*}\ell(\delta)+L_1 &\ge\ell(\delta*\eta)\\&\ge\rho_{Q_0}(G(w),X)\\&\ge\rho_{Q_0}(\alpha_m^*,\alpha_0^*)-\rho_{Q_0}(X,\alpha_0^*)-2K\\&\ge m/B-B-2K-\log(2L/\eta_3).\end{align*}

This concludes the proof of the claim.

4.5 The proof of Theorem 6

Combining Proposition 4.7 and Theorem 2.6, we complete the proof of Theorem 6 as follows.

Proof of Theorem 6. We endow $M_f-\gamma$ with a complete finite-volume hyperbolic structure, which exists by the assumption of $(H_g,\gamma)$ and $(H_g,f(\gamma))$ being pared acylindrical.

Let $\mu$ be the flat geodesic on $T=\partial\mathbb{T}_{\eta_3}(\gamma)$ that represents the filling slope needed to pass from $M_f-\gamma$ to $M_f$ (also known as the meridian of $\gamma$ ). By Theorems 2.6 and 2.7, if ${\rm nl}(\mu)\ge 11$ , then there exists a hyperbolic metric on $M_f$ for which $\gamma$ is a geodesic of length $\ell_{M_f}(\gamma)\le a/{\rm nl}(\mu)^2$ for some universal constant $a>0$ .

By Proposition 4.7, we have

\[\ell(\mu)\ge cd_W(\alpha,\beta)-c,\]

which is larger than $\frac{1}{2}cd_W(\mathcal{D},f\mathcal{D})$ provided that $d_W(\mathcal{D},f\mathcal{D})$ is sufficiently large. Note that ${\rm Area}(T)\le\eta_3\ell(\mu)$ whenever $\eta_3<\ell(\mu)$ (which we have given that $d_W(\mathcal{D},f\mathcal{D}))$ is sufficiently large); hence,

\[{\rm nl}(\mu)=\ell(\mu)/\sqrt{{\rm Area}(T)}\ge\sqrt{\ell(\mu)/\eta_3}.\]

Thus, ${\rm nl}(\mu)\ge 11$ if the subsurface projection of the disk sets to W is sufficiently large. This shows that $M_f$ is hyperbolic.

In order to conclude, it remains to bound the length of $\gamma$ in $M_f$ . This follows again from Theorem 2.6:

\[\ell(\gamma)\le\frac{a}{{\rm nl}(\mu)^2}\le\frac{a}{\ell(\mu)/\eta_3}\le\frac{2a\eta_3/c}{d_W(\mathcal{D},f\mathcal{D})}.\]

5.1 Assembling simple pieces

We identify a tubular neighborhood of the Heegaard surface $\Sigma\subset M_f=H_g\cup_fH_g$ with $\Sigma\times[1,4]$ such that $\Sigma$ is identified with $\Sigma\times\{2.5\}$ .

Given pants decompositions $P_1$ , $P_2$ , $P_3$ and $P_4$ in $\Sigma$ , we consider

\[\mathbb{M}:=M_f-(P_1\times\{1\}\sqcup P_2\times\{2\}\sqcup P_3\times\{3\}\sqcup P_4\times\{4\}).\]

We isolate the five pieces (see Figure 1)

\begin{align*}H_1&:=H_g-(\Sigma\times(1,2.5]\sqcup P_1\times\{1\}),\\\Omega_1&:=\Sigma\times[1,2]-(P_1\times\{1\}\sqcup P_2\times\{2\}),\\Q&:=\Sigma\times[2,3]-(P_2\times\{2\}\sqcup P_3\times\{3\}),\\\Omega_2&:=\Sigma\times[3,4]-(P_3\times\{3\}\sqcup P_4\times\{4\}),\quad\!\! \text{and}\\H_2&:=H_g-(\Sigma\times[2.5,4)\sqcup P_4\times\{4\}).\end{align*}

Figure 1. Gluing.

Lemma 5.1. Provided that

(A) \begin{align}&(H_g-\Sigma\times(1,2.5],P_1\times\{1\}),\nonumber\\&(\Sigma\times[1,2],P_1\times\{1\}\sqcup P_2\times\{2\}),\nonumber\\&(\Sigma\times[2,3],P_2\times\{2\}\sqcup P_3\times\{3\}),\tag{A}\\&(\Sigma\times[3,4],(P_3\times\{3\}\sqcup P_4\times\{4\}),\quad\!\! \text{and}\nonumber\\&(H_g-\Sigma\times[2.5,4),P_4\times\{4\})\nonumber\end{align}

are all pared acylindrical manifolds, there exists a hyperbolic metric on $\mathbb{M}$ such that the restriction of the metric to each of the pieces is isometric to the convex core of the unique maximally cusped hyperbolic structure on the piece.

