CALT-TH-2024-013

Comments Concerning a Hypothetical
Mesoscopic Dark Dimension

John H<EMAIL_ADDRESS>
_Walter Burke Institute for Theoretical Physics
California Institute of Technology
Pasadena, CA 91125, USA_

Abstract

> Motivated by string-theoretic swampland conjectures, the existence of a dark
> fifth dimension, whose size is roughly 1 – 10 microns, has been proposed. A
> great deal of supporting evidence has been presented, and definitive
> experimental tests are likely to be carried out. The basic idea is that the
> four-dimensional spacetime that we observe is confined to a brane that is
> localized in the dark dimension. This short note points out that there are
> two distinct ways to realize such a scenario in string theory/M-theory. In
> the one considered previously, the dark dimension is topologically a circle
> and our observable 4d spacetime is confined to a brane that is localized in
> a GUT-scale region of the circle. An alternative possibility is that the
> dark dimension is a line interval with “end-of-the-world-branes” at each
> end. The latter option would imply the existence of a parallel 4d spacetime,
> with the same gravitational forces as ours, but otherwise its own physical
> laws, microns away from us!

Motivated by string-theoretic swampland conjectures [1], it has recently been
proposed that there is a mesoscopic fifth dimension whose radius $\rho$ is
roughly 1 – 10 microns [2]–[6]. One aspect of this is the proposal that a
tower of light states in our universe is related to the smallness of the
cosmological constant $\Lambda$ [7]. The new idea is that this tower
corresponds to Kaluza–Klein (KK) modes of a graviton in a dark dimension, and
that they constitute dark matter. The size of this dimension is of order
$\Lambda^{-1/4}$. (We always set $\hbar=c=1$.) The unknown coefficient is
responsible for the uncertainty in the prediction for $\rho$. In this way dark
energy and dark matter are connected without invoking an anthropic principle.
This dimension is called dark, because it only supports gravitational forces.
The basic idea is that the four-dimensional spacetime that we observe lives on
a brane that is localized at a point (more precisely, a GUT-scale region) in
the dark dimension (DD). Any additional compact dimensions must be very much
smaller than the DD. Thus, for many purposes, we may assume that the brane is
well approximated by a standard-model 3-brane (SM3-brane) even though it would
be desirable to understand exactly how such a brane arises from a more
fundamental string theory/M-theory starting point. For reasons explained in
[2], the existence of more than one mesoscopic dimension is excluded. As
discussed in [3], astrophysical constraints limit the allowed size of the DD
to 1 – 10 microns, and future large-scale structure surveys could support or
rule out its existence.

A number of arguments were made in [2]–[6], based on various theoretical and
observational facts, in support of the DD proposal. These arguments suggest
that the size of the DD is not very far below the limit (about 30 microns)
obtained by a recent experimental study of Newton’s $1/r^{2}$ gravitational
force law at very short distances [8]. That experiment used “a stationary
torsion-balance detector and a rotating attractor containing test bodies.”
There have also been other related experiments [9]–[12]. An experiment that
can probe even shorter distances is being planned [13][14]. It will study the
orbits of a microscopic planetary system. Precession of the orbit would imply
a central potential different from $1/r$. Other experimental approaches may be
possible, e.g. based on atom interferometry.

It is now conceivable that the DD realization of string theory will receive
experimental support within this decade. Prior to the DD proposal I did not
think that string theory could make novel testable predictions in the
foreseeable future. This DD proposal appears to be well motivated, but even so
it should be considered speculative. The experiments ought to be a very high
priority, because experimental verification would be revolutionary. A negative
result would not be revolutionary, but it would still be fundamental and very
important.

There have been previous theoretical proposals for a relatively large extra
dimension, such as [15] and [16]. They differ in important respects from the
current proposal, which is motivated by swampland considerations as well as
various astrophysical arguments. In particular, the current proposal shows
promise for giving a deeper understanding of why the amounts of dark matter
and dark energy are comparable without invoking the anthropic principle.

If a mesoscopic DD is discovered, it will become important to figure out how
it fits into string theory. We have known since the mid-1990s that the various
versions of superstring theory/M-theory are related by dualities, so that
there really is a unique underlying theory. However, the different versions,
which focus on different corners of the moduli space of vacua, can appear more
or less promising as starting points for phenomenology. There have been two
such classes of starting points that account for almost all studies of string
phenomenology. One class is based on the $E_{8}\times E_{8}$ heterotic string
[17]. The other class is based on F-theory [18], which utilizes certain
nonperturbative solutions of type IIB superstring theory. See [19] for a
recent discussion of the status of string phenomenology.

An F-theory approach to realizing the DD proposal is described in [6]. In that
realization the DD is topologically a circle, and the SM3-brane is localized
in a GUT-scale region of the circle. Among other things, that paper argues
that this picture gives a tower of states, associated to a 7-brane, at
$10^{9}$ – $10^{10}$ GeV. This scale, called the species scale, corresponds to
the 5d Planck scale.

Following the discovery that cancellation of local gauge and gravitational
anomalies in 10d superstring theories is only possible for the gauge groups
$SO(32)$ and $E_{8}\times E_{8}$ [20], a string theory with the latter gauge
group, called the $E_{8}\times E_{8}$ heterotic string, was constructed [17].
For the following decade (prior to the introduction of F-theory) this was the
only version of superstring theory that showed promise for giving realistic
phenomenology. The introduction of Calabi–Yau compactification [21] launched a
great deal of interest in superstring theory in general, and this specific
approach in particular. It was clear from the outset that this setup has many
qualitative successes. However, even today there is no specific construction
that looks exactly right. The math is hard, and the number of possibilities is
very large. The same can be said for the F-theory approach.

Among the many exciting discoveries made in the mid-1990s was the realization
that the $E_{8}\times E_{8}$ heterotic string theory and the type IIA
superstring theory actually have an eleventh dimension that is not visible in
perturbation theory [22][23]. In the type IIA case there is a circular
eleventh dimension ($S^{1}$) whose size is proportional to the string coupling
constant. In the limit of infinite coupling one obtains 11-dimensional
M-theory. In the case of the $E_{8}\times E_{8}$ theory there is an orbifold
eleventh dimension ($S^{1}/Z_{2}$), whose size also is proportional to the
string coupling constant. This dimension can also be viewed as a line interval
with two ends. Each of the ends is attached to an “end-of-the-world” 9-brane
(an EW9-brane). One $E_{8}$ gauge field is associated to each of the two
EW9-branes. This is required by the anomaly analysis [22]. The 11-dimensional
bulk, in between the EW9-branes, is described by 11-dimensional M-theory,
which is the UV completion of 11d supergravity [24].

The eleventh dimension of the $E_{8}\times E_{8}$ theory is a DD. Most studies
of Calabi–Yau compactification of this theory have ignored the eleventh
dimension, which may be a reasonable thing to do for many purposes. However,
it is an important fact that this class of string theory solutions necessarily
implies the existence of a DD. Some of the implications of the DD have been
discussed in [25] – [30]. As explained in [25], one consequence of increasing
the string coupling constant (and hence the size of the DD) is to reduce the
lower bound on the 4d Newton constant $G_{N}$. This is important, since
$G_{N}$ is necessarily too large in the weakly coupled $E_{8}\times E_{8}$
theory. So the requirement that the DD in this theory must be quite large has
been understood for a long time. What is new now is the suggestion, based on
swampland considerations, that the size of this dimension should be several
microns. By the standards of particle physics, this is very large indeed.

Suppose we consider implementing the DD proposal using the $E_{8}\times E_{8}$
theory. Then an essential difference from the F-theory example is that it
requires the existence of two EW-branes. After a suitable compactification of
six dimensions they can be approximated by 3-branes, and one of them can be
identified as the standard model 3-brane (SM3-brane) that describes our 4d
reality at least up to energies of a TeV and possibly much further. In this
setup there is a second parallel 3-brane, just microns away! Should we expect
the physics of the second EW-brane to be the same as or different from “our”
EW-brane? The Calabi–Yau phenomenology works best when the two $E_{8}$’s are
treated differently. This implies that the physics of the two EW-branes is
different, so that seems likely to be the case. In any case, the possibility
of a parallel 4d spacetime just microns away from us is challenging to
visualise and seems quite incredible! However, as the F-theory example
illustrates, this is not an unavoidable prediction of the DD proposal – just
one of the two options.

