# Stable isoperimetric ratios and the Hodge Laplacian of hyperbolic manifolds

Cameron Gates Rudd

###### Abstract

We show that for a closed hyperbolic 3-manifold, the size of the first
eigenvalue of the Hodge Laplacian acting on coexact 1-forms is comparable to
an isoperimetric ratio relating geodesic length and stable commutator length
with comparison constants that depend polynomially on the volume and on a
lower bound on injectivity radius, refining estimates of Lipnowski and Stern.
We use this estimate to show that there exist sequences of closed hyperbolic
3-manifolds with injectivity radius bounded below and volume going to infinity
for which the 1-form Laplacian has spectral gap vanishing exponentially fast
in the volume.

##  1 Introduction

The spectrum of the Hodge Laplacian is a fundamental and well studied
geometric invariant of Riemannian manifolds. The Hodge theorem partitions the
positive spectrum into exact and coexact eigenvalues. For differential forms
of degree one, the exact eigenvalues contain exactly the data of the Laplacian
acting on functions and are well understood. The coexact spectrum however is
considerably more mysterious. Recently, the first coexact eigenvalue of the
Hodge Laplacian of a closed hyperbolic 3-manifold has been related to other
aspects of its geometry and topology. For the function Laplacian, the first
eigenvalue is known to be comparable to the square of the isoperimetric
Cheeger constant. In this paper, we derive a similar estimate for the first
coexact eigenvalue, building on work of Lipnowski and Stern in [LS18]
motivated by torsion growth in finite covers. We use this new estimate to
construct the first examples of hyperbolic 3-manifolds with coexact
eigenvalues exponentially small compared to volume.

Given a hyperbolic 3-manifold $M$, it is natural to try to extract information
about $M$ from its finite covers. A deep and interesting conjecture of
Bergeron-Venkatesh, Lê, and Lück (see [BV13], [Lê18], and [Lüc16]) asks in
part whether the volume of $M$ can be found by studying the torsion in the
homology of a family of finite covers of $M$. In studying this question,
Bergeron, Şengün, and Venkatesh in [BŞV16] relate the growth rate of the
cardinality of the torsion in the first homology of a tower of covers of a
closed arithmetic hyperbolic 3-manifold to the spectrum of the Laplacian on
1-forms. In particular, they prove the following theorem, where the technical
definitions are given below.

######  Theorem 1.1.

([BŞV16]) If a sequence $M_{n}\to M_{0}$ of congruence covers of an arithmetic
hyperbolic 3-manifold $M_{0}$ satisfies the few small eigenvalues, small Betti
numbers, and simple cycles conditions, then the log torsion growth rate is
proportional to the volume. In particular, one has

$\lim\limits_{n\to\infty}\frac{\log\left|H_{1}(M_{n};{\mathbb
Z})_{\emph{torsion}}\right|}{\operatorname{vol}(M_{n})}=\frac{1}{6\pi}.$

The conditions appearing in Theorem 1.1. are:

  1. 1.

(Few small eigenvalues) For all $\varepsilon>0$ there is a $c>0$ such that

$\limsup_{n\to\infty}\frac{1}{\operatorname{vol}(M_{n})}\sum\limits_{0<\lambda<c}|\log\lambda|\leq\varepsilon,$

where $\lambda$ runs over the eigenvalues of the 1-form Laplacian on $M_{n}$.

  2. 2.

(Small Betti numbers)
$b_{1}(M_{n})=o\left(\frac{\operatorname{vol}(M_{n})}{\log\operatorname{vol}(M_{n})}\right)$.

  3. 3.

(Simple cycles) There exists a constant $C$ depending on $M_{0}$ such that
$H_{2}(M_{n};{\mathbb Z})$ admits a basis of surfaces $[S_{i}]$ with bounded
Thurston norm $||[S_{i}]||_{Th}\ll\operatorname{vol}(M)^{C}$.

Bergeron, Şengün, and Venkatesh’s theorem provides a version of Lück
approximation for the limiting $L^{2}$-torsion of sequences of manifolds that
satisfy these conditions. However, no examples of such a sequence are known to
exist. Encouraged by the Betti number approximation theorem of [ABB+17], one
would hope the above conditions also imply the convergence of $L^{2}$-torsion
for families of manifolds that Benjamini-Schramm converge to $\mathbb H^{3}$.
Examples of Brock and Dunfield show that Benjamini-Schramm convergence itself
is insufficient [BD15].

Previous work has focused on the simple cycles condition. In their paper,
Bergeron, Şengün, and Venkatesh conjecture the simple cycles condition is
satisfied for all arithmetic congruence covers, and ask in contrast if there
are families of closed hyperbolic 3-manifolds with injectivity radius bounded
below and volume going to infinity that do not satisfy it. This question was
answered by Brock and Dunfield in [BD17], where they construct a sequence of
closed hyperbolic 3-manifolds $W_{n}$ with injectivity radius bounded below
and volume going to infinity such that $H_{2}(W_{n};{\mathbb Z})\cong{\mathbb
Z}$ for every $n$ and for which the Thurston norm of the generator grows
exponentially in the volume.

In this paper, we continue the study of the few small eigenvalues condition
initiated by Lipnowski and Stern in [LS18]. There, it is shown that for a
family of covers of a triangulated closed hyperbolic $n$-manifold, the first
positive eigenvalue of the Laplacian acting on 1-forms is comparable to a
certain isoperimetric ratio, where the comparison constants depend on the
volume of the cover, the geometry of the base, and the specific triangulation.
In this paper, we prove that for closed hyperbolic 3-manifolds satisfying a
uniform lower bound on injectivity radius, such a comparison can be done with
universal constants and polynomial dependence on volume (Theorems A and B,
with only Theorem A requiring the restriction to dimension 3). We then
leverage the comparison in Theorem A to show that a specific family of closed
hyperbolic 3-manifolds have first positive eigenvalue of the 1-form Laplacian
vanishing exponentially fast in the volume (Theorem C). The manifolds in
Theorem C are not covers of a fixed base, so the estimates of [LS18] do not
apply.

Another motivation for this work comes from recent work of Lin and Lipnowski
in [LL21], where for closed rational homology 3-spheres they leverage a
relationship between the first eigenvalue of the Hodge Laplacian acting on
coexact 1-forms and irreducible solutions to the Seiberg-Witten equations to
determine if certain spaces are $L$-spaces. In particular, if a closed
rational homology 3-sphere $M$ is not an $L$-space, then the first eigenvalue
$\lambda$ of the Hodge Laplacian acting on coexact 1-forms satisfies
$\lambda\leq 2$. Using a version of the Selberg trace formula, they relate
this to the complex length spectrum of $M$. Numerical methods can then be used
to verify if $\lambda\leq 2$. While the constants in our eigenvalue estimates
are rather opaque, so that the isoperimetric ratio we study is not, at least
presently, capable of certifying that $\lambda\leq 2$, the relationship
between the stable isoperimetric ratio and whether a space is an $L$-space
remains tantalizing.

###  1.1 Results

The isoperimetric ratio we study relates the topological complexity of a
surface with boundary to the geometric length of its boundary. One can view
the extremal value of this ratio as an analogue of the two-dimensional Cheeger
constant. The topological complexity measure is given by stable commutator
length. The stable commutator length of a nullhomologous loop $\gamma$ in a
manifold $M$, denoted $\operatorname{{\tt{scl}}}(\gamma)$, is defined to be

$\operatorname{{\tt{scl}}}(\gamma)=\inf\limits_{m\geq
1}\frac{{\tt{cl}}(\gamma^{m})}{m},$

where ${\tt{cl}}(\gamma)$ is the word length of $\gamma$ in the commutator
subgroup of $\pi_{1}M$ with generating set all commutators. Topologically,
stable commutator length corresponds to the stable complexity of a surface
bounding a nullhomologous loop. Denote the subgroup of rationally
nullhomologous loops by $\Gamma_{{\mathbb
Q}}\textquoteright=\ker\left(\pi_{1}M\to H_{1}(M;{\mathbb Q})\right)$. Then,
for $\gamma\in\Gamma\textquoteright_{{\mathbb Q}}$, one has

$\operatorname{{\tt{scl}}}(\gamma)=\inf\left\\{\frac{\chi_{-}(S)}{2m}~{}:~{}S\text{
with }\partial S=\gamma^{m},~{}S\text{ is connected}\right\\},$

where for connected surfaces $\chi_{-}(S)=\max\\{0,-\chi(S)\\}$. One can think
of stable commutator length as a relative version of the Thurston norm. See
the monograph [Cal09] for a detailed exposition of
$\operatorname{{\tt{scl}}}$. Stable commutator length is closely related to
area, by Gauss-Bonnet, providing some justification for the following
nomenclature. Define the stable isoperimetric constant of $M$ to be

$\rho(M)=\inf\limits_{\gamma\in\Gamma\textquoteright_{\mathbb
Q}\setminus\\{1\\}}\frac{|\gamma|}{\operatorname{{\tt{scl}}}(\gamma)}.$

One can also define the stable area of a rationally nullhomologous loop
$\gamma\in\Gamma\textquoteright_{{\mathbb Q}}$ to be the infimal normalized
area of a surface bounding a power of $\gamma$:

$\text{sArea}(\gamma)=\inf\limits_{\partial
S=\gamma^{n}}\frac{\text{Area}(S)}{n}.$

This leads to another notion of stabilized isoperimetric ratio using stable
area in place of stable commutator length:

$\rho_{\text{Area}}(M)=\inf\limits_{\gamma\in\Gamma\textquoteright_{{\mathbb
Q}}}\frac{|\gamma|}{\text{sArea}(\gamma)}.$

Stable area is related to stable commutator length by

$\text{sArea}(\gamma)\leq 4\pi\operatorname{{\tt{scl}}}(\gamma).$

Thus,

$\rho(M)\leq 4\pi\rho_{\text{Area}}(M).$

Very little is known generally about how geodesic length relates to stable
commutator length in hyperbolic manifolds, though estimates relating length,
stable area, and stable commutator length for short curves in hyperbolic
manifolds have been obtained by Calegari in [Cal08].