Since there is only one hyperbolic metric up to isotopy on a triply punctured sphere (such as the connected components of the complement of the pared loci in the boundaries of the pieces), we can arrange the metrics on the pieces such that they agree on the intersections and match together to give a complete finite-volume hyperbolic metric on $\mathbb{M}$ .

In order to pass from $\mathbb{M}$ to the closed 3-manifold $M_f$ , we have to perform Dehn fillings on each cusp. This is the second step of the construction. The filling slopes are completely determined by the identification of $\mathbb{M}$ with the drilled $M_f$ : they are the meridians $\gamma$ of small tubular neighborhoods of the curves in $\alpha\times\{j\}\subset P_j\times\{j\}$ inside $\Sigma\times[1,4]$ . Theorem 2.6 gives us sufficient conditions to guarantee that $M_f$ has a hyperbolic metric. Theorem 2.8 gives sufficient conditions to ensure that $\mathbb{M}$ is K-bilipschitz to $M_f$ away from its cusps. We make this precise in the next subsection.

5.2 A filling criterion

To be able to apply Theorems 2.6, 2.8 and 2.7, we have to certify that the filling slopes have large normalized length. We now discuss a criterion to check this condition.

We define

\begin{align*}\mathbb{N}_1&:=H_1\cup\Omega_1=H_g-P_2\times\{2\},\\\mathbb{N}_2&:=H_2\cup\Omega_2=H_g-P_3\times\{3\},\quad \text{and}\\\mathbb{Q}&:=\Omega_1\cup Q\cup\Omega_2=\Sigma\times[1,4]-(P_1\times\{1\}\sqcup P_2\times\{2\}\sqcup P_3\times\{3\}\sqcup P_4\times\{4\}).\end{align*}

The curves in $P_1$ and $P_4$ represent rank-one cusps on $\partial\mathbb{Q}$ while the curves in $P_2$ and $P_3$ represent rank-two cusps. Similarly, the curves $P_2$ and $P_3$ represent rank-one cusps of $\mathbb{N}_1$ and $\mathbb{N}_2$ while the curves in $P_1$ and $P_4$ are cusps of rank two.

We now try to understand what happens when we Dehn fill the rank-two cusps in $\mathbb{Q},\mathbb{N}_1,\mathbb{N}_2$ . Choosing the filling slopes to be the canonical meridians of $P_1,P_2,P_3,P_4$ in $M_f$ , the Dehn filled manifolds are

\begin{align*}&\mathbb{Q}^{{\rm fill}}=\Sigma\times[1,4]-(P_1\times\{1\}\sqcup P_4\times\{4\}),\\&\mathbb{N}_1^{{\rm fill}}=H_g-(\Sigma\times(2,2.5]\sqcup P_2\times\{2\}),\quad \text{and}\\&\mathbb{N}_2^{{\rm fill}}=H_g-(\Sigma\times[2.5,3)\sqcup P_3\times\{3\}).\end{align*}

We endow $\mathbb{Q}^{{\rm fill}},\mathbb{N}_1^{{\rm fill}},\mathbb{N}_2^{{\rm fill}}$ with a complete hyperbolic metric with totally geodesic boundary and rank-one cusps at the curves $P_1\times\{1\}\sqcup P_4\times\{4\},P_2\times\{2\},P_3\times\{3\}$ . Again, this is possible provided that

(B) \begin{align}&(\Sigma\times[1,4],P_1\times\{1\}\sqcup P_4\times\{4\}),\nonumber\\&(H_g-\Sigma\times(2,2.5],P_2\times\{2\}), \quad \text{and}\tag{B}\\&(H_g-\Sigma\times[2.5,3),P_3\times\{3\}) \nonumber\end{align}

are all pared acylindrical.

Recall that our goal is to show that the filling slopes we singled out on the rank-two cusps of $\mathbb{Q},\mathbb{N}_1,\mathbb{N}_2$ have very large normalized length. First, observe that $\mathbb{Q},\mathbb{N}_1,\mathbb{N}_2$ isometrically embed in their doubles $D\mathbb{Q},D\mathbb{N}_1,D\mathbb{N}_2$ which are finite-volume hyperbolic 3-manifolds. Checking that the filling slopes of the rank-two cusps of $\mathbb{Q},\mathbb{N}_1,\mathbb{N}_2$ have large normalized length is the same as checking that they have large normalized length in $D\mathbb{Q},D\mathbb{N}_1,D\mathbb{N}_2$ .