For strong coupling the $E_{8}\times E_{8}$ heterotic theory is not a string
theory. (A similar remark can be made for the type IIA theory.) In M theory
the basic branes are an M2-brane and its magnetic dual, which is an M5-brane.
A cylindrical M2-brane can be stretched across the dark dimension with its
boundaries attached to the EW-branes. In the weak-coupling limit the two EW-
branes approach one another, and the cylinder collapses to become the
$E_{8}\times E_{8}$ string. However, if the branes are microns apart, it looks
nothing like a string – unless it can be approximated by an open string
stretched between the two EW-branes, but I am unaware of any such proposal. In
the $E_{8}\times E_{8}$ theory the coupling constant $g$ is proportional to
the radius of the dark dimension, $\rho$. More specifically,
$g\sim\rho/l_{s}$, where $l_{s}$ is the string scale, which is very small. If
$\rho$ is several microns, this ratio is enormous. The problem is evident from
the shape of the stretched M2-brane. If there were an S-dual formulation of
the $E_{8}\times E_{8}$ heterotic string theory, whose coupling constant is
$\tilde{g}=1/g$, that could be helpful. The $SO(32)$ heterotic string theory
has an S-dual formulation, namely the type I superstring theory, which is
based on a non-BPS (breakable) string. It seems unlikely that there is an
analogous S-dual of the $E_{8}\times E_{8}$ heterotic string theory. One
problem is that it is not possible to define a $g\to\infty$ limit while
retaining both of the EW-branes, and we don’t want $\rho$ to be infinite.

Suppose that experimental evidence for the existence of a DD is obtained by
measuring the strength of gravity at short distances. How could we decide
whether the DD is a circle or a line segment with a second brane at the other
end of the DD? We would only be sensitive to the gravitational effects of the
other brane. One could imagine a star on our brane orbiting an invisible
object that resides on the other brane. Would it be possible to establish that
the invisible object is not a black hole or something on our brane that we
already understand?

It is conceivable that a very sensitive test of gravity at short distances
could distinguish between the two geometries. If the DD is a circle of radius
$\rho=1/\mu$, then summing Yukawa potentials for the KK excitations of a 5d
graviton gives a potential proportional to

$\frac{1}{r}\sum_{n=-\infty}^{\infty}e^{-|n|\mu r}=\frac{1}{r}\frac{1+e^{-\mu
r}}{1-e^{-\mu r}}\sim\frac{1}{r}(1+2e^{-\mu r}+\ldots)$ (1)

In the case of the orbifold/line segment, one has instead

$\frac{1}{r}\sum_{n=0}^{\infty}e^{-|n|\mu r}=\frac{1}{r}\frac{1}{1-e^{-\mu
r}}\sim\frac{1}{r}(1+e^{-\mu r}+\ldots)$ (2)

The leading large $r$ asymptotic behavior is shown. For small $r$ it is
proportional to $\rho/r^{2}$ in both cases, as expected, though they differ by
a factor of 2.

These formulas probably require corrections for the following reason. A
massless graviton in 5d has five polarizations, forming an irreducible
representation of the SU(2) little group, as does a massive spin two particle
in 4d. The only mismatch here concerns the zero modes. In both cases there are
three extra massless modes in 4d with helicities $+1$, $0$, $-1$ in addition
to the desired graviton with helicities $+2$ and $-2$. They are not wanted,
since it is well established that 4d gravity is mediated by a massless spin 2
graviton. A plausible resolution of this problem is that these three modes
combine to give a massive spin one particle in 4d. The mass of this vector
particle must go to zero in the decompactification limit, if one is to recover
11d M theory. Therefore, it ought to be proportional to $\mu$. The formulas
above need to be modified in order to correctly account for this massive
vector. In principle, the experiments should be able to reveal one massive
spin one particle in addition to the usual graviton and the infinite tower of
massive spin two particles by measuring the strength of gravity at short
distances to high precision.

An analogous problem occurs when M theory has a circular dimension giving
strongly coupled type IIA superstring theory. The type IIA massless spectrum
in 10d contains a graviton, a vector, and a scalar that descend from the 11d
graviton. The presence of the massless vector reflects the $U(1)$ isometry of
the eleventh dimension, in contrast to the cases we have discussed in which
one or two branes break the symmetry. In the type IIA case there is a known
deformation of the theory in which the scalar and vector combine to give a
massive vector [31]. This mass arises when one or more 8-branes are added to
the type IIA background [32]. Thus, it is clear that the vector must be
massive whenever branes break the $U(1)$ symmetry of the dark dimension. This
is the case for the F-theory and M-theory examples under consideration here.

The original papers [2]–[6] make a strong case for KK excitations of gravitons
in the DD comprising the most important component of dark matter. Of course,
it is still possible that there are also other sources of dark matter. Other
sources of dark matter in the DD might include KK excitations of sterile
neutrinos or other bulk degrees of freedom. Possible sources of dark matter on
the SM3-brane are the familiar ones that have received the most attention by
both theorists and experimentalists so far. These include axions and wimps,
though the latter have been largely excluded. If there is a second EW-brane,
matter that is confined to that brane would also be dark matter.

All known black holes are much larger than the proposed DD. Black holes that
are smaller than the DD would be localized within that dimension, away from
any branes, and they would be very difficult to detect. The implications of
the DD proposal for primordial black holes have been discussed in [33]-[35].
These papers point out that in the DD scenario 5d black holes have a much
longer lifetime due to Hawking decay than 4d ones of the same mass. Therefore,
they suggest that primordial black holes that are smaller than the DD could be
another significant component of dark matter.

One may wonder whether the DD proposal has any bearing on the $\Lambda$CDM
model of cosmology. In particular, the time evolution of various quantities is
important. Since $G_{N}\sim G_{5}/\rho$, where $G_{5}$ is the 5d Newton
constant, and $\Lambda\sim\rho^{-4}$, it follows that

$\frac{\dot{G_{N}}}{G_{N}}=\frac{\dot{G_{5}}}{G_{5}}-\frac{\dot{\rho}}{\rho}{\quad{\rm
and}\quad}\frac{\dot{\Lambda}}{\Lambda}=-4\frac{\dot{\rho}}{\rho}.$ (3)

Variation of $G_{N}$ is constrained by various observations, such as binary
pulsars, and variation of $G_{5}$ seems unlikely in the epoch where the
$\Lambda$CDM model is supposed to apply. Therefore significant variation of
$\rho$ and $\Lambda$ appears to be unlikely in this approach. Even so, the DD
model may help to address the current stress in the $\Lambda$CDM model. An
additional element is the fact that the DD model predicts that Kaluza–Klein
excitations of the graviton constitute a spectrum of decaying dark matter
particles. While there have been studies of of decaying dark matter as a
possible mechanism to address the stress, the details of how it occurs in the
DD model are different from the way it has been modeled in these studies.
There may also be other effects, such as Casimir forces, that are relevant.

In conclusion, while the existence of a dark dimension of the sort discussed
here is certainly speculative, it is a well-motivated and experimentally
testable proposal. It is based on plausible string theory considerations that
should be experimentally testable. Its confirmation would be revolutionary for
particle physics and astrophysics. If the existence of a DD is confirmed, an
important question would be whether or not there is a parallel 3-brane microns
away from the SM3-brane on which we reside.

## Acknowledgments

I am grateful to Cumrun Vafa and Temple He for helpful discussions. This
material is based upon work supported by the U.S. Department of Energy, Office
of Science, Office of High Energy Physics, under Award Number DE-SC0011632.