Our first theorem relates the coexact spectral gap to stable isoperimetric
ratios of arbitrary nullhomologous curves.

######  Theorem A.

Let $M$ be a closed hyperbolic 3-manifold with injectivity radius bounded
below by $\varepsilon>0$ and let $\lambda$ denote the first eigenvalue of the
Hodge Laplacian acting on coexact 1-forms. Then there is a constant
$A=A(\varepsilon)$ such that for any nontrivial element
$\gamma\in\Gamma\textquoteright_{\mathbb Q}$, one has

$\sqrt{\lambda}\leq
A\operatorname{vol}(M)\frac{|\gamma|}{\operatorname{{\tt{scl}}}(\gamma)},$

where $|\gamma|$ denotes the geodesic length of $\gamma$.

Theorem A has the following obvious corollary:

######  Corollary.

Let $M$ be a hyperbolic $3$-manifold with injectivity radius bounded below by
$\varepsilon>0$ and let $\lambda$ be the first eigenvalue of the Hodge
Laplacian acting on coexact 1-forms. Then for the constant $A=A(\varepsilon)$
from Theorem A,

$\sqrt{\lambda}\leq A\operatorname{vol}(M)\rho(M).$

The analogue of Theorem A in [LS18] studies a cochain version of the Hodge
Laplacian introduced by Dodziuk in [Dod76] for triangulated manifolds. This
chochain Laplacian is called the Whitney Laplacian, and is induced by the
Hodge Laplacian by embedding cochains into the $L^{2}$-de Rham complex.

######  Theorem 1.2.

([LS18] Theorem 1.4) Let $M_{0}$ be a closed hyperbolic $n$-manifold. Let
$K_{0}$ be a sufficiently fine triangulation. Let $M$ be a finite cover of
$M_{0}$. Let $\lambda_{W}(M)_{d^{*}}$ be the first coexact eigenvalue for the
Whitney cochain Laplacian associated to the pullback of the triangulation
$K_{0}$ to $M$. Then if some multiple of $\gamma\in\pi_{1}(M)$ bounds a
surface, then

$\left(\frac{\operatorname{{\tt{scl}}}(\gamma)}{|\gamma|}\right)^{2}\leq
W_{M_{0}}\frac{\operatorname{vol}(M)}{\lambda_{W}(M)_{d^{*}}},$

for a constant $W_{M_{0}}$ depending on the triangulation $K_{0}$.

Under a well behaved sequence of subdivisions, Dodziuk and Patodi in [DP76]
showed the spectrum of the Whitney Laplacian converges to the spectrum of the
Hodge Laplacian. For a fixed but very fine triangulation, the eigenvalue
comparison is somewhat delicate. In this setting, Lipnowski and Stern relate
the Whitney Laplacian’s first coexact eigenvalue to the Hodge Laplacian’s
first coexact eigenvalue in the following way:

######  Theorem 1.3.

([LS18] Theorem 1.5) Let $M_{0}$ be a closed hyperbolic $n$-manifold. Let
$K_{0}$ be a sufficiently fine triangulation of $M_{0}$. Let $M$ be a finite
cover of $M_{0}.$ Let $\lambda_{W}(M)_{d^{*}}$ be the first coexact eigenvalue
for the Whitney cochain Laplacian associated to the pullback of the
triangulation $K_{0}$ to $M$. Then,

$\frac{1}{\lambda_{W}(M)_{d^{*}}}\leq\max\left\\{\frac{4G_{M_{0}}^{2}\operatorname{vol}(M)}{\lambda_{d^{*}}(M)},G_{M_{0}}^{2}C_{M_{0}}^{2}\operatorname{vol}(M)\right\\}.$

The constants $C_{M_{0}}$ and $G_{M_{0}}$ depend only on $K_{0}$.

In the course of proving Theorem A, we too require a comparison of this sort.
By using a smoothed version of the Whitney embedding of cochains into the de
Rham complex and triangulations with uniformly controlled geometry (these are
called deeply embedded triangulations and are introduced in Section 2), we
prove the following Whitney-Hodge eigenvalue comparison.

######  Theorem 1.4.

Let $M$ be a closed hyperbolic 3-manifold with
$\operatorname{inj}(M)>\varepsilon$. Let $\lambda$ denote the first coexact
eigenvalue for the Hodge Laplacian acting on $1$-forms and let $\lambda_{W}$
denote the first coexact eigenvalue for the smoothened Whitney Laplacian on
1-cochains associated to a deeply embedded triangulation. There is a constant
$G=G(\varepsilon)$ such that

$\lambda\leq G\operatorname{vol}(M)\lambda_{W}.$

The smoothened version of the Whitney map needed for the above proposition is
obtained by replacing the barycentric coordinates associated to a
triangulation with certain smooth partitions of unity indexed by the vertices
of a triangulation (an idea of Dodziuk’s [Dod81]), which we call barycentric
partitions of unity. One can then show that the forms built from these pieces
have well behaved Hodge decompositions. This analysis leads to the above
eigenvalue comparison for the smoothened Whitney Laplacian constructed from a
barycentric partition of unity.

Our second theorem uses the isoperimetric constant $\rho(M)$ to provide a
lower bound on the first coexact eigenvalue of the 1-form Laplacian.

######  Theorem B.

Let $M$ be a closed hyperbolic $n$-manifold with
$\operatorname{inj}(M)>\varepsilon$. Let $\lambda$ be the first positive
eigenvalue for the Hodge Laplacian acting on coexact 1-forms and let
$H>\lambda$. Then there is a constant $P(H,\varepsilon,n)>0$ such that

$\frac{P\rho(M)}{\operatorname{vol}(M)^{7/2+1/n}}\leq\sqrt{\lambda}.$

Theorem B corresponds to Theorem 1.5 below from [LS18], which uses stable area
in place of stable commutator length. Note that Theorem B above remains true
if one replaces $\rho(M)$ with $\rho_{\text{Area}}(M)$.

######  Theorem 1.5.

([LS18] Theorem 1.3 ) Let $M_{0}$ be a closed hyperbolic $n$-manifold and let
$M$ be a finite cover of $M_{0}.$ Then there are constants $A_{0}$ and $C$,
where $A_{0}$ is depends only on $M_{0}$ and $C$ is a constant that is
uniformly bounded when the injectivity radius of $M$ is bounded below and
$\lambda_{1}^{1}(M)$ is bounded above, for which

$\frac{1}{\lambda_{1}^{1}(M)_{d^{*}}}\leq
A_{0}C^{2}\operatorname{vol}(M)^{3/2}\operatorname{diam}(M)^{2}\left(1+\rho_{\emph{Area}}(M)^{-1}\right).$

Our approach to proving Theorems A and B is grounded in the following dual
characterizations of $\operatorname{{\tt{scl}}}$. By Bavard duality, stable
commutator length is related to the defect norm for quasimorphisms and the
Gersten filling norm for singular 1-chains. The Gersten filling norm for a
nullhomologous loop $\gamma$ is given by the infimal $\ell^{1}$-norm of a
singular 2-chain whose boundary is a fundamental cycle for $\gamma^{m}$,
normalized by $m$. A quasimorphism for a group $\Gamma$ is a map from
$\Gamma\to\mathbb R$ that is nearly a homomorphism in the sense that its
coboundary is a bounded map on $\Gamma^{2}$. The defect of a quasimorphism is
the sup norm of its coboundary. Bavard duality says

$\operatorname{{\tt{scl}}}(\gamma)=4\operatorname{{\tt{fill}}}(\gamma)=\frac{1}{2}\sup\limits_{q}\frac{q(\gamma)}{D(q)},$

where the supremum is over all quasimorphisms $q$ and $D(q)$ is the defect of
$q$.

One can therefore use the characterization of $\operatorname{{\tt{scl}}}$ as a
filling norm when bounding it from above and, similarly, the quasimorphism
point of view when bounding it from below. For Theorem A, we relate the
filling norm to the spectrum of the Hodge Laplacian via the Whitney Laplacian
and Poincaré duality (which forces us to restrict to dimension 3). For Theorem
B, we use de Rham quasimorphisms, which are given by integrating coclosed
forms over geodesics. Studying the de Rham quasimorphism of a coexact
eigenform gives the connection to the spectrum of the Hodge Laplacian.

Our methods primarily differ from [LS18] in that instead of studying covers of
a fixed manifold with a specific triangulation, we use that closed hyperbolic
manifolds with injectivity radius greater than some $\varepsilon>0$ can all be
triangulated so that the simplices come from a compact collection, and instead
of using an $L^{2}$ discretization of the eigenvalue problem, we use a smooth
discretization. The local structure of these triangulations can then be
compared in a uniform way, thereby allowing us to relate various combinatorial
and geometric norms. By working in the smooth setting instead of the $L^{2}$
setting, we are able to make use of geometric estimates that require higher
regularity. This leads to the more direct Whitney-Hodge eigenvalue comparison
of Proposition 1.4.

As an application of the spectral gap estimate of Theorem A, we modify the
construction in [BD17] to construct a family of closed hyperbolic manifolds
for which we have control over the stable isoperimetric constant and which
have injectivity radius uniformly bounded below.

######  Theorem C.

There is a family $\\{W_{n}\\}$ of closed hyperbolic 3-manifolds with
injectivity radius bounded below by some $\varepsilon>0$ and volume growing
linearly in $n$ such that the 1-form Laplacian spectral gap vanishes
exponentially fast in relation to volume:

$\sqrt{\lambda(W_{n})}\leq
B\operatorname{vol}(W_{n})e^{-r\operatorname{vol}(W_{n})}$

where $r$ and $B$ are positive constants and $\lambda(W_{n})$ is the first
positive eigenvalue of the 1-form Laplacian on $W_{n}$.

The manifolds $W_{n}$ in Theorem $C$ are obtained by taking a hyperbolic
3-manifold with totally geodesic boundary and gluing it to itself using a
particular psuedo-Anosov with several useful properties. By [BMNS16], this
family has geometry that up to bounded error can be understood in terms of a
simple model family and therefore has the desired injectivity radius lower
bound and linear volume growth.