Also note that

\[\mathbb{Q},\mathbb{N}_1,\mathbb{N}_2\subset\mathbb{Q}^{{\rm fill}},\mathbb{N}_1^{{\rm fill}},\mathbb{N}_2^{{\rm fill}}\subset D\mathbb{Q}^{{\rm fill}},D\mathbb{N}_1^{{\rm fill}},D\mathbb{N}_2^{{\rm fill}}.\]

Again, the doubles are finite-volume hyperbolic 3-manifolds.

The idea is as follows. Suppose that we can find hyperbolic metrics on $\mathbb{Q}^{{\rm fill}},\mathbb{N}_1^{{\rm fill}},\mathbb{N}_2^{{\rm fill}}$ such that the curves

\[P_2\times\{2\}\sqcup P_3\times\{3\},P_1\times\{1\},P_4\times\{4\}\subset\mathbb{Q}^{{\rm fill}},\mathbb{N}_1^{{\rm fill}},\mathbb{N}_2^{{\rm fill}}\]

are very short geodesics. Then, $D\mathbb{Q},D\mathbb{N}_1,D\mathbb{N}_2$ are exactly the complete finite-volume hyperbolic structures on the complement of those geodesic links $D\mathbb{Q}^{{\rm fill}}-P_1\times\{1\}\sqcup P_4\times\{4\},D\mathbb{N}_1^{{\rm fill}}-P_2\times\{2\},D\mathbb{N}_2^{{\rm fill}}-P_3\times\{3\}$ .

By Theorem 2.7, the condition of having large normalized length for the meridians of $P_1\times\{1\}\sqcup P_4\times\{4\},P_2\times\{2\},P_3\times\{3\}$ is equivalent to the condition of being very short for those geodesics.

The previous discussion leads us to make the following definition.

1. (1) $P_1\times\{1\}$ is isotopic to its geodesic realization in $\mathbb{N}_1^{\rm fill}$ and each component of this geodesic link has length at most $\eta$ ;

2. (2) $P_4\times\{4\}$ is isotopic to its geodesic realization in $\mathbb{N}_2^{\rm fill}$ and each component of this geodesic link has length at most $\eta$ ; and

3. (3) $P_2\times\{2\}\sqcup P_3\times\{3\}$ is isotopic to its geodesic realization in $\mathbb{Q}^{\rm fill}$ and each component of this geodesic link has length at most $\eta$ .

By the above discussion we have the following. For $L>0$ , if $P_1,P_2,P_3,P_4$ satisfy the filling criterion with parameter

\[\eta:=\min\bigg\{\frac{2\pi}{L^2+16.17},\frac{0.1396}{3g-3}\bigg\},\]

then the normalized length of the filling slopes of $\mathbb{M}$ corresponding to $P_1\times\{1\},P_2\times\{2\},P_3\times\{3\},P_4\times\{4\}\subset M_f$ is at least L. Hence, by Theorems 2.6 and 2.7 (combined with Theorem 2.8 for the furthermore part), we established the following.

Proposition 5.2. Fix $K\in(1,2)$ . There exists $\eta>0$ such that the following holds. Suppose that there are four pants decompositions $P_1,P_2,P_3,P_4$ such that the filling criterion with parameter $\eta$ is satisfied. Then, $M_f$ admits a hyperbolic metric such that $\Gamma:=\bigsqcup_{j=1,2,3,4}{P_j\times\{j\}}$ is a geodesic link and each component has length at most $\eta$ . Furthermore, if $\mathbb{M}$ denotes the unique finite-volume hyperbolic structure on $M_f-\Gamma$ , then we have a K-bilipschitz diffeomorphism of pairs

\[\bigg(\mathbb{M}-\bigsqcup_{\alpha\in P_1\cup P_2\cup P_3\cup P_4}{\mathbb{T}_{\eta_3}(\alpha)}\bigg)\cong\bigg(M_f-\bigsqcup_{\alpha\in P_1\cup P_2\cup P_3\cup P_4}{\mathbb{T}_{\eta_3}(\alpha)}\bigg).\]

We conclude with a small remark. The model manifold technology of Minsky [Reference MinskyMin10] and Brock, Canary and Minsky [Reference Brock, Canary and MinskyBCM12], provides several tools to locate and measure the length of the geodesic representatives of $P_2\times\{2\}$ and $P_3\times\{3\}$ in $\mathbb{Q}^{{\rm fill}}$ . However, the same technology is not available for handlebodies. This is the place where the difficulties arise. We get around the lack of a model for handlebodies by restricting ourselves to the geometry of the collars of $\mathbb{N}_1^{{\rm fill}},\mathbb{N}_2^{{\rm fill}}$ where, in some cases, we have a good amount of control (as in [Reference Hamenstädt and ViaggiHV22]).