## References

  * [1] N. B. Agmon, A. Bedroya, M. J. Kang and C. Vafa, “Lectures on the string landscape and the Swampland,” [arXiv:2212.06187 [hep-th]].
  * [2] C. Vafa, “Swamplandish Unification of the Dark Sector,” [arXiv:2402.00981 [hep-ph]].
  * [3] G. Obied, C. Dvorkin, E. Gonzalo and C. Vafa, “Dark Dimension and Decaying Dark Matter Gravitons,” [arXiv:2311.05318 [astro-ph.CO]].
  * [4] J. A. P. Law-Smith, G. Obied, A. Prabhu and C. Vafa, “Astrophysical Constraints on Decaying Dark Gravitons,” [arXiv:2307.11048 [hep-ph]].
  * [5] E. Gonzalo, M. Montero, G. Obied and C. Vafa, “Dark dimension gravitons as dark matter,” JHEP 11, 109 (2023) [arXiv:2209.09249 [hep-ph]].
  * [6] M. Montero, C. Vafa and I. Valenzuela, “The dark dimension and the Swampland,” JHEP 02, 022 (2023) [arXiv:2205.12293 [hep-th]].
  * [7] D. Lüst, E. Palti and C. Vafa, “AdS and the Swampland,” Phys. Lett. B 797, 134867 (2019) [arXiv:1906.05225 [hep-th]].
  * [8] J. G. Lee, E. G. Adelberger, T. S. Cook, S. M. Fleischer and B. R. Heckel, “New Test of the Gravitational $1/r^{2}$ Law at Separations down to 52 $\mu$m,” Phys. Rev. Lett. 124, no.10, 101101 (2020) [arXiv:2002.11761 [hep-ex]].
  * [9] A. A. Geraci, S. J. Smullin, D. M. Weld, J. Chiaverini and A. Kapitulnik, “Improved constraints on non-Newtonian forces at 10 microns,” Phys. Rev. D 78, 022002 (2008) [arXiv:0802.2350 [hep-ex]].
  * [10] W. H. Tan, S. Q. Yang, C. G. Shao, J. Li, A. B. Du, B. F. Zhan, Q. L. Wang, P. S. Luo, L. C. Tu and J. Luo, “New Test of the Gravitational Inverse-Square Law at the Submillimeter Range with Dual Modulation and Compensation,” Phys. Rev. Lett. 116, no.13, 131101 (2016)
  * [11] T. Westphal, H. Hepach, J. Pfaff and M. Aspelmeyer, “Measurement of gravitational coupling between millimetre-sized masses,” Nature 591, no.7849, 225-228 (2021) [arXiv:2009.09546 [gr-qc]].
  * [12] C. P. Blakemore, A. Fieguth, A. Kawasaki, N. Priel, D. Martin, A. D. Rider, Q. Wang and G. Gratta, “Search for non-Newtonian interactions at micrometer scale with a levitated test mass,” Phys. Rev. D 104, no.6, L061101 (2021) [arXiv:2102.06848 [hep-ex]].
  * [13] J. Baeza-Ballesteros, A. Donini and S. Nadal-Gisbert, “Dynamical measurements of deviations from Newton’s $1/r^{2}$ law,” Eur. Phys. J. C 82, no.2, 154 (2022) [arXiv:2106.08611 [hep-ph]].
  * [14] J. Baeza-Ballesteros, A. Donini, G. Molina-Terriza, F. Monrabal and A. Simón, “Towards a realistic setup for a dynamical measurement of deviations from Newton’s $1/r^{2}$ law: the impact of air viscosity,” [arXiv:2312.13736 [hep-ph]].
  * [15] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, “New dimensions at a millimeter to a Fermi and superstrings at a TeV,” Phys. Lett. B 436, 257-263 (1998) [arXiv:hep-ph/9804398 [hep-ph]].
  * [16] L. Randall and R. Sundrum, “A Large mass hierarchy from a small extra dimension,” Phys. Rev. Lett. 83, 3370-3373 (1999) [arXiv:hep-ph/9905221 [hep-ph]].
  * [17] D. J. Gross, J. A. Harvey, E. J. Martinec and R. Rohm, “The Heterotic String,” Phys. Rev. Lett. 54, 502-505 (1985)
  * [18] C. Vafa, “Evidence for F theory,” Nucl. Phys. B 469, 403-418 (1996) [arXiv:hep-th/9602022 [hep-th]].
  * [19] F. Marchesano, G. Shiu and T. Weigand, “The Standard Model from String Theory: What Have We Learned?,” [arXiv:2401.01939 [hep-th]].
  * [20] M. B. Green and J. H. Schwarz, “Anomaly Cancellation in Supersymmetric D=10 Gauge Theory and Superstring Theory,” Phys. Lett. B 149, 117-122 (1984)
  * [21] P. Candelas, G. T. Horowitz, A. Strominger and E. Witten, “Vacuum configurations for superstrings,” Nucl. Phys. B 258, 46-74 (1985)
  * [22] P. Horava and E. Witten, “Heterotic and type I string dynamics from eleven-dimensions,” Nucl. Phys. B 460, 506-524 (1996) [arXiv:hep-th/9510209 [hep-th]].
  * [23] P. Horava and E. Witten, “Eleven-dimensional supergravity on a manifold with boundary,” Nucl. Phys. B 475, 94-114 (1996) [arXiv:hep-th/9603142 [hep-th]].
  * [24] E. Cremmer, B. Julia and J. Scherk, “Supergravity Theory in Eleven-Dimensions,” Phys. Lett. B 76, 409-412 (1978).
  * [25] E. Witten, “Strong coupling expansion of Calabi-Yau compactification,” Nucl. Phys. B 471, 135-158 (1996) [arXiv:hep-th/9602070 [hep-th]].
  * [26] T. Banks and M. Dine, “Couplings and scales in strongly coupled heterotic string theory,” Nucl. Phys. B 479, 173-196 (1996) [arXiv:hep-th/9605136 [hep-th]].
  * [27] E. Caceres, V. S. Kaplunovsky and I. M. Mandelberg, “Large volume string compactifications, revisited,” Nucl. Phys. B 493, 73-100 (1997) [arXiv:hep-th/9606036 [hep-th]].
  * [28] P. Horava, “Gluino condensation in strongly coupled heterotic string theory,” Phys. Rev. D 54, 7561-7569 (1996) [arXiv:hep-th/9608019 [hep-th]].
  * [29] A. Lukas, B. A. Ovrut, K. S. Stelle and D. Waldram, “The Universe as a domain wall,” Phys. Rev. D 59, 086001 (1999) [arXiv:hep-th/9803235 [hep-th]].
  * [30] A. Lukas, B. A. Ovrut, K. S. Stelle and D. Waldram, “Heterotic M theory in five-dimensions,” Nucl. Phys. B 552, 246-290 (1999) [arXiv:hep-th/9806051 [hep-th]].
  * [31] L. J. Romans, “Massive N=2a Supergravity in Ten-Dimensions,” Phys. Lett. B 169, 374 (1986)
  * [32] E. Bergshoeff, M. de Roo, M. B. Green, G. Papadopoulos and P. K. Townsend, “Duality of type II 7 branes and 8 branes,” Nucl. Phys. B 470, 113-135 (1996) [arXiv:hep-th/9601150 [hep-th]].
  * [33] L. A. Anchordoqui, I. Antoniadis and D. Lust, “Dark dimension, the swampland, and the dark matter fraction composed of primordial black holes,” Phys. Rev. D 106, no.8, 086001 (2022) [arXiv:2206.07071 [hep-th]].
  * [34] L. A. Anchordoqui, I. Antoniadis and D. Lust, “The dark universe: Primordial black hole $\leftrightharpoons$ dark graviton gas connection,” Phys. Lett. B 840, 137844 (2023) [arXiv:2210.02475 [hep-th]].
  * [35] L. A. Anchordoqui, I. Antoniadis and D. Lust, “The Dark Dimension, the Swampland, and the Dark Matter Fraction Composed of Primordial Near-Extremal Black Holes,” [arXiv:2401.09087 [hep-th]].


# Large volume fibered knots in 3-manifolds

J. Robert Oakley

###### Abstract.

We prove that for hyperbolic fibered knots in any closed, connected, oriented
3-manifold the volume and genus are unrelated. As an application we answer a
question of Hirose, Kalfagianni, and Kin about volumes of mapping tori that
are double branched covers.

##  1\. Introduction

Let $M$ denote a closed, connected, oriented 3-manifold. Alexander in [Ale23]
proved that every such $M$ contains a fibered link. This result has been
strengthened in many ways such as the link being a knot [Mye78] and the
monodromy being right-veering and pseudo-Anosov [CH08]. More recently, this
construction has been used in [BFPS23] to show that for hyperbolic fibered
knots in the 3-sphere, volume and genus are unrelated. In this paper we
generalize the approach in [BFPS23] to show that for hyperbolic, fibered knots
in closed, connected, oriented 3-manifolds the volume and genus are unrelated.
In particular we prove the following theorem.