Using this model family, we show that one can find curves with uniformly
bounded length whose stable commutator length grows exponentially in the
volume. This task is rather delicate, as it is quite difficult to ensure that
the _stable_ commutator length of a given loop is large— one typically does
not know if passing to a power of the loop causes a drastic simplification in
commutator length. To overcome this, we require an algebraic condition on the
gluing map to control which surfaces bound the loops we study. We then use the
following homological isoperimetric estimate to control stable commutator
length.

######  Theorem 1.6.

Let $M$ be a compact oriented hyperbolic 3-manifold with totally geodesic
boundary $\partial M=S$. Let $\gamma$ be a geodesic multicurve in $S$ that is
rationally nullhomologous in $M$. Let $||\cdot||_{s,S}$ be the stable norm on
$H_{1}(S)$ induced by the Riemannian metric on $S.$ Then there is a constant
$D>0$ depending only on $M$ such that

$||[\gamma]||_{s,S}\leq D\operatorname{{\tt{scl}}}_{M}(\gamma).$

Theorem A then implies the first positive eigenvalue vanishes exponentially
fast. We stress that while the manifolds $W_{n}$ are constructed from the same
basic building blocks, they are not otherwise related to one another. In
particular, they are not finite covers of some fixed base manifold, so the
estimates of [LS18] do not apply; it is therefore vital that the constants
appearing in Theorem A depend only on an injectivity radius lower bound.

###  1.2 Brief Outline

In Section 2, we show existence and study the local properties of the
triangulations we use throughout the paper. We also introduce the smooth
Whitney cochain map used to define the approximation of the Laplacian. In
Section 3, we relate various chain and cochain norms and compare these to
various geometric norms. In Section 4, we compare the eigenvalues of the
approximation of the Laplacian to the genuine eigenvalues of the Laplacian,
then use this comparison and the estimates from Section 3 to prove Theorem A.
In Section 5, we prove Theorem B. Finally, in Section 6, we compare the
homological length and stable commutator length of certain curves and
construct the example of Theorem C. Section 6 makes use of Theorem A, but is
otherwise independent from the rest of the paper.

######  Remark 1.

Throughout the paper, numerous constants are used. Constants defined inside
proofs have no meaning outside the local setting of the proof. The letter $C$
is repeatedly reused in Sections 2 and 3 to denote a constant coming from a
Sobolev type estimate and can at any time be taken to be the maximum among all
constants denoted by $C$. Such constants depend only on injectivity radius and
local choices of things like bump functions, unless specifically noted.

## Acknowledgments

Countless thanks are due to my advisor Nathan Dunfield for suggesting the
questions considered in this paper and for invaluable support. Thanks to
Michael Lipnowski and Joel Hass for answering some questions. Thanks to the
anonymous referee for helpful comments that improved this paper. This work was
partially supported by US NSF grant DMS-1811156.

##  2 Triangulations and Whitney forms

The purpose of this section is to outline the basic properties of the
triangulations we use in this paper and how the Whitney map relates these
triangulations to the de Rham complex.

###  2.1 Deeply embedded triangulations

In this section we study certain triangulations, called deeply embedded
triangulations, of hyperbolic manifolds with injectivity radius bounded below
that enjoy useful combinatorial and geometric properties that will facilitate
the estimates in sections 3 through 5. While we focus on the hyperbolic
setting, we give an account motivated by potential generalizations to the
variable negative curvature setting.

The triangulations we use are generally obtained via Delaunay complexes
associated to collections of points. To obtain a Delaunay complex in a
Riemannian manifold $M$, take a finite collection of points $P\subset M$ and
consider the Voronoi celluation consisting of cells

$V_{p}=\\{x\in M~{}:~{}d(x,p)\leq d(x,q)~{}\text{for all }p\neq q\in P\\}$

for $p\in P.$ Dual to the Voronoi celluation is the Delaunay complex. The
cells of the Delauney complex are the convex hulls of tuples of points whose
corresponding Voronoi cells have nonempty intersections. In [BDG18], it is
shown that if the collection of points $P$ satisfies certain density and
separation conditions, then there is a quantifiably small perturbation of the
point set $P$ whose Delaunay complex is a triangulation. The simplices of this
triangulation are geodesic. The precise conditions are as follows.

Let $M$ be a closed Riemannian manifold with distance function $d$. Given a
pair $1\geq\mu>0,~{}\varepsilon>0$, a $(\mu,\varepsilon)$-net is a collection
$P$ of points in $M$ for which the following hold:

  1. 1.

($P$ is $\varepsilon$-dense) For all $x\in M$, there is a $p\in P$ such that
$d(x,p)<\varepsilon$.

  2. 2.

($P$ is $\mu$-separated) All distinct $p,q\in P$ satisfy
$d(p,q)\geq\mu\varepsilon$.

Theorem 3 of [BDG18] says that if $\mu$ and $\varepsilon$ satisfy several
inequalities relating to the injectivity radius and sectional curvatures, a
$(\mu,\varepsilon)$-net can be perturbed to
$(\mu\textquoteright,\varepsilon\textquoteright)$-net such that the resulting
Delaunay complex is indeed a triangulation. This theorem, specialized to
closed hyperbolic $n$-manifolds, becomes:

######  Theorem 2.1.

[BDG18] Let $M$ be a closed hyperbolic $n$-manifold and $P$ a
$(\mu,\varepsilon)$-net such that

$\varepsilon\leq\min\left\\{\frac{\operatorname{inj}(M)}{4},~{}\Psi(\mu)\right\\},$

where $\operatorname{inj}(M)$ is the injectivity radius of $M$ and $\Psi$ is a
function of the net parameter $\mu$. The function $\Psi$ is described in
[BDG18] and is independent of the manifold $M$. Then there is a point set
$P\textquoteright$ that is a
$(\mu\textquoteright,\varepsilon\textquoteright)$-net with resulting Delaunay
complex a triangulation. Moreover, $\mu\textquoteright$ and
$\varepsilon\textquoteright$ satisfy the following (see equation (2) in
[BDG18]): :

$\displaystyle\varepsilon$
$\displaystyle\leq\varepsilon\textquoteright\leq\frac{5}{4}\varepsilon,$
$\displaystyle\frac{2}{5}\mu$ $\displaystyle\leq\mu\textquoteright\leq\mu.$

The separation and density conditions of a
$(\mu\textquoteright,\varepsilon\textquoteright)$-net ensure that the
resulting Delaunay triangulation has edge lengths lying in the closed interval
$[\frac{2}{5}\mu\varepsilon,2\varepsilon]$. We now specialize to the case
$\mu=1$. Note that when $n=3$, $\Psi(1)=2\times 3^{121.5}\times 5^{-81}$,
which is roughly 45.15.

Set $\epsilon_{0}=\min\\{\varepsilon/10,\Psi(1)\\}$ and define
$\mathcal{G}_{\varepsilon}$ to be the space of hyperbolic $n$-simplices with
edge lengths in the interval $[\frac{2\epsilon_{0}}{5},2\epsilon_{0}]$.
Because the space of all hyperbolic $n$-simplices is parametrized by edge
lengths, the space $\mathcal{G}_{\varepsilon}$ is compact. Proposition 2.2
below establishes that every closed hyperbolic manifold $M$ with injectivity
radius bounded below by $\varepsilon$ admits a triangulation whose simplices
are isometric to those in $\mathcal{G}_{\varepsilon}$.

In a triangulation $K$, the combinatorial 1-neighborhood of a simplex is the
union of the stars of the vertices of that simplex. A triangulation $K$ of a
hyperbolic manifold $M$ is an $\varepsilon$-deeply embedded triangulation if:

  1. 1.

Every simplex is geodesic and contained in $\mathcal{G}_{\varepsilon}$,

  2. 2.

The combinatorial 1-neighborhood of every simplex lifts isometrically to
$\mathbb H^{n}$.

Throughout, when referring to deeply embedded triangulations, we generally
suppress reference to some fixed $\varepsilon$.

######  Proposition 2.2.

Let $M$ be a closed hyperbolic $n$-manifold and let
$0<\varepsilon<\operatorname{inj}(M)$. Then there is a deeply embedded
triangulation $K$ of $M$. That is, there is a geodesic triangulation $K$ of
$M$ whose simplices come from $\mathcal{G}_{\varepsilon}$ such that the
combinatorial 1-neighborhood of every simplex isometrically embeds in $\mathbb
H^{n}$.

###### Proof.

Set $\epsilon_{0}=\min\\{\varepsilon/10,\Psi(1)\\}$. Take a maximal collection
of points $P\subset M$ such that the balls $B_{\epsilon_{0}/2}(p)$ for $p\in
P$ are all disjoint. By maximality, the $B_{\epsilon_{0}}(p)$ balls then cover
$M$. Since the $B_{\epsilon_{0}/2}(p)$ balls are all disjoint, we have that
for all $p,q\in P$, $d(p,q)\geq\epsilon_{0}=\mu{\epsilon_{0}}$, so $P$ is
$\mu$-separated. Since the $B_{\epsilon_{0}}(p)$ balls cover, every point
$x\in M$ is ${\epsilon_{0}}$-close to some point $p\in P$, so $P$ is
${\epsilon}$-dense. The collection $P$ therefore is a
$(\mu,{\epsilon_{0}})$-net, with $\mu=1$ and ${\epsilon_{0}}$ satisfying the
hypotheses of Theorem 2.1. Consequently, there is a perturbation of $P$ that
is a $(\mu\textquoteright,{\epsilon_{0}}\textquoteright)$-net whose Delaunay
complex is a triangulation. Since, as remarked above, the edge lengths of
simplices in this Delaunay triangulation lies in the interval
$[\frac{2}{5}{\epsilon_{0}},2{\epsilon_{0}}]$, the simplices come from
$\mathcal{G}_{\varepsilon}$. The edge length bound along with the fact
${\epsilon_{0}}<\operatorname{inj}(M)/10$ ensures the diameter of any vertex
star (which will be less than 3 times the length of the longest edge of a
simplex) will be less than $2{\epsilon_{0}}$. Thus, the star of every vertex
embeds isometrically in $\mathbb H^{n}$ via the local inverse of the
exponential map. ∎

We will also use cell complexes that are dual to deeply embedded
triangulations. Every simplicial triangulation $K$ of a closed Riemannian
manifold admits a dual celluation $K^{*}$ comprised of cells $\sigma^{*}$ dual
to the simplices $\sigma$ of the triangulation $K$ in the following sense (for
a reference, see [Bre97] chapter VI.6): Take the first barycentric subdivision
$\tau(K)$ of $K$, then the $n$-cells of the dual celluation $K^{*}$ are the
closed stars of the vertices of the original triangulation $K$ in the
barycentric subdivision. This celluation is naturally triangulated by the the
barycentric subdivision triangulation $\tau(K)$. Like $K$, the dual celluation
can be uniformly controlled.