6.1 A family of hyperbolic mapping tori

For every pants decomposition $P\subset\Sigma$ , we first construct hyperbolic mapping tori

\[T_\psi=\Sigma\times[0,1]/(x,0)\sim(\psi(x),1),\]

with the property that the multicurve $P\times\{0\}\subset\Sigma\times\{0\}$ is a very short geodesic link. In particular, the infinite cyclic covering of $T_\psi$ (see Figure 2) is a hyperbolic structure ${\hat T}_\psi$ on $\Sigma\times\mathbb{R}$ where the multicurves $\psi^nP\times\{n\}\subset\Sigma_n:=\Sigma\times\{n\}$ are very short geodesic links for every $n\in\mathbb{Z}$ .

Figure 2. Infinite cyclic covering.

Definition (Mapping class with a short pants decomposition). Let $\eta>0$ and let P be a pants decomposition of $\Sigma$ . If $\psi\in{\mathrm{Mod}}(\Sigma)$ is a mapping class such that its mapping torus

\[T_\psi:=(\Sigma\times[0,1]/(x,0)\sim(\psi x,1))\]

has a hyperbolic metric with respect to which $P\times\{0\}\subset T_\psi$ is a geodesic link of length at most $\eta$ , then we call $\psi$ a pseudo-Anosov mapping class with a short pants decomposition P of length at most $\eta$ .

These objects are abundant. Concretely, we have the following.

Proof. Let $\phi\in{\rm Mod}(\Sigma)$ be a mapping class such that

\[(\Sigma\times[0,1],P\times\{0\}\sqcup\phi P\times\{1\})\]

is pared acylindrical. For example, one can choose $\phi$ to be a large power of any pseudo-Anosov. Consider the convex core Q of the maximally cusped structure $Q(P,\phi P)$ . The boundary $\partial Q$ consists of totally geodesic hyperbolic three punctured spheres that are paired according to $\phi$ . We glue them together isometrically as prescribed by the pairing. The glued manifold is a finite-volume hyperbolic 3-manifold diffeomorphic to

\[T_\phi-P\times\{0\}=(\Sigma\times[0,1]/(x,0)\sim(\phi x,1))- P\times\{0\}.\]

The curves in $P\times\{0\}$ represent rank-two cusps. Observe that for each $\alpha\in P$ any choice of corresponding boundary torus $\partial\mathbb{T}(\alpha)\subseteq T_\phi-P\times\{0\}$ has a preferred meridian $m_\alpha$ (given as the boundary of an essential disk in $\mathbb{T}(\alpha)\subseteq T_\phi$ ) and a preferred longitude $l_\alpha$ (given as a curve in the fiber $\Sigma\times\{0\}\subseteq T_\phi$ that is parallel to $\alpha$ ).

If we perform Dehn surgeries with slopes $m_\alpha+kl_\alpha$ , the resulting closed manifold will be diffeomorphic to the mapping torus $T_\psi$ , where $\psi=\phi\delta_P^k$ and $\delta_P$ is a Dehn multitwist about the pants decomposition P. By Thurston’s hyperbolic Dehn surgery [Reference Benedetti and PetronioBP92, Chapter E.6]) (or by invoking Theorems 2.6 and 2.7), if k is large enough, then the resulting manifold carries a hyperbolic metric for which the core curves of the added solid tori are very short geodesics. Hence, for k large enough, $\psi$ is a pseudo-Anosov mapping class with a short pants decomposition P of length at most ${\epsilon}$ .

6.2 The criterion

We now describe some criteria (Propositions 6.2 and 6.3) to copy and paste the geometry of a hyperbolic mapping torus with a short pants decomposition into the collar geometry of maximally cusped handlebodies H and trivial bundles $\Sigma\times[0,1]$ . These structures will automatically satisfy the filling criterion for a suitable choice of pants decompositions (Proposition 6.4).

In the statement of the following proposition we will think of the repelling lamination of a pseudo-Anosov as an element of the Gromov boundary $\partial \mathcal{C}$ of $\mathcal{C}$ , see [Reference KlarreichKla18].

Consider the maximally cusped structure $N=H(P)$ on ${\rm int}(H)$ . Identify a collar of $\Sigma=\partial H$ with $\Sigma\times[0,2]$ where $\partial H=\Sigma\times\{0\}$ . Then the geodesic realization of $\psi^{-1}P\times\{1\}$ in N is isotopic to $\psi^{-1}P\times\{1\}$ and has length at most $2\eta$ .