######  Theorem 1.1.

Let $M$ be a closed, connected, oriented 3-manifold. There exists some
$g_{0}>1$ such that for all $g\geq g_{0}$ and $V>0$ there exists a knot
$K\subseteq M$ such that $M-K$ is fibered over the circle with genus $g$ and
$M-K$ is hyperbolic with $\text{vol}(M-K)>V.$

As an application of theorem 1.1 we answer a question asked by Hirose,
Kalfagianni, and Kin in [HKK23]. Let
$\mathfrak{D}_{g}(M)\subseteq\operatorname{Mod}(S_{g})$ be the subset of the
mapping class group of a closed surface of genus $g$ consisting of elements
whose mapping tori are 2-fold branched covers of $M$ branched along a link.
Hirose, Kalfagianni, and Kin asked the following question in [HKK23].

######  Question 1.

For $g$ sufficiently large, does $\mathfrak{D}_{g}(M)$ contain an infinite
family of pseudo-Anosov mapping classes whose mapping tori have arbitrarily
large volume?

Combining theorem 1.1 and theorem 10 of [HKK23] yields the following
affirmative answer to question 1.

######  Corollary 1.2.

For any closed, connected, oriented 3-manifold $M$ there exists some $g_{0}>1$
such that for any $g>g_{0}$ the set $\mathfrak{D}_{2g}(M)$ contains an
infinite family of pseudo-Anosov elements whose mapping tori have arbitrarily
large volume.

###  1.1. Outline of the proof

The strategy of the proof is to construct for every closed, connected,
oriented 3-manifold $M$ and for every sufficiently large genus $g$ of the
fiber, a family of fibered hyperbolic knots of genus $g$ with linearly growing
volume. We use three main tools to construct these families. The first is the
theory of branched covers in dimensions 2 and 3. Using branched covers of the
3-sphere branched over a knot or link in braid position to obtain a fibered
link in a 3-manifold goes back to Alexander. Here we use an improvement on
Alexander’s theorem due to Hilden and Montesinos [Hil76, Mon76]. This
refinement allows us to consider simple 3-fold branched covers. That the
branched covers are degree 3 is crucial for control of the number of preimages
of the braid axis which will become the fibered link. This control over the
degree of the cover comes at the expense of the covers considered by Hilden
and Montesinos being irregular. Fortunately these irregular covers have the
property of being _simple_ (see definition 2.1). Simple branched covers are
well studied in dimension 2 (see for example [Ful01, GK87, Win15]). This
allows us to control the monodromy of the fibered link.

In section 3 we use this control of the monodromy and the second tool, open
book decompositions and stabilization, to ensure that our fibered link becomes
a fibered knot with pseudo-Anosov monodromy (see section 3 for a brief
discussion of open book decompositions). To that end we use the equivalence
between open book decompositions and fibered links as well as the technique
introduced by Colin and Honda in [CH08] to transform our fibered 2-component
link with possibly reducible monodromy, obtained from the branched cover
construction described above, into a fibered knot with pseudo-Anosov
monodromy.

Crucially, even after this transformation we maintain the control over the
monodromy. This allows us to use our third tool, subsurface projections and
Brock’s work in [Bro03] relating translation distance of a pseudo-Anosov in
the pants graph and the volume of the associated mapping torus. The control
over the monodromy that we have maintained throughout the construction of
these fibered hyperbolic knots ensures that in addition to the monodromies
being pseudo-Anosov, they factor as $(A)(S^{-1}F^{n}S)(F^{-n})$ where $A$ is
constant as $n$ varies and $F^{\pm n}$ are pseudo-Anosov on essential
subsurfaces. Then we apply the work of Clay–Leininger–Mangahas [CLM12] to see
that the monodromy has linearly growing subsurface projections to the supports
of $F^{-n}$ and $S^{-1}F^{n}S$. The Masur–Minsky distance formula for the
pants graph [MM00] then implies that the monodromies have linearly growing
translation distance in the pants graph. Finally, the work of Brock [Bro03]
implies that this linearly growing translation distance corresponds to
linearly growing volume of the associated mapping tori.

###  1.2. Acknowledgments

The author thanks his advisor Dave Futer for pointing him towards this problem
as well as his guidance. The author thanks the NSF for its support via grant
DMS-1907708.

##  2\. Branched Cover Construction

Let $M$ be a closed, connected, oriented 3-manifold. We will begin by
constructing a fibered 2-component link in $M.$ We need the following
definition.

######  Definition 2.1.

A branched covering $p:M\longrightarrow N$ of degree $d$ is called simple if
each point in $N$ has at least $d-1$ preimages in $M$. In particular, points
away from the branch locus in $N$ will have $d$ preimages while points on the
branch locus in $N$ will have $d-1$ preimages in $M.$

Due to a theorem of Hilden and Montesinos, [Hil76, Mon76], there exists a
3-fold simple branched covering $p:M\longrightarrow S^{3}$ branched along a
knot $K.$ Since the branched covers constructed are simple, points on the
branch locus $K\subseteq S^{3}$ have two preimages in $M.$ One preimage has
branching index 1 while the other has branching index 2. By a theorem of
Alexander [Ale23] we can represent $K$ as the closure of a braid
$\Pi\in\mathfrak{B}_{k}$ for some $k.$

\begin{overpic}[width=170.71652pt]{Braid.png} \put(8.25,95.0){1}
\put(25.0,95.0){$2g+1-k$} \put(70.0,95.0){$2g+1$} \put(56.0,76.0){\huge$\Pi$}
\put(45.0,61.0){\huge$\Phi^{n}$}
\put(57.0,53.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}$\gamma_{n}$}
\put(0.0,49.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
$\omega_{n}$} \put(22.0,36.0){\huge$\Sigma$}
\put(57.0,27.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}$\delta_{n}$}
\put(41.0,15.0){\huge$\Phi^{-n}$} \end{overpic} Figure 1. The braid whose
closure will be the branching locus of our branched cover. The braid $\Pi$ is
the braid word representing the knot $K$ which we branch over. The braid
$\Phi$ is a pseudo-Anosov on its support. The word $\Sigma$ totally consists
of stabilizations and destabilizations. Note that $\gamma_{n}$ and
$\delta_{n}$ and the disks they bound do not change as $n$ increases.

Fix $g>1$ such that $2g+1>k.$ Let $\sigma_{i}$ denote the positive half-twist
between the $i^{\text{th}}$ and $(i+1)^{\text{th}}$ strands. We will consider
$\Pi\in\mathfrak{B}_{k}$ under the natural inclusion into
$\mathfrak{B}_{2g+1}$ where $\Pi$ is on the last $k$ strands (see figure 1).
In a variation on the construction of [BFPS23] we now define the following
braids.

$\displaystyle\Phi$
$\displaystyle=(\sigma_{2g+2-k})^{-1}(\sigma_{2g+3-k})(\sigma_{2g+4-k})^{-1}(\sigma_{2g+5-k})$
$\displaystyle\Sigma$
$\displaystyle=(\sigma_{2g+1-k})(\sigma_{2g-k})^{-1}\cdots(\sigma_{2})^{\pm}(\sigma_{1})^{\mp}$
$\displaystyle\beta_{n}$ $\displaystyle=\Pi\Phi^{n}\Sigma\Phi^{-n}$

Let $\widehat{\beta_{n}}$ be the braid closure of $\beta_{n}.$ Let
$\omega_{n}$ be its braid axis encircling the braid after the $\Phi^{n}$
factor and before the $\Sigma$ factor of $\beta_{n}.$ Let
$\Lambda_{n}=\widehat{\beta_{n}}\cup\omega_{n}.$ Note that $\omega_{n}$ bounds
a disk $\Omega_{n}$ in $S^{3}$ meeting $\widehat{\beta_{n}}$ in $2g+1$ points.
Hence, $S^{3}-\Lambda_{n}$ is a punctured disk bundle over the circle with
monodromy $\beta_{n}.$

######  Lemma 2.2.

The braid closure $\widehat{\beta_{n}}$ is the knot $K.$

###### Proof.

Note that $\widehat{\Pi}$ is $K,$ and that $\widehat{\beta_{n}}$ is stabilized
along the first $2g+1-k$ strands. Destabilizing smooths the crossings coming
from $\Sigma$ and and deletes the first $2g+1-k$ strands. Now the $\Phi^{n}$
and $\Phi^{-n}$ factors cancel. We are left with the $k$-strand braid closure
$\widehat{\Pi}$ which by definition is $K.$ ∎

We will now introduce some notation. Let $\gamma_{n}$ and $\delta_{n}$ be the
simple closed curves encircling the
$2g+2-k,\>2g+3-k,\>2g+4-k,\>2g+5-k,\>\text{and}\>2g+6-k$ strands of
$\beta_{n}$ taken immediately before and after the $\Sigma$ factor of
$\beta_{n}$ (see figure 1). Let $\Gamma_{n}$ and $\Delta_{n}$ denote the disks
bounded by $\gamma_{n}$ and $\delta_{n}$ respectively. Note that $\Gamma_{n}$
and $\Delta_{n}$ each intersect $\beta_{n}$ in 5 points. Let
$w_{n}\>\text{and}\>W_{n}$ denote
$p^{-1}(\omega_{n})\>\text{and}\>p^{-1}(\Omega_{n})$ respectively.