The controlled geometry of deeply embedded triangulations and their dual
celluations primarily manifests in the compactness of
$\mathcal{G}_{\varepsilon}$ and the following local bound.

######  Proposition 2.3.

Let $M$ be a closed hyperbolic $n$-manifold of injectivity radius
$\operatorname{inj}(M)>\varepsilon$ with a deeply embedded triangulation $K$.
Then there is a positive constant $N=N(\varepsilon)$ such that the number of
$k$-simplices in the star of a $j$-simplex is less than $N$.

###### Proof.

The edge length bounds provide a lower bound on the angle between two edges
meeting at a vertex that span a face via the hyperbolic law of cosines. This
implies that the number of $n$-simplices meeting at a vertex $v$ is bounded
uniformly, since for any ball around the vertex, there is a uniform lower
bound on the volume of the intersection of an $n$-simplex containing the
vertex $v$ and the ball. It follows that there is an $N$ such that the number
of $k$-simplices in the star of a vertex is less than $N$ for $k=0,\dots,n$.∎

Proposition 2.3 and the compactness of the space $\mathcal{G}_{\varepsilon}$
of simplices together imply that the geometry and combinatorics of the
1-neighborhood of a simplex in a deeply embedded triangulation is uniformly
controlled. In particular, let $K$ be a deeply embedded triangulation of $M$.
By Proposition 2.3, any vertex in $K$ is contained in at most $N$ simplices.
Therefore, there are finitely many possible finite simplicial complexes that
appear as the combinatorial 1-neighborhood of a simplex in a deeply embedded
triangulation. Let $\mathcal{C}$ be the finite set of such possible complexes.

For any complex $a\in\mathcal{C}$, say with $|a|$ many $n$-simplices, a
hyperbolic structure on $a$ is given by identifying each $n$-simplex in $a$
with a model simplex in $\mathcal{G}_{\varepsilon}$ so that the face gluing
maps are isometries. The possible geometric structures on $a$ are parametrized
by a subspace $\mathcal{S}_{\varepsilon}(a)$ of
$\mathcal{G}_{\varepsilon}^{|a|}$. Because the gluing conditions are closed,
and because $\mathcal{G}_{\varepsilon}$ is compact, the space
$\mathcal{S}_{\varepsilon}(a)$ is compact. By taking the disjoint union over
the finite list of possible complexes $a\in\mathcal{C}$, there is a compact
space $\mathcal{S}_{\varepsilon}$ that parametrizes the geometry of the
combinatorial 1-neighborhood of a simplex in a deeply embedded triangulation.

We now turn to relating closed geodesics in $M$ to cellular paths in $K$.

######  Proposition 2.4.

Let $M$ be a closed hyperbolic $n$-manifold with injectivity radius
$\operatorname{inj}(M)>\varepsilon$ with a deeply embedded triangulation $K$.
Let $\gamma$ be a closed geodesic curve in $M$. Then there is a constant
$J=J(\varepsilon)$ such that the number $v$ of cells in the dual cell complex
$K^{*}$ that $\gamma$ intersects (counted with multiplicity) satisfies

$v\leq J|\gamma|.$

###### Proof.

Suppose $\gamma$ moves from an $n$-cell $\sigma$ to an $n$-cell
$\sigma\textquoteright$, intersecting the ($n-1)$-skeleton of $K^{*}$ at a
point $p\in\sigma\cap\sigma\textquoteright$. Consider the closed radius
$\varepsilon$-ball at the point $p$, $V=\bar{B}_{\varepsilon}(p).$ Let $x$ be
the point at which $\gamma$ enters $V$ and let $y$ be the point at which it
exits $V$. Then the geodesic subarc of $\gamma$ running from $x$ to $y$ has
length $2\varepsilon$. Since the triangulation $K$ has simplices from
$\mathcal{G}_{\varepsilon}$, the restrained combinatorics of the dual
celluation ensures that the ball $V$ intersects a universally bounded number
of dual cells. Let $R(\varepsilon)$ denote this bound.

Consider the sequence $x_{n},y_{n}$ of points such that $x_{1}=x$ and
$y_{1}=y$ from above for the first simplex crossing, and $x_{n}$ is obtained
by taking the simplex crossing that happens after $y_{n}$. Then each pair
$x_{n},y_{n}$ corresponds to a geodesic sub arc of $\gamma$ that intersects at
most $R(\varepsilon)$ simplices. Thus,
$\nu\leq(\frac{|\gamma|}{2\varepsilon}+1)R(\varepsilon)$. It therefore follows
that $\frac{v}{R(\varepsilon)}2\varepsilon\leq|\gamma|+2\varepsilon.$ Since
$\varepsilon\leq|\gamma|$, we have $|\gamma|+2\varepsilon\leq 3|\gamma|$, and
the stated linear bound follows with $J=\frac{3R(\varepsilon)}{2\varepsilon}$.
∎

The next result compares the lengths of closed geodesics in $M$ to
approximating paths in the 1-skeleton of dual celluation $K^{*}$. To measure
the complexity of paths in $K^{*}$, let $||\cdot||_{G}$ be the $\ell^{1}$-norm
on chains and $\operatorname{{\tt{len}}}(\cdot)$ the word length of the
cellular path. For a cellular path $c$, let $||c||_{G}$ be the $\ell^{1}$-norm
of the corresponding chain.

######  Proposition 2.5.

There is a constant $L=L(\varepsilon)>0$ such that for any closed geodesic
curve $\gamma$ in $M$, there is a cellular path $c$ in $K^{*}$ homotopic to
$\gamma$ such that $||c||_{G}\leq\operatorname{{\tt{len}}}(c)\leq L|\gamma|$.

###### Proof.

Fix a base point and orientation for $\gamma$ such that the base point lies on
a face of a top dimensional cell. The curve $\gamma$ can be replaced by a
homotopic curve whose length is bounded by a constant times the geodesic
length of $\gamma$ and which intersects the boundary of every simplex at
vertices. This follows from Proposition 2.4 which gives that there is a bound
on the number of simplices $\gamma$ intersects (counting these intersections
with multiplicity) that depends linearly on the length of $\gamma$ and the
fact the simplices of the triangulation have bounded diameter. Using the
orientation and basepoint, we obtain a sequence of vertices with line segments
between them that lie entirely in a cell. We can further modify $\gamma$ by
replacing these curve segments with curves that lie in the 1-skeleton by
traversing the 1-simplex joining the two boundary vertices. Since the edges in
the celluation $K^{*}$ have bounded length, this again adds bounded length to
the curve. Let $c$ denote the cellular path we have constructed. By the
previous considerations, there is a constant $L$ depending only on
$\varepsilon$ giving the comparison $\operatorname{{\tt{len}}}(c)\leq
L|\gamma|$. The inequality $||c||_{G}\leq\operatorname{{\tt{len}}}(c)$ is
trivial.

∎

The geometry of geodesic simplices in hyperbolic manifolds can also be
understood using barycentric coordinates via Thurston’s straightening map (see
[Thu97] page 124). Identify $\mathbb H^{n}$ with the upper sheet of the
hyperboloid in Minkowski space $\mathbb R^{n,1}$ with quadratic form
$Q=x_{0}^{2}+x_{1}^{2}+\cdots+x_{n-1}^{2}-x_{n}^{2}$ and consider a singular
simplex $\sigma:\Delta\to\mathbb H^{n}$. Let
$b_{0},\dots,b_{n}:\Delta\to[0,1]$ be the barycentric coordinates on the
standard Euclidean simplex $\Delta$ with vertices $e_{0},\dots,e_{n}$. Then
for $v\in\Delta$, write $v=\sum b_{i}(v)e_{i}$. The straightening of $\sigma$
is the singular simplex in $\mathbb H^{n}$ given by centrally projecting the
affine simplex $\sum b_{i}(v)\sigma(v_{i})$ from the origin to the upper sheet
of the hyperboloid. This process endows each geodesic simplex with a natural
barycentric coordinate.

If $\pi:\mathbb H^{n}\to M$ is the projection map and $\sigma$ is a singular
simplex in $M$, let $\text{st}(\sigma):\Delta\to M$ be the composition of the
straightening of $\sigma$ applied to some lift of $\sigma$ and the projection
map. This is well-defined and independent of the lift because the isometry
group of $\mathbb H^{n}$ acts linearly on $\mathbb R^{n,1}$ preserving the
quadratic form $Q$.

For a geodesic simplex $\sigma$ in $\mathbb H^{n}$, let
$V_{\sigma}:\sigma\to\Delta$ be the map from $\sigma$ to the standard simplex
given by the barycentric coordinates induced by straightening. Geodesic
simplices $\sigma,\sigma^{\prime}$ can be compared using the composition
$V_{\sigma\textquoteright}^{-1}\circ V_{\sigma}$. Using the straightening
construction and the barycentric coordinates, one sees that the maps
$V_{\sigma}$ depend continuously on $\sigma$ in the sense that if $\sigma$ is
a straight simplex in $\mathbb H^{n}$ and $\sigma\textquoteright$ is obtained
by perturbing the vertices of $\sigma$, then the composition map
$V_{\sigma\textquoteright}^{-1}\circ V_{\sigma}$ is almost an isometry, where
the failure to be an isometry is controlled by the size of the vertex
perturbation.