We prove Proposition 6.2 at the end of the section. The idea is that if m is large enough then the geometry of N near the boundary of its convex core looks like the geometry of the region $[\Sigma_{-2},\Sigma_0]\subset{\hat T}_\psi$ and there the geodesic realization of $\psi^{-1}P$ lies as a short geodesic link on the level surface $\Sigma_1$ (see Figure 3).

Figure 3. Collars handlebodies.

Similarly, we also use the following variation (see also Figure 4).

Figure 4. Collars I-bundles.

Consider the maximally cusped structure $Q=Q(P,R)$ on ${\rm int}(\Sigma\times[0,3])$ . Then the geodesic realization of $\psi P\times\{1\}$ in Q is isotopic to $\psi P\times\{1\}$ and has length at most $2\eta$ .

Having in mind the application to the random 3-manifold setup, we combine the two propositions in the following criterion which produces many examples of pants decompositions that satisfy the assumptions of Proposition 5.2.

Let f be a gluing map. Set $S_0:=o$ , $S_5:=fo$ . Suppose that there are four points $S_1,S_2,S_3,S_4\in[o,fo]\subset\mathcal{C}$ with the following properties.

1. (i) Along the geodesic segment [o,fo] we have $S_j<S_{j+1}$ and $d_{\mathcal{C}}(S_j,S_{j+1})\ge h$ .

2. (ii) There exist mapping classes $\tau$ and $\nu$ such that $[S_1,S_2]$ and $[S_3,S_4]$ $\delta$ -fellow travel the translates $\tau l_\psi$ and $\nu l_\phi$ of the uniform quasi-axes $l_\psi,l_\phi:\mathbb{R}\to\mathcal{C}$ of $\psi$ and $\phi$ , respectively.

3. (iii) We have $d_{\mathcal{C}}([S_1,S_2],\mathcal{D})\ge h$ and $d_{\mathcal{C}}([S_3,S_4],f\mathcal{D})\ge h$ .

Then, there exist four pants decompositions

\[(P_1,P_2,P_3,P_4)\]

such that:

• – they satisfy the filling criterion with parameter $\eta$ for the gluing map f;

• – we have $d_{\mathcal{C}}(P_2,[S_1,S_2]),d_{\mathcal{C}}(P_3,[S_3,S_4])\le B$ ;

• – we have $P_2=\tau\psi\tau^{-1}P_1$ and $P_4=\nu\phi\nu^{-1}P_3$ .

In § 7, we use Proposition 6.4 to detect whether, for a random gluing map f, $M_f$ can be described as one of the examples we constructed simply by staring at the geometry of the segment [o,fo], where $o\in\mathcal{D}$ is some basepoint that we will fix once and for all.

6.3 The proof of Proposition 6.4

Assuming Propositions 6.2 and 6.3, we give a proof of Proposition 6.4.

Proof of Proposition 6.4 . Let $P_n,R_n:=\tau\psi^nP,\nu\phi^nR$ .

Let $L:=\max\{L(\psi),L(\phi)\}>0$ be the maximum of the translation length of $\psi,\phi$ on $\mathcal{C}$ (it is positive as $\psi,\phi$ are pseudo-Anosov). Note that $L(\tau\psi\tau^{-1})=L(\psi)$ and $L(\nu\phi\nu^{-1})=L(\phi)$ . Let $h:=2nL$ , with n much larger than the $m(\psi)$ and $m(\phi)$ as given by Propositions 6.2 and 6.3.

As the geodesic representatives of $P_n,R_n$ are very short in the infinite cyclic coverings ${\hat T}_\psi,{\hat T}_\phi$ of the hyperbolic mapping tori $T_\psi,T_\phi$ , it follows from work of Minsky [Reference MinskyMin10] combined with the structural results of Masur and Minsky [Reference Masur and MinskyMM99, Reference Masur and MinskyMM00], that there exists a uniform constant B, only depending on $\Sigma$ , and $t,r\in\mathbb{R}$ such that

\[d_{\mathcal{C}}(P,l_\psi(t)),d_{\mathcal{C}}(R,l_\phi(r))\le B\]

for some t,r and, thus, by equivariance

\[d_{\mathcal{C}}(P_j,\tau l_\psi(t+jL(\psi))),d_{\mathcal{C}}(R_i,\tau l_\phi(r+iL(\phi)))\le B\]

for every $j,i\in\mathbb{Z}$ . Up to reparametrization, we can assume $t=r=0$ and simplify the notation by introducing

\[\alpha:=l_\psi(0),\quad \beta:=l_\phi(0)\]

and

\[\alpha_j:=\tau l_\psi(jL(\psi)),\quad \beta_i:=\nu l_\phi(iL(\phi)).\]

By the above discussion $d_{\mathcal{C}}(P_j,\alpha_j),d_{\mathcal{C}}(R_i,\beta_i)\le B$ for every $i,j\in\mathbb{Z}$ . (Note that the $\alpha_j$ and $\beta_i$ are linearly aligned along $\tau l_\psi$ and $\nu l_\phi$ .)