######  Lemma 2.3.

The manifold $N_{n}=M-w_{n}$ is an $S_{g-1,2}-$bundle over the circle.

###### Proof.

That $N_{n}$ is fibered over the circle follows from the fact that
$S^{3}-\Lambda_{n}$ is a punctured disk bundle over the circle. Since $p$ is a
simple, 3-fold cover and $\Omega_{n}$ intersects $\beta_{n}$ in $2g+1$ points,
a Riemann–Hurwitz formula calculation shows that $W_{n}$ is homeomorphic to
$S_{g-1,2}.$ ∎

##  3\. Hyperbolicity and Volume

To prove the main theorem we need to construct from $N_{n}$ the desired knot
complements, verify their hyperbolicity, and show that their volumes grow
linearly with $n$. In service of this we will change our perspective to a
2-dimensional perspective. We will focus on the fiber surface $W_{n}\subseteq
N_{n}$ which is fixed as the monodromy of the fibration varies. In particular
we specify a marking of the disk $\Omega=\Omega_{n}\subseteq S^{3}$. We let
$S^{3}=\mathbb{R}^{3}\cup{\infty}.$ Let $\Omega$ be the unit disk in the plane
$\mathbb{R}^{2}\times{0}.$ Moreover, by fixing a simple 3-fold branched
covering of the disk, we fix a marking on the fiber surface $W=W_{n}$ as shown
in figure 2 below. Moreover, we note that
$\gamma_{n},\delta_{n},\Gamma_{n},\>\text{and}\>\Delta_{n}$ do not depend on
$n,$ so from now on we will set $\gamma=\gamma_{n},$ $\delta=\delta_{n},$
$\Gamma=\Gamma_{n},$ and $\Delta=\Delta_{n}.$ Let $b_{n}=P\cdot F^{n}\cdot
S\cdot F^{-n}$ be the monodromy of $N_{n},$ where $P,\>F,\>\text{and}\>S$ are
the lifts of $\Pi,\>\Phi,\text{and}\>\Sigma$ respectively.

Let $c,\>d,$ and $w_{n}$ denote
$p^{-1}(\gamma),\>p^{-1}(\delta),\>\text{and}\>p^{-1}(\omega_{n})$
respectively. Also, let $C^{+}$ and $D^{+}$ denote $p^{-1}(\Gamma)$ and
$p^{-1}(\Delta)$ respectively.

\begin{overpic}[width=170.71652pt]{Simplecover.png}
\put(25.0,73.0){\color[rgb]{1,0.6015625,0.33203125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.6015625,0.33203125}
\small$p^{-1}(\eta)$}
\put(23.0,24.0){\color[rgb]{1,0.6015625,0.33203125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.6015625,0.33203125}
\small$\eta$}
\put(41.0,72.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
\tiny$p^{-1}(b)$}
\put(41.0,49.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
\tiny$p^{-1}(b)$}
\put(44.0,20.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
\tiny$b$}
\put(51.0,20.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
\tiny$\alpha$}
\put(60.0,20.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\tiny$\omega_{n}$}
\put(60.0,74.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\tiny$w_{n}$}
\put(60.0,48.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\tiny$w_{n}$}
\put(58.0,70.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
\tiny$p^{-1}(\alpha)$}
\put(50.0,42.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
\tiny$p^{-1}(\alpha)$} \par\end{overpic} Figure 2. The unique simple cover
from $S_{3,2}$ to a disk. The gray curves divide the covering surface into
three ”fundamental domains” for the cover. See the papers by Winarski and
Fuller (section 3.2 of[Win15] and [Ful01]) for more on a similar simple
branched cover. \begin{overpic}[width=227.62204pt]{Cutsimplecover.png}
\put(9.0,57.0){\color[rgb]{1,0.6015625,0.33203125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.6015625,0.33203125}
\tiny$p^{-1}(\eta)$}
\put(74.0,58.0){\color[rgb]{1,0.6015625,0.33203125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.6015625,0.33203125}
\tiny$p^{-1}(\eta)$}
\put(38.0,16.0){\color[rgb]{1,0.6015625,0.33203125}\definecolor[named]{pgfstrokecolor}{rgb}{1,0.6015625,0.33203125}
\tiny$\eta$}
\put(51.0,11.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
\tiny$b$}
\put(59.0,12.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
\tiny$\alpha$}
\put(10.0,40.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
\tiny$p^{-1}(\alpha)$}
\put(86.0,62.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
\tiny$p^{-1}(\alpha)$}
\put(86.0,36.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
\tiny$p^{-1}(\alpha)$}
\put(20.0,11.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\tiny$\omega_{n}$}
\put(95.0,59.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\tiny$w_{n}$}
\put(95.0,39.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\tiny$w_{n}$} \par\end{overpic} Figure 3. After cutting the disk
$\Omega=\Omega_{n}$ along $\alpha$ and the covering surface along
$p^{-1}(\alpha)$ we obtain two components of the covering surface. One
component covers the cut disk trivially. The other component covers the cut
disk by an order 2 involution that identifies the two boundary components.

######  Lemma 3.1.

$C^{+}$ and $D^{+}$ are each homeomorphic to the disjoint union of a disk and
$S_{2,1}.$ Furthermore, $C^{+}\cap D^{+}$ is the disjoint union of a $S_{1,2}$
and a disk.

###### Proof.

If we restrict $p$ to a single fiber of $N_{n}$ we get a simple 3-fold
branched cover $p^{\prime}:S_{g-1,2}\longrightarrow D^{2}$ branched over
$2g+1$ points. It follows from the work of Gabai and Kazez in [GK87] that
3-fold simple branched covers of the disk are unique up to equivalence. By
uniqueness of the cover we may assume it takes on the configuration shown in
figure 2 in the disk fiber of $S^{3}$ immediately following $\Pi.$ That is,
the distinguished branch point labeled $b$ may be taken to lie on the
$(2g+1)^{\text{st}}$ strand of the braid in the fiber $\Omega_{n}.$ In
particular, every simple closed curve that bounds any 5 of the branch points
excluding the distinguished point $b$ on the far right of the disk is taken by
a homeomorphism of the disk to the simple closed curve $\eta$ as shown in
figure 2. Observe that $c$ and $d$ will consist of two simple closed curves
since $\gamma$ and $\delta$ each bound an odd number of branch points. One of
the two preimages is completely contained in the upper fundamental domain of
the cover and bounds a disk as in the left panel of figure 3. For example
every simple closed curve that bounds five branch points other than the
distinguished red branch point in figure 2 is homeomorphic to the curve $\eta$
by the change of coordinates principle. Again by a Riemann–Hurwitz formula
calculation we conclude that the other component is homeomorphic to $S_{2,1},$
a surface of genus two with one boundary component. Note that
$\Gamma\cap\Delta$ is a disk that intersects $\beta_{n}$ in 4 points. Thus,
$C^{+}\cap D^{+}$ consists of a 2 disconnected components. One is the
intersection of two disks in the upper fundamental domain that must be
connected by the properties of the cover and hence is a disk. The other
component is the double branched cover of a disk, branched over 4 points all
of which have branching index 2. By another Riemann–Hurwitz calculation this
is a surface of genus 1 with 2 boundary components. ∎

Observe that $b_{n}$ factors as $(PS)\cdot(S^{-1}F^{n}S)\cdot F^{-n}.$ Let $C$
and $D$ denote the $S_{2,1}$ component of $C^{+}$ and $D^{+}$ respectively.

######  Lemma 3.2.

$F$ is a partial pseudo-Anosov with $\text{supp}(F)=D$ and
$\text{supp}(S^{-1}FS)=C$.