######  Proposition 2.6.

Let $\sigma$ and $\sigma\textquoteright$ be geodesic simplices from
$\mathcal{G}_{\varepsilon}$ embedded in $\mathbb H^{n}$. Then the map
$V_{\sigma\textquoteright}^{-1}\circ V_{\sigma}$ is $\kappa$-biLipschitz for
some $\kappa=\kappa(\varepsilon)>0$ that does not depend on $\sigma$ and
$\sigma\textquoteright$.

###### Proof.

The biLipschitz constant for the comparison map between any given simplex and
the Euclidean simplex depends continuously on the simplex. The result then
follows from the compactness of $\mathcal{G}_{\varepsilon}$. ∎

We also have uniform control over the geometry of the combinatorial
1-neighborhood of a simplex.

######  Proposition 2.7.

There is a constant $\mathcal{L}=\mathcal{L}(\varepsilon)$ such that if $s$
and $s\textquoteright$ are two complexes of the same combinatorial type in
$\mathcal{S}_{\varepsilon}$, then $s$ and $s\textquoteright$ are
$\mathcal{L}$-biLipschitz equivalent.

###### Proof.

Define maps from the combinatorial 1-neighborhood $s$ of a simplex $\sigma$ to
an abstract Euclidean model by gluing the maps $V_{\sigma^{\prime}}$ together
according to the combinatorics of $s$. Since the gluing maps are isometries,
this is well defined. Since the map restricted to each simplex
$\sigma^{\prime}$ is uniformly biLipschitz equivalent to the model simplex,
and since there are a uniformly bounded number of simplices in $s$, it follows
that $s$ is uniformly biLipschitz equivalent to the Euclidean model. ∎

Lastly, we note that the lower bound on volumes of simplices in deeply
embedded triangulations implies the following.

######  Proposition 2.8.

There is a constant $T=T(\varepsilon)$ such that if $M$ is a closed hyperbolic
$n$-manifold with $\operatorname{inj}(M)>\varepsilon$ and $K$ is a deeply
embedded triangulation of $M$, then the number $V_{K}$ of simplices in $K$
satisfies

$V_{K}\leq T\operatorname{vol}(M).$

###  2.2 Whitney forms

The combinatorial geometry of the triangulation $K$ is related to the
Riemannian geometry of $M$ by way of the Whitney form map $W:C^{\bullet}(K)\to
L^{2}\Omega^{\bullet}(M)$ relating the cochain complex $C^{\bullet}(K)$, with
$\mathbb R$ coefficients, to the $L^{2}$-de Rham complex. It will be useful to
view $C^{\bullet}(K)$ as a subcomplex of the singular cochain complex. With
this in mind, we often identify singular simplices with their images.

The Whitney map is readily defined using the basis for $C^{\bullet}(K)$ dual
to the basis of simplices. This basis consists of the cochains
$\delta_{\sigma}$ that take the value 1 on the simplex $\sigma$ and zero on
all other simplices. The Whitney form $W(\delta_{\sigma})$ associated to the
cochain $\delta_{\sigma}$ dual to an oriented simplex
$\sigma=[v_{0},\dots,v_{q}]$ is given by

$W(\delta_{\sigma})=q!\sum_{k=0}^{q}(-1)^{k}b_{k}db_{0}\wedge\cdots\wedge
db_{k-1}\wedge db_{k+1}\wedge\cdots\wedge db_{q},$

where $b_{k}:M\to[0,1]$ is the barycentric coordinate associated to the vertex
$v_{k}$. See [Dod76] for more details.

An $L^{2}$-form in the image of $W$ is called a Whitney form. The support of a
Whitney form $W(\delta_{\sigma})$ is contained in the closed star of the
simplex $\sigma$. The barycentric coordinates used to define the Whitney forms
are not smooth, however, they are smooth in the compliment of the
$(n-1)$-skeleton of $K$. One can define the exterior derivative of a Whitney
form in a weak sense, which yields a differential that is well defined as an
$L^{2}$-form. With this exterior derivative, the Whitney map becomes a chain
map. For any cochain $f$ and simplex $\sigma$, the restriction of the Whitney
cochain $\omega=W(f)$ to $\sigma$, denoted $\omega|_{\sigma}$, can be uniquely
extended to a smooth form on the boundary of $\sigma$. This extension however
is not unique when $\sigma$ lies in the boundary of multiple simplices. In
addition to the restriction of Whitney forms, we have the restriction for
cochains. If $f=\sum a_{i}\delta_{\sigma_{i}}$ is a cochain and $\sigma$ is a
simplex, then
$f|_{\sigma}=\sum\limits_{\sigma_{i}\subset\sigma}a_{i}\delta_{\sigma_{i}}.$
This cochain restriction satisfies
$\omega|_{\sigma}=W(f|_{\sigma})|_{\sigma}$.

A Whitney form associated to a geodesic simplex $\sigma$ in $M$ is the
corresponding Whitney form on the standard Euclidean simplex pulled back to
$\sigma$ via the map $V_{\sigma}$. Geometric norms on cochains determined by
the Whitney map can be compared with various combinatorial norms. This is done
using the $L^{p}$-change of variables formula for $k$-forms, see [Ste13].

######  Proposition 2.9.

Let $s\in\mathcal{S}_{\varepsilon}\cup\mathcal{G}_{\varepsilon}$. Let $W(f)$
be the Whitney form associated to a cochain $f\in C^{k}(s;\mathbb R)$. Let
$||\cdot||$ be some fixed norm on the real vector space $C^{k}(s;\mathbb R)$
and let $||\cdot||_{p,s}$ be the $p$-norm associated to $s$ on
$\Omega^{k}(s)$, where $p={\infty}$ or $p=2$. Then there is a constant
$\mathcal{A}=\mathcal{A}(\varepsilon,||\cdot||)>0$ such that

$\mathcal{A}^{-1}||W(f)||_{p,{s}}\leq\mathcal{|}|f||\leq\mathcal{A}||W(f)||_{p,s}.$

The constant $\mathcal{A}$ only depends on the chosen norm and combinatorial
type of $s,$ not on the geometric structure of $s$.

###### Proof.

By Proposition 2.7, there is a constant $\mathcal{L}$ (note
$\kappa\leq\mathcal{L}$, so that if we’re working with simplices,
$\mathcal{L}$ works as Lipschitz constant) such that any pair
$s,s\textquoteright\in\mathcal{S}_{\varepsilon}\cup\mathcal{G}_{\varepsilon}$
in the same combinatorial type are $\mathcal{L}$-biLipschitz equivalent. For
each combinatorial type, fix a model
$s_{a}\in\mathcal{S}_{\varepsilon}\cup\mathcal{G}_{\varepsilon}$ and let
$\mu:s\to s_{a}$ be the $\mathcal{L}$-biLipschitz comparison map described
above. Then, because Whitney forms are obtained by pulling back the standard
Whitney forms via the model map, we can apply the $L^{2}$-change of variables
formula for $k$-forms and use the biLipschitz comparison to get

$\frac{||W(f)||_{2,{s}}}{||f||}\leq\mathcal{L}^{n/2}\frac{||W(f)||_{2,s_{a}}}{||f||}.$

Similarly, applying the $L^{\infty}$-change of variables formula for
$k$-forms, gives

$\frac{||W(f)||_{{\infty},s}}{||f||}\leq\mathcal{L}^{k}\frac{||W(f)||_{\infty,s_{a}}}{||f||}.$

For $p=2$, set

$\mathcal{A}_{2}=\mathcal{L}^{n/2}\max\limits_{a}\sup\limits_{g\in
C^{k}(s_{a})}\frac{||W(g)||_{2,s_{a}}}{||g||}$

and for $p=\infty$, set

$\mathcal{A}_{\infty}=\mathcal{L}^{n}\max\limits_{a}\sup\limits_{g\in
C^{k}(s_{a})}\frac{||W(g)||_{2,s_{a}}}{||g||},$

where the maximum runs over all combinatorial types $a$. Then,
$A=\max\\{\mathcal{A}_{2},\mathcal{A}_{\infty}\\}$ gives the first inequality.
The second inequality is obtained from an identical argument via the lower
bound in biLipschitz comparison. Let $\mathcal{A}$ be the maximum of these two
constants. ∎

This estimate is enough to reproduce the statements in [LS18] for deeply
embedded triangulations. However, to obtain the cleaner discrete-smooth
eigenvalue comparison in Proposition 4.2, we need to study a smooth analogue
of the Whitney map. The smooth Whitney map was introduced by Dodzuik in
[Dod81]. The map is defined by replacing the barycentric coordinates with a
smooth partition of unity indexed by the vertices of a triangulation. The
particular partition of unity is provided by the following proposition.

######  Proposition 2.10.

([Dod81], Lemma 2.11) If $M$ admits a deeply embedded triangulation $K$, then
there exists a $C^{\infty}$ partition of unity $\beta_{i}$ indexed by the
vertices of $K$ and subordinate to the covering of $M$ by open stars of
vertices of $K$ (indeed, compactly supported in each open star). Moreover,
each $\beta_{i}$ has covariant derivatives satisfying the pointwise bound
$|\nabla^{k}\beta_{i}|<C$ for some constant $C=C(\varepsilon)$, for $k\leq n$.