By assumption (ii) of Proposition 6.4 and the choice of $h\ge2nL(\psi),2nL(\phi)$ , we can find $a,b\in\mathbb{Z}$ such that the following hold:

(1) \begin{equation}\begin{aligned}-\ &\tau l_\psi[(a-n)L(\psi),(a+n)L(\psi)] \text{ is in a } \delta\text{-neighborhood of } [S_1,S_2];\\-\ &\nu l_\phi[(b-n)L(\phi),(b+n)L(\phi)] \text{ is in a } \delta\text{-neighborhood of } [S_3,S_4].\end{aligned}\end{equation}

Again, up to renumbering, we can assume $a=b=0$ .

We now show that

\[(P_{-1},P_0,R_0,R_1)\]

satisfies the filling criterion with parameter $2\eta>0$ , where $\eta$ is the minimum of the $\eta$ in Propositions 6.2 and 6.3.

We first make sure that the eight building blocks (A) and (B) are pared acylindrical (and, hence, admit maximally cusped structures). As the maps $\psi,\phi$ are pseudo-Anosov, the pared acylindrical assumptions are automatically satisfied for $(P_{-1},P_0)$ and $(R_0,R_1)$ . In order to check the others, we use Lemma 2.5.

Let us check that $d_{\mathcal{C}}(P_{-1},\mathcal{D}),d_{\mathcal{C}}(P_0,\mathcal{D})\ge 3$ : we know that $P_{-1},P_0$ are uniformly close to $\alpha_{-1},\alpha_0$ which, in turn, are $\delta$ -close to $[S_1,S_2]$ . By assumption (iii), we have $d_{\mathcal{C}}([S_1,S_2],\mathcal{D})\ge h$ . Thus, provided that h is large enough, the claim holds. The same argument shows that $d_{\mathcal{C}}(R_0,f\mathcal{D}),d_{\mathcal{C}}(R_1,f\mathcal{D})\ge3$ provided that h is sufficiently large.

Let us now check that $d_{\mathcal{C}}(P_{-1},R_1),d_{\mathcal{C}}(P_0,R_0)\ge 3$ : consider $(P_0,R_0)$ first, the argument for the other pair $(P_{-1},R_1)$ is completely analogous. We know that $P_0,R_0$ are uniformly close to $\alpha_0,\beta_0$ , which, in turn, are $\delta$ -close to points on $[S_1,S_2]$ and $[S_3,S_4]$ . Thus, up to a uniform additive constant, $d_{\mathcal{C}}(P_0,R_0)$ is larger than $d_{\mathcal{C}}(S_2,S_3)\ge h$ .

Hence, the pared acylindrical assumptions are satisfied and the eight pieces admit maximally cusped structures. We now check the length-isotopy properties as given by the filling criterion.

Consider the maximally cusped structures $N_1,N_2$ , and Q, where $N_1,N_2$ are hyperbolic structure on the interior of handlebodies with boundary $\Sigma$ and disk sets $\mathcal{D},f\mathcal{D}$ with cusps at $P_0,R_0$ , while Q is a hyperbolic structure on the interior of $\Sigma\times[0,3]$ with cusps at $P_{-1}\times\{0\}\sqcup R_1\times\{3\}$ . Identify a collar of the boundary $\Sigma$ of the two handlebodies with $\Sigma\times[0,2]$ in such a way that the boundary is $\Sigma\times\{0\}$ . Then:

1. (1) the collection $P_{-1}\times\{1\}$ is isotopic to its geodesic realization in $N_1$ and has length at most $\eta$ ;

2. (2) the collection $R_1\times\{1\}$ is isotopic to its geodesic realization in $N_2$ and has length at most $\eta$ ;

3. (3) the collection $P\times\{1\}\sqcup R\times\{2\}$ is isotopic to its geodesic realization in Q and has length at most $\eta$ .

We use Proposition 6.2 to check parts (1) and (2). We use Proposition 6.3 to check part (3).