###### Proof.

Recall that $p^{-1}(\text{supp}(\Phi))=p^{-1}(\Gamma)=C^{+}.$ By lemma 3.1
$C^{+}$ is the disjoint union of $C$ and a disk. Moreover, the component of
$C^{+}$ that contains branching index one points is the disk component again
by lemma 3.1 (see also figure 3). We observe that $C^{+}-C$ and $D^{+}-D$ are
both disks and hence any mapping class restricted to either one is trivial.
Note that $p$ restricted to the subsurface $C$ that contains only index two
points is a double branched cover of $\Gamma.$ Since $\Phi$ is a pseudo-Anosov
on $\Gamma$ it preserves two transverse singular foliations on $\Gamma.$ The
foliations lift through the branched cover to give singular transverse
foliations on the surface $C.$ Any possible 1-prong singularities at punctures
of $\Omega_{n}$ are double covered in $C$ and $D$ and hence become 2-prong
singularites. Therefore, $F$ is a partial pseudo-Anosov with
$\text{supp}(F)=D$ and $\text{supp}(S^{-1}FS)=C.$ See section 14.1 of [FM12]
for more on pseudo-Anosov mapping classes arising from branched covers.

∎

We will need the following notation. Let $S$ be a surface and let
$\alpha,\>\beta\subseteq S$ be essential simple closed curves on $S.$ Let
$d_{S}(\alpha,\beta)$ denote distance in the curve graph of $S$ between the
vertices of the curve graph that represent the simple closed curves $\alpha$
and $\beta.$ Now, let $Y\subseteq S$ be a subsurface of $S.$ Given a curve
$\eta\subseteq S$ isotope $\eta$ so that it intersects $Y$ minimally. Define
$\pi_{Y}(\eta)$ to be the arcs of $\eta\cap Y$ or the curve $\eta$ in the case
that $\eta\subseteq Y.$ If $\eta$ projects to a collection of arcs in $Y,$
then we arbitrarily choose a single arc $\eta^{\prime}$ of $\pi_{Y}(\eta)$ and
”close it up” into a curve in the following way. If $\eta^{\prime}$ has
endpoints on a single component of $\partial Y$ then let
$\overline{\eta^{\prime}}$ denote the curve obtained by concatenating the
endpoints of $\eta^{\prime}$ with a sub arc of $\partial Y.$ If
$\eta^{\prime}$ has endpoints on distinct components of $\partial Y$ then a
neighborhood $P\subseteq Y$ of $\eta^{\prime}\cup\partial Y$ is homeomorphic
to a pair of pants. Let $\overline{\eta^{\prime}}$ be the component of
$\partial P$ disjoint from $\partial Y$ (see figure 4). Now let
$\alpha,\beta\subseteq S$ be two simple closed curves that project to arcs in
$Y.$ Let
$d_{Y}(\alpha,\beta)=d_{\mathcal{C}(Y)}(\overline{\alpha^{\prime}},\overline{\beta^{\prime}}),$
if $\beta$ projects to a closed curve then let
$d_{Y}(\alpha,\beta)=d_{\mathcal{C}(Y)}(\overline{\alpha^{\prime}},\beta),$
and if both $\alpha$ and $\beta$ project to closed curves then
$d_{Y}(\alpha,\beta)=d_{\mathcal{C}(Y)}(\alpha,\beta).$ Note that we made
several choices in defining distance in a subsurface. For example we could
have chosen a different (but disjoint) arc or closed up the arc differently.
Hence, our definition is only coarsely well-defined up to a small additive
error.

######  Lemma 3.3.

$b_{n}$ has linearly growing translation distance in the curve complexes of
$C$ and $D.$ In particular there exists some $n_{0}>0$ such that $b_{n}$ is
not periodic for all $n\geq n_{0}.$

###### Proof.

By lemma 3.2, $S^{-1}F^{n}S$ and $F^{-n}$ are each pseudo-Anosov on their
support, namely $C$ and $D$. By lemma 3.1 $C\cap D$ is homeomorphic to a
surface with genus 1 and 2 boundary components. Hence, Theoreom 5.2 of [CLM12]
implies that $(S^{-1}F^{n}S)\cdot F^{-n}$ has linearly growing translation
distance in the curve complexes of each of $C$ and $D$. Therefore, there
exists some $n_{0}$ such that for all $n\geq n_{0},\>b_{n}$ has large
translation distance in each of $C$ and $D,$ and hence is infinite order. ∎

While we have shown that our monodromies are eventually not periodic, we need
monodromies that are eventually pseudo-Anosov. Since, this may not be true for
$b_{n}$, we need to modify our manifolds $N_{n}$ slightly. To that end we
introduce a more 2-dimensional way of discussing fibered links in closed
oriented 3-manifolds. We will use the equivalent notion of an open book
decomposition of our closed, oriented 3-manifold $M.$ An open book
decomposition of $M$ is a pair $(S,\varphi)$ where $S$ is the fiber surface
($\partial S\neq\varnothing$) and $\varphi$ is the monodromy. The mapping
torus $M_{\varphi}$ of $\varphi$ is homeomorphic to a link complement in $M.$
Then $M$ is homeomorphic to the quotient of $M_{\varphi}$ under the
identification $(x,t)\sim(x,t^{\prime})$ where $x\in\partial S$ and
$t,t^{\prime}\in[0,1].$ The quotient of $\partial M_{\varphi}$ under the above
identification is the fibered link in $M$ which is called the binding of
$(S,\varphi).$ We now describe an operation on an open book decomposition
$(S,\varphi)$ of $M$ called a stabilization.

######  Definition 3.4.

Let $(S,\varphi)$ be an open book decomposition of $M,$ and let
$(\gamma,\partial\gamma)\subseteq(S,\partial S)$ be a properly embedded arc. A
stabilization along $\gamma$ of the open book decomposition $(S,\varphi)$ is
the open book decomposition $(S^{\prime},\varphi^{\prime})$ of $M.$ where
$S^{\prime}$ is obtained by attaching a 1-handle to $S$ that connects the
endpoints of $\gamma$ on $\partial S$ and $\varphi^{\prime}=\varphi\circ
T_{\gamma^{\prime}}$ where $\gamma^{\prime}$ is the curve that agrees with
$\gamma$ in $S$ and intersects the co-core of the 1-handle once (see figure 4)
and $T_{\gamma^{\prime}}$ denotes the Dehn twist along $\gamma^{\prime}.$

\begin{overpic}[width=227.62204pt]{stabilization.PNG}
\put(53.0,55.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
$\overline{\gamma}$}
\put(89.0,19.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
$\gamma^{\prime}$} \par\end{overpic} Figure 4. The surface obtained by
stabilizing $S_{2,2}.$ The curve $\gamma^{\prime}$ is the curve we twist along
while stabilizing. The curve $\overline{\gamma}$ is the result of the ”closing
up” of the arc $\gamma$ which agrees with $\gamma^{\prime}$ on $S_{2,2}.$

The following proposition is essentially theorem 1.1 of [CH08]. In the same
way as described when we defined distance in the curve graph of a subsurface,
we associate to any arc $\gamma$ a simple closed curve $\overline{\gamma}.$

######  Proposition 3.5.

Let $(S_{g-1,2},\varphi)$ be an open book decomposition of a closed, oriented
3-manifold $M$ such that $\varphi$ is not periodic. Let $\overline{\gamma}$ be
the simple closed curve associated to $\gamma$ as described above. If
$d_{S_{g-1,2}}(\overline{\gamma},\varphi(\overline{\gamma}))=N>16$ then
stabilization along the arc $\gamma$ yields an open book decomposition
$(S_{g,1},\varphi\circ T_{\gamma^{\prime}})$ such that $\varphi\circ
T_{\gamma^{\prime}}$ is pseudo-Anosov. Recall that $\gamma^{\prime}$ is the
extension of $\gamma$ to the stabilized surface and $d_{S}(\cdot,\cdot)$
denotes the distance in curve graph of $S.$

###### Proof.

We note here that while theorem 1.1 of [CH08] contains the hypothesis that the
monodromy $\varphi$ is right-veering this is not needed to show that
$\varphi\circ T_{\overline{\gamma}}$ is pseudo-Anosov. All that is required is
that $\varphi$ is not periodic. While the lemma is essentially theorem 1.1 of
[CH08], the required distance between $\overline{\gamma}$ and its image under
$\varphi$ is not made explicit there. For the sake of completeness, we sketch
their proof and make the distance explicit.