###### Proof.

Let $s\in\mathcal{S}_{\varepsilon}$ be the combinatorial 1-neighborhood of a
simplex. Denote the vertices of $s$ by $v_{0},\dots,v_{n}$ and let $b_{i}$ be
the standard barycentric coordinates associated to the vertex $v_{i}$. Define

$\bar{b}_{i}(x)=\begin{cases}0&b_{i}(x)\leq 1/(n+2),\\\
\frac{(n+2)b_{i}(x)-1}{n+1}&b_{i}(x)\geq 1/(n+2).\\\ \end{cases}$

Observe that $\sum\limits_{i}\bar{b}_{i}(y)\geq\frac{1}{(n+2)}$. Define

$\delta(s)=\inf\limits_{\begin{subarray}{c}x\in\operatorname*{supp}(\bar{b}_{i})\\\
y\in\partial\operatorname{{\text{star}}}(v_{i})\end{subarray}}d(x,y),$

where $d$ is the distance function induced by the Riemannian metric. Set
$\delta=\frac{1}{2}\inf\limits_{s\in\mathcal{S}_{\varepsilon}}\delta(s)$ and
notice $\delta>0$. For any point $x\notin\operatorname{{\text{star}}}(v_{i})$,
the ball $B_{\delta}(x)$ is disjoint from the support of $\bar{b}_{i}$. Let
$\eta$ be a smooth cutoff function such that $\eta(r)=1$ when $|r|<\delta/2$
and $\eta(r)=0$ when $|r|>3\delta/4$. The function $\eta(d(x,y))$ is smooth on
the open star of a vertex, so the operator given by integrating against
$\eta(d(x,y))$ is smoothing. Therefore, if we define

$\tilde{b}_{i}(x)=\int_{B_{3\delta/4}(x)}\eta(d(x,y))\bar{b}_{i}(y)dy,$

the result is a smooth function. Notice $\tilde{b}_{i}$ is supported in the
interior of the star of $v_{i}$ by virtue of our choice of $\delta$. We now
define smoothed barycentric partitions of unity for a smooth manifold with
deeply embedded triangulation $K$ by normalizing the functions $\tilde{b}_{i}$
associated to the vertices of $K$:

$\beta_{i}(x)=\left(\sum\limits_{j}\tilde{b}_{j}(x)\right)^{-1}\tilde{b}_{i}(x).$

Notice that if $\tilde{b}_{i}(x)\neq 0$, then this normalizing sum really just
runs over the vertices of $\operatorname{{\text{star}}}(v_{i})$. This
normalizing constant can be bounded from below:

$\displaystyle\sum\limits_{j}\tilde{b}_{j}(x)$
$\displaystyle\geq\sum\limits_{j}\int_{B_{\delta/2}(x)}\bar{b}_{j}(y)dy$
$\displaystyle=\int_{B_{\delta/2}(x)}\sum\limits_{j}\bar{b}_{j}(y)dy$
$\displaystyle\geq\operatorname{vol}(B_{\delta/2}(x))\frac{1}{(n+2)(n+1)}.$

The covariant derivative bound follows from repeated application of the
quotient rule and the corresponding bounds for $\bar{b}_{i}$, which depends
only on the derivatives of cutoff function $\eta$ and the covariant
derivatives of the metric. The choice of $\delta$ ensures each function
$\beta_{i}$ is compactly supported in the star of $v_{i}$. ∎

Our aim now is to establish a version of Proposition 2.9 for these partitions
of unity that will allow us to relate the geometric norms induced by the
smooth Whitney map to combinatorial norms.

Let $M$ be a hyperbolic $n$-manifold with a deeply embedded triangulation $K$.
Denote by $\operatorname{{\text{star}}}_{0}(\sigma)$ the combinatorial
1-neighborhood of an $n$-simplex $\sigma$ in $K$. Fix some $n$-simplex
$\sigma$ in $K$ and set $s=\operatorname{{\text{star}}}_{0}(\sigma)$. Because
$K$ is deeply embedded, for any point $p\in s$, the ball
$B=B_{\varepsilon}(p)$ contains $s$ and lifts isometrically to $\mathbb
H^{n}$. Identify $s$ with such a lift. The functions $\beta_{i}$ associated to
the vertices of $\sigma$ are supported in $s$ and their value in any simplex
$\sigma$ depends only on the geometry of $s$. This enables us to isolate the
local properties of the barycentric partition of unity functions. By
perturbing the vertices of $s$ in $\mathbb H^{n}$ and modifying the various
simplices making up the complex $s$ accordingly, we can then see how these
functions relate to the geometry of combinatorial 1-neighborhoods as encoded
by the space $\mathcal{S}_{\varepsilon}$.

######  Lemma 2.11.

Let $\sigma$ be an $n$-simplex with vertices $v_{0},\dots,v_{n}$ from
$\mathcal{G}_{\varepsilon}$ contained in combinatorial 1-neighborhood
$s=\operatorname{{\text{star}}}_{0}(\sigma)\in\mathcal{S}_{\varepsilon}$ with
additional vertices $v_{n+1},\dots,v_{m}$. The functions $\beta_{i}$
associated to the vertices of $\sigma$ constructed in Proposition 2.10 and
their covariant derivatives $\nabla\beta_{i}$ vary continuously in
$L^{2}(\mathbb H^{n})$ when the perturbation of $s$ in
$\mathcal{S}_{\varepsilon}$ is realized as above in $\mathbb H^{n}$.

###### Proof.

As above, we can embed the combinatorial 1-neighborhood $s$ in $\mathbb H^{n}$
and the functions $\tilde{b}_{i}$ for each vertex $v_{i}$ of $\sigma$
described in Proposition 2.10 are well defined smooth functions on $\mathbb
H^{n}$ supported on $s$. To define the barycentric partition of unity, we also
need the functions $\tilde{b}_{k}$ for vertices $v_{k}$ in $s$ that are not in
$\sigma$ to be well defined on the support of the functions $\tilde{b}_{i}$
for vertices $v_{i}$ of $\sigma$. For a vertex $v_{k}$ in $s$ that is not part
of $\sigma$ and a point $x$ in the support of $\tilde{b}_{i}$ for $v_{i}$ a
vertex of $\sigma$, we have that the ball $B_{3\delta/4}$ used in the
definition of $\tilde{b}_{k}$ is contained in $s$, so $\tilde{b}_{k}(x)$ only
depends on $s$.

From the definition, one sees that each function $\tilde{b}_{i}$ varies
continuously in $L^{2}(\mathbb H^{n})$ as the complex $s$ in $\mathbb H^{n}$
is varied by perturbing the vertices and modifying the various simplices
making up the complex $s$ accordingly. One similarly can see from the
definition of $\tilde{b}_{i}(x)$ that $\nabla\tilde{b}_{i}(x)$ varies
continuously as $s$ is varied as above. The functions $\beta_{i}$ in the
barycentric partition of unity are defined by normalizing the functions
$\tilde{b}_{i}$:

$\beta_{i}(x)=\left(\sum\limits_{v_{k}\in
s^{(0)}}\tilde{b}_{k}(x)\right)^{-1}\tilde{b}_{i}(x).$

By the remark above, $\tilde{b}_{k}$ is well defined on the support of
$\tilde{b}_{i}$ for every vertex $v_{k}$ of $s$. Since each $\tilde{b}_{j}$
and $\nabla\tilde{b}_{j}$ varies continuously with $s$, the same holds for
$\beta_{j}$ and $\nabla\beta_{j}$. ∎

To a partition of unity indexed by the vertices of a triangulation and
subordinate to the covering by open stars of vertices, one can define a
generalized Whitney mapping, given by the same formula as the standard Whitney
map but with the smooth barycentric partitions of unity in place of the
standard barycentric coordinates. Let $\beta=(\beta_{i})$ be the barycentric
partition of unity. The Whitney form $W_{\beta}(\delta_{\sigma})$ associated
to the cochain $\delta_{\sigma}$ dual to an oriented simplex
$\sigma=[v_{0},\dots,v_{q}]$ is given by

$W_{\beta}(\delta_{\sigma})=q!\sum_{k=0}^{q}(-1)^{k}\beta_{k}d\beta_{0}\wedge\cdots\wedge
d\beta_{k-1}\wedge d\beta_{k+1}\wedge\cdots\wedge d\beta_{q},$

Like the standard Whitney map, these generalized Whitney maps satisfy:

  1. 1.

For a chain $a$ and cochain $f$ of the same degree,
$\int_{a}W_{\beta}(f)=f(a).$

  2. 2.

For any cochain $f$, $dW_{\beta}(f)=W_{\beta}(df)$.

  3. 3.

If $p$ is contained in the interior of an $n$-simplex $\sigma$, and any
cochain $f$, $W_{\beta}(f)_{p}=W_{\beta}(f|_{\sigma})_{p}$.

######  Lemma 2.12.

Fix a simplex $\sigma\in\mathcal{G}_{\varepsilon}$ contained in its
combinatorial 1-neighborhood $s\in\mathcal{S}_{\varepsilon}$. The Whitney map
$W_{\beta}:C^{\bullet}(\sigma)\to L^{2}\Omega^{\bullet}(\mathbb H^{n})$ varies
continuously as the geometry of the star $s$ varies in $\mathbb H^{n}$ as
described above. Consequently, $||W_{\beta}(f)||_{2,s}$ and
$||W_{\beta}(f)||_{2,\sigma}$ vary continuously with the geometric structure
on $s$ in $\mathcal{S}_{\varepsilon}$.

###### Proof.

We work with the combinatorial 1-neighborhood $s$ embedded in a ball
$B\subset\mathbb H^{n}$ as above. Because $\beta_{i}$ and $\nabla\beta_{i}$
vary continuously with $s$ and since $||\nabla\beta_{i}||_{2,B}$ is comparable
to $||d\beta_{i}||_{2,B}$, it suffices to show that exterior products of
$d\beta_{i}$ vary continuously in the $L^{2}$-norm. The degree 0 case and
degree 1 case are immediate from the continuity in Lemma 2.11. We treat only
the degree 2 case as the other higher degree cases are handled similarly.
Assume $||\beta_{i}-\beta_{i}\textquoteright||_{2,B}<\epsilon$ and
$||d\beta_{i}-d\beta_{i}\textquoteright||_{2,B}<\varepsilon$. Then we have

$\displaystyle||d\beta_{0}\wedge d\beta_{1}-d\beta\textquoteright_{0}\wedge
d\beta_{1}\textquoteright||_{2,B}$ $\displaystyle=||d\beta_{0}\wedge
d\beta_{1}-d\beta\textquoteright_{0}\wedge
d\beta_{1}\textquoteright+d\beta_{0}\wedge
d\beta_{1}\textquoteright-d\beta_{0}\wedge d\beta_{1}\textquoteright||_{2,B}$
$\displaystyle\leq||d\beta_{0}\wedge d\beta_{1}-d\beta_{0}\wedge
d\beta_{1}\textquoteright||_{2,B}+||d\beta_{0}\textquoteright\wedge
d\beta_{1}\textquoteright-d\beta_{0}\wedge d\beta_{1}\textquoteright||_{2,B}$
$\displaystyle=||d\beta_{0}\wedge
d(\beta_{1}-\beta_{1}\textquoteright)||_{2,B}+||d(\beta_{1}\textquoteright-\beta_{1})\wedge
d\beta_{1}\textquoteright||_{2,B}$ $\displaystyle\leq
C||d\beta_{0}||_{2,B}||d(\beta_{1}-\beta_{1}\textquoteright)||_{2,B}+C||d\beta_{1}\textquoteright||_{2,B}||d(\beta_{1}\textquoteright-\beta_{1})||_{2,B},$
$\displaystyle\leq 2C\epsilon,\text{~{}after increasing $C$.}$