Note that $\tau^{-1}$ (respectively, $\nu^{-1}$ ) induces a natural orientation-preserving isometry between the maximally cusped hyperbolic structure on the handlebody with boundary $\Sigma$ and disk set $\mathcal{D}$ (respectively, $f\mathcal{D}$ ) associated with the pants decomposition $P_0$ (respectively, $R_0$ ) and the maximally cusped hyperbolic structure on the handlebody with boundary $\Sigma$ and disk set $\tau^{-1}\mathcal{D}$ (respectively, $\nu^{-1}f\mathcal{D}$ ) associated with the pants decomposition $\tau^{-1}P_0$ (respectively, $\tau^{-1}R_0$ ).

Therefore, according to Proposition 6.2 to prove parts (1) and (2) we only need that:

1. (a) $(\lambda_\psi^-\,|\,\tau^{-1}\mathcal{D})_P\ge m(\psi)$ ;

2. (b) $(\lambda_\phi^+\,|\,\nu^{-1}f\mathcal{D})_R\ge m(\phi)$ .

Here $\lambda_\psi^-,\lambda_\phi^+$ are the repelling laminations of $\psi,\phi^{-1}$ thought as points on the Gromov boundary of $\mathcal{C}$ and the left-hand sides of the inequalities are Gromov products based at P,R.

Properties (a) and (b) are similar. We just prove property (a). By properties of the Gromov product on Gromov hyperbolic spaces (such as $\mathcal{C}$ ) we have

\[(\lambda_\psi^-\,|\,\tau^{-1}\mathcal{D})_P\ge\min\{(\tau^{-1}P_{-n}\,|\,\tau^{-1}\mathcal{D})_P,(\lambda_\psi^-\,|\,\tau^{-1}P_{-n})_P\}-K\]

for some uniform constant K. Regarding $(\tau^{-1}P_{-n}\,|\,\tau^{-1}\mathcal{D})_P=(P_{-n}\,|\,\mathcal{D})_{P_0}$ , we claim that, up to some uniform additive constant, it is larger than $d_{\mathcal{C}}(P_{-n},P_0)$ . The latter, in turn, is comparable to $d_{\mathcal{C}}(\alpha_{-n},\alpha_0)\simeq nL(\psi)$ by the above observations. In order to prove the claim, observe that $[P_{-n},P_0]$ lies uniformly close to $[S_1,S_2]$ (with $P_{-n}$ closer to $S_1$ and $P_0$ closer to $S_2$ ), which is far from $\mathcal{D}$ (with $S_2$ further away than $S_1$ ). By hyperbolicity of the curve graph and quasi-convexity of the disk set, this implies that the segment $[P_0,P_{-n}]$ uniformly fellow travels the shortest geodesic connecting $P_0$ to $\mathcal{D}$ . This implies the claim. Regarding the second term, we have

\[(\lambda_\psi^-\,|\,\tau^{-1}P_{-n})_P=(\lambda_\psi^-\,|\,\psi^{-n}P)_P.\]

since, by definition, $\tau^{-1}P_{-n}=\psi^{-n}P$ . The Gromov product is computed up to a uniform additive error by

\[(\lambda_\psi^-\,|\,\psi^{-n}P)_P=\tfrac{1}{2}[d_{\mathcal{C}}(l_\psi(-kL(\psi)),P)+d_\mathcal{C}(P,\psi^{-n}P)-d_{\mathcal{C}}(\psi^{-n}P,l_\psi(-kL(\psi))],\]

where k is sufficiently large. We recall that $l_\psi(-kL(\psi))=\psi^{-k}l_\psi(0)=\psi^{-k}\alpha$ and, by our choices, $d_\mathcal{C}(\alpha,P)\le B$ is uniformly bounded. Thus, the right-hand side is, up to a uniform additive constant, greater than or equal to

\[\tfrac{1}{2}[d_{\mathcal{C}}(\psi^{-k}\alpha,\alpha)+d_\mathcal{C}(\alpha,\psi^{-n}\alpha)-d_{\mathcal{C}}(\psi^{-n}\alpha,\psi^{-k}\alpha)],\]

which is comparable to $L(\psi)n$ as $\alpha=l_\psi(0)$ and $\psi$ acts on $l_\psi$ by translations by $L(\psi)$ . Thus, we proved that up to a uniform additive constant we have that $(\lambda_\psi^-\,|\,\tau^{-1}\mathcal{D})_P$ is at least $nL(\psi)$ .

We now discuss point (3).