Let $\varphi^{\prime}=\varphi\circ T_{\gamma^{\prime}}.$ We argue that
$\varphi^{\prime}$ is psuedo-Anosov by showing that it is neither reducible
nor periodic. We will first argue that $\varphi^{\prime}(\delta)\neq\delta$
for any multicurve $\delta\subseteq S_{g,1}.$

First suppose that $\delta\subseteq S_{g-1,2}\subseteq S_{g,1}.$ If
$i(\varphi(\delta),\gamma^{\prime})=n$ then there is a representative, $g$ of
$\varphi^{\prime}(\delta)$ that intersects the co-core of the 1-handle, $a,$
$n$ times. If $i(\varphi^{\prime}(\delta),a)<n$ then there is a bigon
consisting of a subarc of $a$ and an arc of $g.$ Then one checks that this
implies there is a bigon consisting of an arc of $\gamma$ and an arc of
$\varphi(\delta).$ See the third paragraph of the proof of case 1 of theorem
1.1 of [CH08] for this argument. This is a contradiction if $n>0.$ If $n=0$
then $i(\varphi(\beta),\overline{\gamma})=0$ for any component $\beta$ of
$\delta.$ Hence, $d(\varphi(\beta),\overline{\gamma})=1$ for all such $\beta.$
However, $d(\overline{\gamma},\varphi(\overline{\gamma}))=N$ and therefore,
$d(\varphi(\overline{\gamma}),\varphi(\beta))\geq N-1$ for every $\beta.$
Since $n=0,$ we see that $i(\varphi^{\prime}(\delta),\overline{\gamma})=0.$
This yields contradiction so long as $N>2.$

Now suppose that $\delta\nsubseteq S_{g-1,2}.$ Let $i(\delta,a)=k>0.$ Let
$B=S_{g,1}-S_{g-1,2}$ denote the stabilization band. Normalize $\delta$ so
that it intersects $\gamma^{\prime}$ and $a$ transversely and efficiently. We
then subdivide $\delta$ into arcs
$\delta_{1},\dots,\delta_{k},\delta_{1}^{\prime},\dots,\delta_{k}^{\prime}$ so
that $\delta_{i}\subseteq S_{g-1,2}$ and $\delta_{i}^{\prime}\subseteq B.$ The
$\delta_{i}^{\prime}$ are all linear arcs in $B.$ If
$i(\delta_{i}^{\prime},\gamma^{\prime})=0$ we call such an arc vertical. If
$i(\delta_{i}^{\prime},\gamma^{\prime})>0$ we say it has positive slope or
negative slope. See figure 5. We normalize and subdivide $\varphi(\delta)$ in
the same way. Up to isotopy we may assume that there is no triangle contained
in $S_{g-1,2}$ with boundary an arc of $\gamma^{\prime},$ an arc of $\delta,$
and an arc of $\partial S_{g-1,2}.$ Now let $m$ denote the number of arcs of
$\varphi(\delta)\cap B$ that are not vertical. Let $n$ denote the number of
intersections between $\varphi(\delta)$ and $\gamma^{\prime}$ that are outside
of $B$ so that $i(\varphi(\delta),\gamma^{\prime})=m+n.$ Note that this
definition of $n$ agrees with the definition in the above paragraph if
$\varphi(\delta)$ does not intersect the band $B.$ Colin and Honda show that
$i(\varphi^{\prime}(\delta),a)=k\pm m+n$ where the sign depends on whether the
non-vertical arcs have positive or negative slope. If $m\neq n$ then
$i(\varphi^{\prime}(\delta),\gamma^{\prime})\neq i(\delta,a)=k,$ so
$\varphi^{\prime}(\delta)\neq\delta.$ It remains to deal with the case that
$m=n.$

Suppose that $m=n.$ Then
$i(\varphi^{\prime}(\delta),\gamma^{\prime})=m+n=2m\leq 2k$ by the definitions
of $m$ and $k.$ Suppose for the sake of contradiction that
$\varphi^{\prime}(\delta)=\delta.$ Then
$i(\delta,\gamma^{\prime})=i(\varphi^{\prime}(\delta),\gamma^{\prime}).$
Therefore $i(\delta,\gamma^{\prime})\leq 2k,$ and hence there is some $1\leq
i\leq k$ such that $i(\delta_{i},\gamma)\leq 2k/k=2.$ Now we note that

$i(\overline{\delta_{i}},\overline{\gamma})\leq 2+4=6.$

By Hempel’s lemma (lemma 2.1 of [Hem97])

$d(\overline{\delta_{i}},\overline{\gamma})\leq\lfloor
2\log_{2}(6)+2\rfloor=7.$

Observe that $\overline{\delta_{i}}$ and $\overline{\delta_{j}}$ cannot
jointly fill the surface, so

$d(\overline{\delta_{i}},\overline{\delta_{j}})\leq 2\>\forall i,j$

Therefore,

$d(\overline{\delta_{j}},\overline{\gamma})\leq 2+7=9\>\forall j.$

Applying $\varphi$ yields

$d(\varphi(\overline{\delta_{j}}),\varphi(\overline{\gamma}))\leq 9\>\forall
j.$

Thus,

$d(\varphi(\overline{\delta_{j}}),\overline{\gamma})\geq N-9.$

Applying Hempel’s lemma again,

$i(\varphi(\overline{\delta_{j}}),\overline{\gamma})\geq 2^{(N-9)/2-1}.$

Hence,

$i(\varphi(\delta_{j}),\gamma)\geq 2^{(N-9)/2-1}-4.$

Therefore,

$i(\varphi(\delta),\gamma^{\prime})\geq(2^{(N-9)/2-1}-4)k>2k$

this is a contradiction so long as $N>16$ which proves that $\varphi^{\prime}$
is not reducible.

It just remains to show that $\varphi^{\prime}$ is not periodic. Colin and
Honda show that $\varphi^{\prime}$ is not periodic by considering one of the
connected components, $\eta,$ of $\partial S_{g-1,2}$ as a curve in $S_{g,1}.$
They argue that $i((\varphi^{\prime})^{n}(\eta),a)\rightarrow\infty$ as
$n\rightarrow\infty.$ ∎

\begin{overpic}[width=227.62204pt]{stabilization_slopes.png}
\put(10.0,60.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
\small$\delta$}
\put(63.0,60.0){\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}
\small$\delta$}
\put(18.0,5.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
\small$\gamma^{\prime}$}
\put(80.0,5.0){\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
\small$\gamma^{\prime}$}
\put(34.0,30.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\small$a$}
\put(89.0,30.0){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}
\small$a$} \end{overpic} Figure 5. On the left is the case that $\delta$ has
negative slope, and on the right is the case that $\delta$ has positive slope.
In both cases $k=5,$ $m=3,$ and $n=4.$

We now apply the above lemma to $M-w_{n}$ to obtain the following:

######  Proposition 3.6.

Let $(S_{g-1,2},b_{n})$ be the open book decomposition of $M$ corresponding to
the fibered link $w_{n}\subseteq M.$ There exists an arc of stabilization
$\gamma$ and $n_{1}\geq n_{0}>0$ such that $b_{n}\circ T_{\gamma^{\prime}}$ is
pseudo-Anosov for all $n\geq n_{1},$ and hence $M-\overline{w_{n}}$ is
hyperbolic, where $\overline{w_{n}}$ is the image of $w_{n}$ after
stabilization.