This immediately gives that $||W_{\beta}(f)||_{2,s}$ varies continously with
$s$. Let $\xi_{\sigma}$ be the characteristic function of $\sigma$ in $\mathbb
H^{n}$. Then $\xi_{\sigma}$ varies continuously in $L^{2}$ when $\sigma$ is
varied by perturbing its vertices. The norm
$||W_{\beta}(f)||_{2,\sigma}=||W_{\beta}(f)\xi_{\sigma}||_{2,B}$ therefore
also varies continuously when $s$ is varied in $\mathcal{S}_{\varepsilon}$. ∎

######  Proposition 2.13.

There is a constant $A=A(\varepsilon)>0$ such that if
$\sigma\in\mathcal{G}_{\varepsilon}$ has combinatorial 1-neighborhood
$s\in\mathcal{S}_{\varepsilon}$ and $f\in C^{\bullet}(\sigma)$ is any cochain,
then there are comparisons

$A^{-1}||W(f)||_{2,\sigma}\leq||W_{\beta}(f)||_{2,\sigma}\leq
A||W(f)||_{2,\sigma}.$

###### Proof.

We embed $s$ in a ball $B$ in $\mathbb H^{n}$ as above. The $L^{2}$-norm
induced by $\beta$ for a fixed geometric structure on $\sigma$ is continuous
on the vector space of cochains $C^{\bullet}(\sigma)$. For any cochain $f\in
C^{\bullet}(\sigma)$, the form $W_{\beta}(f)$ varies continuously in
$L^{2}\Omega^{\bullet}(B)$ as the combinatorial 1-neighborhood $s$ varies in
$\mathcal{S}_{\varepsilon}$. It follows that $||W_{\beta}(f)||_{2,\sigma}$ is
continuous as a function on the component of $\mathcal{S}_{\varepsilon}\times
C^{\bullet}(\sigma)$ corresponding to the combinatorial type of the
combinatorial 1-neighborhood $s$.

Since $W_{\beta}$ sends nonzero cochains to nonzero forms, for any $f\neq 0$,
one has $0<||W_{\beta}(f)||_{2,\sigma}$. Let $||\cdot||_{E}$ be the usual
$\ell^{2}$ norm on $C^{\bullet}(\sigma)$. The norm
$\left\lVert\cdot\right\rVert_{E}$ is fixed as the geometric structure on
$\sigma$ varies. For $s\textquoteright\in\mathcal{S}_{\varepsilon}$, let
$\beta\textquoteright$ be the corresponding barycentric partition of unity
defined by the combinatorial 1-neighborhood $s\textquoteright$. By definition,
each $s\textquoteright\in\mathcal{S}_{\varepsilon}$ is the combinatorial
1-neighborhood of some simplex, let $\sigma\textquoteright$ be this simplex.

Using the continuity in Lemma 2.11 and the compactness of
$\mathcal{S}_{\varepsilon}$, we conclude the constants

$A_{\bullet}=\inf\limits_{s\textquoteright\in\mathcal{S}_{\varepsilon}}\inf\limits_{\begin{subarray}{c}f\in
C^{\bullet}(\sigma\textquoteright)\\\
||f||_{E}=1\end{subarray}}||W_{\beta\textquoteright}(f)||_{2,\sigma\textquoteright}$

and

$B_{\bullet}=\sup\limits_{s\textquoteright\in\mathcal{S}_{\varepsilon}}\sup\limits_{\begin{subarray}{c}f\in
C^{\bullet}(\sigma\textquoteright)\\\
||f||_{E}=1\end{subarray}}||W_{\beta\textquoteright}(f)||_{2,\sigma\textquoteright},$

are strictly positive real numbers independent of the particular geometric
structure on $s$ or $\sigma$ giving the desired comparison between $||f||_{E}$
and $||W_{\beta}(f)||_{2,\sigma}$.

We can then compare $||W(f)||_{2,\sigma}$ and $||f||_{E}$ using Proposition
2.9. For $f\in C^{\bullet}(\sigma)$, Proposition 2.9 gives a constant
$\mathcal{A}$ such that
$\mathcal{A}^{-1}||W(f)||_{2,\sigma}\leq||f||_{E}\leq\mathcal{A}||W(f)||_{2,\sigma}$.
Combining these comparisons with the comparison of
$||W_{\beta}(f)||_{2,\sigma}$ and $||f||_{E}$ above gives the claimed result
for $||W_{\beta}(f)||_{2,\sigma}$. ∎

The upshot of this is that the smooth barycentric partition of unity induces a
norm on the cochain complex of a simplex that locally is uniformly comparable
to the $L^{2}$-norm induced by the standard barycentric partition of unity.
This gives an analogue of Proposition 2.9 for the $L^{2}$-norm induced by the
barycentric partition of unity. We also require such a comparison for the
$L^{\infty}$-norm. The upgraded version of Proposition 2.9 appears below as
Proposition 2.16.

For a smooth manifold $Y$, possibly with boundary, we denote the Sobolev
spaces of differential $\bullet$-forms by $H^{k}_{\nabla}\Omega^{\bullet}(Y)$
and their norms by $||\cdot||_{H^{k}_{\nabla}(Y)}$, where

$||\omega||_{H^{k}_{\nabla}}=\sum\limits_{i=0}^{k}||\nabla^{i}\omega||_{2}.$

When $\bullet=0$, we drop $\Omega^{0}$ from this notation. When $Y$ has
boundary, the marking $\mathring{H}^{k}_{\nabla}$ denotes the subspace of
forms that are approximated by smooth forms supported in the interior of $Y$.

######  Lemma 2.14.

There is a constant $R(\varepsilon)>0$ such that for any $n$-simplex
$\sigma\in\mathcal{G}_{\varepsilon}$ with combinatorial 1-neighborhood
$s\in\mathcal{S}_{\varepsilon}$, the map $W_{\beta}:C^{\bullet}(\sigma)\to
H^{n}_{\nabla}\Omega^{\bullet}(B)$ satisfies

$||W_{\beta}(f)|_{\sigma}||_{H^{n}_{\nabla}(\sigma)}\leq||W_{\beta}(f)||_{H^{n}_{\nabla}(B)}\leq
R||W_{\beta}(f)||_{2,\sigma},$

where we have identified the combinatorial 1-neighborhood $s$ isometrically
with a domain $D$ in $\mathbb H^{n}$ and $B$ is a ball of radius $\varepsilon$
based at $p\in\sigma$.

###### Proof.

The first inequality follows from the definition of the Sobolov norm.

We now observe that the covariant derivative bounds for a smooth barycentric
partition of unity imply that $||W_{\beta}(f)||_{H^{n}_{\nabla}(B)}$ is
bounded by some constant times $||f||_{G,\sigma}$, where
$||\cdot||_{G,\sigma}$ is the $\ell^{1}$-norm on $C^{\bullet}(\sigma)$. For a
cochain $f=\sum a_{i}\delta_{\sigma_{i}}$, let
$\omega_{i}=W_{\beta}(\delta_{\sigma_{i}})$ so that $W_{\beta}(f)=\sum
a_{i}\omega_{i}$. We can then compute,

$\displaystyle||W_{\beta}(f)||_{H^{k}_{\nabla}(B)}$ $\displaystyle=||\sum
a_{i}\omega_{i}||_{H^{k}_{\nabla}(B)}$
$\displaystyle\leq\sum_{j}||\nabla^{j}\sum_{i}a_{i}\omega_{i}||_{2,B}$
$\displaystyle\leq\sum_{j}\sum_{i}|a_{i}|||\nabla^{j}\omega_{i}||_{2,B}.$

Each summand above satisfies $||\nabla^{j}\omega_{i}||_{2,B}<C$ for a constant
$C$ depending on the covariant derivative bounds of the barycentric partition
of unity. There is a constant $T$ such that the number of $\bullet$-faces of
an $n$-simplex is less than $T$. Thus,

$\displaystyle||W_{\beta}(f)||_{H^{k}_{\nabla}(B)}$
$\displaystyle\leq\sum_{j}\sum_{i}|a_{i}|||\nabla^{j}\omega_{i}||_{2,B}$
$\displaystyle\leq\sum_{j}C\sum_{i}|a_{i}|$ $\displaystyle\leq
TC\sum_{i}|a_{i}|$ $\displaystyle=TC||f||_{G,\sigma}.$

For a single simplex, this combinatorial $\ell^{1}$-norm is comparable to the
$\beta$-induced $L^{2}$-norm by Proposition 2.9 and Proposition 2.13. Thus,

$||W_{\beta}(f)||_{H^{n}_{\nabla}(B)}\leq R||W_{\beta}(f)||_{2,\sigma}.$

∎

The following version of the Sobolev inequality, a consequence of Theorem 1 in
[Can75], ensures that control over certain Sobolev norms implies pointwise
norm control. Let $|\cdot|$ be the pointwise norm induced by the Riemannian
metric.

######  Theorem 2.15.

(Cantor-Sobolev) Suppose $M$ is a hyperbolic $n$-manifold with injectivity
radius bounded below by $\varepsilon$. Let $r<\varepsilon$, and $l\geq
0,~{}k\geq 0$ be such that $l+n/2<k$. Then if $\omega$ is in the Sobolev space
$H^{k}_{\nabla}\Omega^{\bullet}(M)$, there is a constant $C=C(r)$ such that
for every $p\in M$,

$|\nabla^{l}\omega(p)|\leq C||\omega||_{H_{\nabla}^{k}(B_{r}(p))}.$

This discussion provides us with the following upgraded version of Proposition
2.9.