Again, observe that $\tau^{-1}$ (respectively, $\nu^{-1}$ ) induces a natural orientation-preserving isometry between the maximally cusped hyperbolic structure on $\Sigma\times[0,3]$ associated with the pants decomposition $P_1,R_1$ and the maximally cusped hyperbolic structure on the same manifold associated with the pants decomposition $\tau^{-1}P_1,\tau^{-1}R_1$ (respectively, $\nu^{-1}P_1,\nu^{-1}R_1$ ). Hence, according to Proposition 6.3, for point (3) in order to know that $P\times\{1\}$ and $R\times\{2\}$ are isotopic to their geodesic realizations in $Q=Q(P_{-1},R_1)$ we need:

1. (a’) $(\lambda_\psi^+\,|\,\tau^{-1}R_1)_P\ge m(\psi)$ ;

2. (b’) $(\lambda_\phi^-\,|\,\nu^{-1}P_1)_R\ge m(\phi)$ .

Properties (a’) and (b’) are also similar and follow exactly the same lines as the previous computation done for properties (a) and (b). In order to avoid repetitions, we omit these computations and focus instead on an additional issue, namely that we also need to know that $P\times\{1\}$ and $R\times\{2\}$ can be simultaneously isotoped to their geodesic realizations. This can be done as follows: let $\beta_P,\beta_R$ be the geodesic realizations of $P\times\{1\},R\times\{2\}$ in $Q(P_{-1},R_1)$ . By Theorem 4.8, the distance between two ${\epsilon}$ -Margulis neighborhoods $\mathbb{T}_{\epsilon}(\alpha),\mathbb{T}_{\epsilon}(\beta)$ is bounded by $d_{\mathcal{C}}(\alpha,\beta)$ .

This implies that the ${\epsilon}$ -Margulis neighborhoods around the curves of $\beta_P$ have (electric) distance from $\partial_+\mathcal{CC}_0(Q)$ uniformly bounded in terms of

\[D_P:=\max\{d_{\mathcal{C}}(\gamma,\gamma')\,|\,\gamma\subset P_{-1},\gamma'\subset P\}.\]

Similarly, the (electric) distance of the ${\epsilon}$ -Margulis tubes around the curves in $\beta_R$ from $\partial_-\mathcal{CC}_0(Q)$ is bounded in terms of

\[D_R:=\max\{d_{\mathcal{C}}(\gamma,\gamma')\,|\,\gamma\subset R_1,\gamma'\subset R\}.\]

On the other hand the (electric) distance between $\partial_+\mathcal{CC}_0(Q)$ and $\partial_-\mathcal{CC}_0(Q)$ is bounded from below by $d_\mathcal{C}(P_{-1},R_1)$ . Recall that $d_{\mathcal{C}}(P_{-1},R_1)\ge d_{\mathcal{C}}(S_2,S_3)\ge h$ up to some uniform additive constant (as $P_{-1},R_1$ are uniformly close to $[S_1,S_2],[S_3,S_4]$ , respectively). If h is much larger than $D_P+D_R$ , then we can isotope $\beta_P$ into $\partial_+\mathcal{CC}_0(Q)$ and $\beta_R$ into $\partial_-\mathcal{CC}_0(Q)$ simultaneously. This comes from the following.

Sketch of proof. First, as the distance between $\partial_+\mathcal{CC}_0(Q)$ and $\partial_-\mathcal{CC}_0(Q)$ is very large, we can find a pleated surface that separates $\beta_P$ from $\beta_R$ . This can be done as follows. By the filling theorem of Canary [Reference CanaryCan96] and Thurston [Reference ThurstonThu80], one can find a pleated surface S’ passing uniformly near a point in $\mathcal{CC}_0(Q)$ whose distance from $\partial\mathcal{CC}_0(Q)$ is equal to $d(\partial_+\mathcal{CC}_0(Q),\partial_-\mathcal{CC}_0(Q))/2$ . Such a pleated surface has uniformly bounded intrinsic (electric) diameter, in particular, it is disjoint from the $D_P,D_R$ -(electric) neighborhoods $U_+,U_-$ of $\partial_+\mathcal{CC}_0(Q),\partial_-\mathcal{CC}_0(Q)$ provided that $d(\partial_+\mathcal{CC}_0(Q),\partial_-\mathcal{CC}_0(Q))\asymp d_{\mathcal{C}}(P_{-1},R_1)\ge h$ is large enough. As $\mathcal{CC}_0(Q)\simeq\Sigma\times\{0\}$ and the pleated surface S’ is homotopic to $\Sigma\times\{0\}$ , we have that S’ separates such neighborhoods $U_+,U_-$ . Using standard techniques (see Freedman, Hass and Scott [Reference Freedman, Hass and ScottFHS83]), we can find in a small neighborhood of S’, still disjoint from $U_+,U_-$ , an embedded surface $S\subset\mathcal{CC}_0(Q)$ homotopic to S’. The fact that S is boundary parallel comes from the classification of incompressible surfaces in $\mathcal{CC}_0(Q)\simeq\Sigma\times[0,1]$ .