###### Proof.

We first explain how to choose an arc of stabilization $\gamma$ such that
$d_{S_{g-1,2}}(\overline{\gamma},b_{n_{0}}(\overline{\gamma}))>16$ following
[CH08]. First let $\gamma_{0}$ be any properly embedded arc connecting the two
boundary components of $S_{g-1,2}.$ Fix a pseudo-Anosov
$f\in\operatorname{Mod}(S_{g-1,2})$ that does not share a stable or unstable
lamination with $b_{n_{0}}.$ Let $\nu$ be the stable lamination of $f.$ The
sequence $f^{i}(\overline{\gamma_{0}})\rightarrow\nu$ as $i\rightarrow\infty.$
We claim that
$d_{S_{g-1,2}}(f^{i}(\overline{\gamma_{0}}),b_{n_{0}}(f^{i}(\overline{\gamma_{0}})))\rightarrow\infty.$
If
$d_{S_{g-1,2}}(f^{i}(\overline{\gamma_{0}}),b_{n_{0}}(f^{i}(\overline{\gamma_{0}})))$
does not approach $\infty,$ then up to passing to a subsequence
$d_{S_{g-1,2}}(f^{i}(\overline{\gamma_{0}}),b_{n_{0}}(f^{i}(\overline{\gamma_{0}})))=N$
for some constant $N.$ Let
$f^{i}(\overline{\gamma_{0}})=v_{i,0},v_{i,1},\dots,v_{i,N-1},v_{i,N}=b_{n_{0}}(f^{i}(\overline{\gamma_{0}}))$
be a geodesic in the curve graph. Note that $v_{i,0}$ and $v_{i,1}$ are
disjoint. Up to passing to a subsequence, $v_{i,1}$ converges to a lamination
$\nu^{\prime}$ such that $i(\nu,\nu^{\prime})=0.$ Similarly, $v_{i,N-1}$
converges to a lamination $\lambda$ such that $i(\lambda,b_{n_{0}}(\nu))=0.$
Since $\nu$ is minimal, $\nu=\nu^{\prime}$ and $b_{n_{0}}(\nu)=\lambda.$
Hence, we have $v_{i,1}\rightarrow\nu$ and $v_{i,N-1}\rightarrow
b_{n_{0}}(\nu)$ as $i\rightarrow\infty,$ but
$d_{S_{g-1,2}}(v_{i,1},v_{i,N-1})<N.$ After repeating this process finitely
many times, we conclude that $b_{n_{0}}(\nu)=\nu.$ This contradicts the choice
of $f.$ Therefore, there exists some $m>0$ such that
$d_{S_{g-1,2}}(f^{m}(\overline{\gamma_{0}}),b_{n_{0}}(f^{m}(\overline{\gamma_{0}})))>16.$
Moreover, there exists some $l_{0}>0$ so that for all $l\geq l_{0}$
$d_{S_{g-1,2}}(f^{l}(\overline{\gamma_{0}}),\partial C)>18.$ Let
$k=\max\\{m,l\\}.$ Let $\gamma$ be the arc such that
$f^{k}(\overline{\gamma_{0}})=\overline{\gamma}.$

Note that although the above only shows how to choose an arc of stablization
for $b_{n_{0}}$ we may choose such an arc of stabilization uniformly for all
open book decompositions $(S_{g-1,2},b_{n})$ where $n$ is large enough. To see
why this is true consider the subsurface $C.$ Observe that
$d_{C}(\overline{\gamma},b_{n}(\overline{\gamma}))\approx
d_{C}(\overline{\gamma},S^{-1}F^{n}SF^{-n}(\overline{\gamma})).$ By lemma 3.3
there exists $n_{1}\geq n_{0}>0$ such that
$d_{C}(\overline{\gamma},S^{-1}F^{n}SF^{-n}(\overline{\gamma}))\gg 0$ for all
$n\geq n_{1}.$ Hence, by the bounded geodesic image theorem (theorem 3.1 of
[MM00]) the geodesic in $\mathcal{C}(S_{g-1,2}),$ the curve graph of
$S_{g-1,2},$ between $\overline{\gamma}$ and $b_{n}(\overline{\gamma})$ must
pass within distance 1 of $\partial C$ for all $n\geq n_{1}.$ Therefore we
have the following inequality.

$d_{S_{g-1,2}}(\overline{\gamma},\partial C)-1\leq
d_{S_{g-1,2}}(\overline{\gamma},b_{n}({\overline{\gamma})})$

Now note that $d_{S_{g-1,2}}(\overline{\gamma},\partial C)$ is unchanged as
$n\rightarrow\infty,$ and hence
$d_{S_{g-1,2}}(\overline{\gamma},b_{n}({\overline{\gamma})})>16$ for all
$n\geq n_{1}.$ Thus, by proposition 3.5 the family of open book decompositions
$(S_{g,1},b_{n}\circ T_{\gamma^{\prime}})$ have pseudo-Anosov monodromies for
all $n\geq n_{1}.$ ∎

######  Remark 3.7.

Note that stabilization does not affect the conclusions of lemmas 3.2 and 3.3.
We consider the subsurfaces $C$ and $D$ included into the stabilized surface
and the mapping class $F$ as a mapping class on the stabilized surface. Then,
$b_{n}\circ T_{\gamma^{\prime}}$ has linearly growing translation distance in
the curve graphs of $C$ and $D,$ and $F$ is a partial pseudo-Anosov on
$S_{g,1},$ the stabilized surface.

######  Proposition 3.8.

In fixed genus $g,$ as $n$ tends to infinity, the knot complements
$M_{n}=M-\overline{w_{n}}$ are eventually hyperbolic, with volumes tending to
infinity.

###### Proof.

As is shown in proposition 3.6 $b_{n}^{\prime}=b_{n}\circ T_{\gamma^{\prime}}$
are pseudo-Anosov for $n\geq n_{1}.$ Hence, their mapping tori, $M_{n},$ are
hyperbolic for $n\geq n_{1}.$ As is noted in remark 3.7 and lemma 3.3,
$b_{n}^{\prime}$ has linearly growing subsurface projections to $C_{n}$ and
$D_{n}.$ Now, by applying the Masur–Minsky distance formula for the pants
graph (see theorem 6.12 of [MM00]) we see that $b_{n}^{\prime}$ has linearly
growing translation distance in the pants graph of $S_{g,1}$. By the work of
Brock [Bro03], the mapping tori of $b_{n}^{\prime}$ have linearly growing
volume. These mapping tori are precisely the fibered knot complements $M_{n}.$
∎

For fixed $g$ the family $\overline{w_{n}}$ are the promised family of fibered
knots in $M$ satisfying the conclusion of Theorem 1.1.

## References

  * [Ale23] James W Alexander.  A lemma on systems of knotted curves.  Proceedings of the National Academy of Sciences, 9(3):93–95, 1923\.
  * [BFPS23] Kenneth L. Baker, David Futer, Jessica S. Purcell, and Saul Schleimer.  Large volume fibred knots of fixed genus.  Mathematical Research Letters, 2023.  to appear.
  * [Bro03] Jeffrey F. Brock.  Weil-petersson translation distance and volumes of mapping tori.  Comm. Anal. Geom., 11(5):987–999, 2003.
  * [CH08] Vincent Colin and Ko Honda.  Stabilizing the monodromy of an open book decomposition.  Geom. Dedicata, 132(1):95–103, 2008.
  * [CLM12] Matt T Clay, Christopher J Leininger, and Johanna Mangahas.  The geometry of right-angled artin subgroups of mapping class groups.  Groups, Geometry, and Dynamics, 6(2):249–278, 2012.
  * [FM12] Benson Farb and Dan Margalit.  A primer on mapping class groups, volume 49 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 2012.
  * [Ful01] Terry Fuller.  On fiber-preserving isotopies of surface homeomorphisms.  Proc. Amer. Math. Soc., 129(4):1247–1254, 2001.
  * [GK87] David Gabai and William H. Kazez.  The classification of maps of surfaces.  Invent. Math., 90:219–242, 1987.
  * [Hem97] J. P. Hempel.  3-manifolds as viewed from the curve complex.  Topology, 40:631–657, 1997.
  * [Hil76] Hugh M. Hilden.  Three-fold branched coverings of $S^{3}$.  Am. J. Math., 98(4):989–997, 1976.
  * [HKK23] Susumu Hirose, Efstratia Kalfagianni, and Eiko Kin.  Volumes of fibered 2-fold branched covers of 3-manifolds.  Journal of Topology and Analysis, 2023.  to appear.
  * [MM00] Howard A. Masur and Yair N. Minsky.  Geometry of the complex of curves. ii. hierarchical structure.  Geom. Funct. Anal., 10(4):902–974, 2000.
  * [Mon76] José M. Montesinos.  Three-manifolds as 3-fold branched covers of $S^{3}$.  Q. J. Math., 27(1):85–94, 1976.
  * [Mye78] Robert Myers.  Open book decompositions of 3-manifolds.  Proc. Amer. Math. Soc., 72(2):397–402, 1978.
  * [Win15] Rebecca R. Winarski.  Symmetry, isotopy, and irregular covers.  Geom. Dedicata, 177(1):213–227, 2015.