######  Proposition 2.16.

Let $s\in\mathcal{S}_{\varepsilon}$ be the combinatorial 1-neighborhood of an
$n$-simplex $\sigma\in\mathcal{G}_{\varepsilon}$. Let $\beta$ be the smooth
barycentric partition of unity in Proposition 2.13 or the standard barycentric
coordinate on $s$. Let $W_{\beta}(f)$ be the resulting generalized Whitney
form associated to a cochain $f\in C^{\bullet}(\sigma;\mathbb R)$. Let
$||\cdot||$ be some fixed norm on the real vector space
$C^{\bullet}(\sigma;\mathbb R)$ and let $||\cdot||_{p,s}$ be the $p$-norm
associated to $s$ on $\Omega^{\bullet}(s)$ and likewise for
$||\cdot||_{p,\sigma}$, where $p={\infty}$ or $p=2$. Then there is a constant
$\mathcal{B}=\mathcal{B}(\varepsilon,||\cdot||)>0$ (independent of $\sigma$
and $s$) such that

$\mathcal{B}^{-1}||W_{\beta}(f)||_{2,{\sigma}}\leq||f||\leq\mathcal{B}||W_{\beta}(f)||_{2,\sigma}$

and

$\mathcal{B}^{-1}||W_{\beta}(f)||_{\infty,\sigma}\leq\mathcal{B}^{-1}||W_{\beta}(f)||_{\infty,s}\leq||f||\leq\mathcal{B}||W_{\beta}(f)||_{\infty,s}.$

###### Proof.

Identify $s$ with a domain $D$ in $\mathbb H^{n}$ and let $B$ be a ball of
fixed radius $r=r(\varepsilon)$ that contains $s$ as in Lemma 2.11 and assume
the basepoint of the ball is the point at which $||W_{\beta}(f)||_{\infty,s}$
is realized.

From the Sobolev inequality, Propositions 2.9 and 2.13, and Lemma 2.14, for
any cochain $f$,

$\displaystyle||W_{\beta}(f)||_{\infty,\sigma}$
$\displaystyle\leq||W_{\beta}(f)||_{\infty,s}$ $\displaystyle\leq
C||W_{\beta}(f)||_{H^{n}_{\nabla}(B)}$ $\displaystyle\leq CR||f||_{G,\sigma}$
$\displaystyle\leq\mathcal{A}CR||W_{\beta}(f)||_{2,\sigma}.$

Then the comparison

$||W_{\beta}(f)||_{2,\sigma}\leq||W_{\beta}(f)||_{2,s}\leq\sqrt{\operatorname{vol}(s)}||W_{\beta}(f)||_{\infty,s}$

implies the smooth barycentric partition of unity induced $L^{2}$-norm
$||W_{\beta}(f)||_{2,\sigma}$ and $L^{\infty}$-norm
$||W_{\beta}(f)||_{\infty,s}$ are comparable for any simplex
$\sigma\in\mathcal{G}_{\varepsilon}$ with combinatorial 1-neighborhood
$s\in\mathcal{S}_{\varepsilon}$.

The only constant appearing in the comparison depending on $s$ is
$\sqrt{\operatorname{vol}(s)}$, which can be uniformly bounded by a constant
depending only on $\varepsilon$ by compactness of $\mathcal{S}_{\varepsilon}$.
As remarked after the proof of Proposition 2.13, the $L^{2}$-norm version of
the desired estimate follows from Proposition 2.9 and Proposition 2.13. The
above comparisons of the $L^{2}$-norm and the $L^{\infty}$-norm imply the
claim. ∎

##  3 Norm estimates

In this section, we use deeply embedded triangulations and the Whitney maps
described in the previous section to compare various geometric and
combinatorial norms on forms and cochains. Throughout, let be $M$ a closed
hyperbolic manifold of dimension $n>2$ with injectivity radius bounded below
by $\varepsilon>0$ and a fixed deeply embedded triangulation $K$. Let $\beta$
be the smooth barycentric partition of unity associated to $K$.

We require various comparisons of the following norms on cochain and chain
complexes associated to $M$ and $K$. The relevant norms are:

  1. 1.

The combinatorial Gromov norm $||\cdot||_{G}$ on any chain or cochain complex
given by $||\sum a_{i}\sigma_{i}||_{G}=\sum|a_{i}|$ and $||\sum
a_{i}\delta_{\sigma_{i}}||_{G}=\sum|a_{i}|$.

  2. 2.

The combinatorial Euclidean norm $||\cdot||_{E}$, which is the usual
$\ell^{2}$ norm on chains and cochains given by $||\sum
a_{i}\sigma_{i}||_{E}=\sqrt{\sum|a_{i}|^{2}}$ and $||\sum
a_{i}\delta_{\sigma_{i}}||_{E}=\sqrt{\sum|a_{i}|^{2}}$.

  3. 3.

The combinatorial max norm $||\cdot||_{\max}$ on any chain or cochain complex
given by $||\sum a_{i}\sigma_{i}||_{\max}=\max|a_{i}|$.

  4. 4.

The Whitney induced $L^{2}$-norm $||~{}\cdot~{}||_{2}$ on the cochain complex
$C^{\bullet}(K)$, given by $||f||_{2}=\sqrt{\int_{M}W_{\beta}(f)\wedge\star
W_{\beta}(f)}$.

  5. 5.

The Whitney induced $L^{\infty}$-norm $||\cdot||_{\infty}$ on the cochain
complex $C^{\bullet}(K)$, given by taking the essential supremum of the
pointwise Riemannian metric operator norms
$||f||_{\infty}=\operatorname*{ess\,sup}\limits_{p\in
M}||W_{\beta}(f)_{p}||_{\infty}.$

Given a norm $||\cdot||$ on the cochain complex $C^{\bullet}(K)$, let
$||\cdot||^{*}$ denote the dual norm on the linear dual chain complex
$C_{\bullet}(K)$ induced by the integration pairing:

$||a||^{*}=\sup\limits_{\begin{subarray}{c}||f||=1\\\ f\in
C^{\bullet}(K)\end{subarray}}\int_{a}W_{\beta}(f).$

######  Proposition 3.1.

There is a constant $B=B(\varepsilon)>0$ such that the norms $||\cdot||_{G}$
and $||\cdot||_{2}$ on $C^{\bullet}(K)$ satisfy

$||\cdot||_{G}\leq B\sqrt{\operatorname{vol}(M)}||\cdot||_{2},$

and the norms $||\cdot||_{G}$ and $||\cdot||_{2}^{*}$ on $C_{\bullet}(K)$
satisfy

$||\cdot||_{G}\leq B\sqrt{\operatorname{vol}(M)}||\cdot||_{2}^{*}.$

###### Proof.

Let $f=\sum\limits_{F}a_{F}\delta_{F}$ be a cochain. Then for any $n$-simplex
$\sigma$, $f|_{\sigma}=\sum\limits_{F\subset\sigma}a_{F}\delta_{F}$ and

$||f||^{2}_{2}=\sum\limits_{\sigma\in
K^{(n)}}||W_{\beta}(f|_{\sigma})|_{\sigma}||_{2}^{2}=\sum\limits_{\sigma\in
K^{(n)}}||W_{\beta}(f)|_{\sigma}||_{2}^{2}.$

Apply Proposition 2.16 to obtain $D=\mathcal{B}(\varepsilon,||\cdot||_{G})$.
This gives $||f|_{\sigma}||_{G}\leq D||f|_{\sigma}||_{2,\sigma},$ where
$||\cdot||_{2,\sigma}$ is the $L^{2}$-norm on the simplex $\sigma$ associated
to the smooth barycentric coordinate $\beta$. Then, by applying the Euclidean
$\ell^{1}$-$\ell^{2}$-comparison to the cochain complex and using the fact
there is a constant $T$ such that the number of $n$-simplices in $K$ is less
than $T\operatorname{vol}(M)$, we find

$\displaystyle||f||_{G}$ $\displaystyle\leq\sum\limits_{\sigma\in
K^{(n)}}||f|_{\sigma}||_{G}$ $\displaystyle\leq\sum\limits_{\sigma\in
K^{(n)}}D||f|_{\sigma}||_{2,\sigma}$ $\displaystyle\leq
D\sqrt{T\operatorname{vol}(M)\sum\limits_{\sigma\in
K^{(n)}}||W_{\beta}(f)|_{\sigma}||_{2}^{2}}$ $\displaystyle\leq
D\sqrt{T\operatorname{vol}(M)}||f||_{2}.$

For the second inequality, notice $||\cdot||_{G}$ is the usual $\ell^{1}$-norm
on a finite dimensional vector space, so its dual norm is the max norm
$||\cdot||_{\max}$.

Apply Proposition 2.16 and set
$D\textquoteright=\mathcal{B}(\varepsilon,||\cdot||_{\max}),$ so that if
$\sigma$ is the simplex in which $||f||_{\infty}$ is realized, then

$||f||_{\infty}=||f|_{\sigma}||_{\infty}\leq
D\textquoteright||f|_{\sigma}||_{\max}\leq D\textquoteright||f||_{\max}.$

Then,

$||f||_{2}\leq\sqrt{\operatorname{vol}(M)}||f||_{\infty}\leq
D\textquoteright\sqrt{\operatorname{vol}(M)}||f||_{\max}.$

Dualizing gives

$||\cdot||_{G}=||\cdot||_{\max}^{*}\leq
D\textquoteright\sqrt{\operatorname{vol}(M)}||\cdot||_{2}^{*},$

since

$\displaystyle||a||_{G}$ $\displaystyle=||a||_{\max}^{*}$
$\displaystyle=\sup\limits_{||f||_{\max}\leq 1}\int_{a}W_{\beta}(f)$
$\displaystyle=\sup\limits_{||D\textquoteright\sqrt{\operatorname{vol}(M)}f||_{\max}\leq
1}\int_{a}D\textquoteright\sqrt{\operatorname{vol}(M)}W_{\beta}(f)$
$\displaystyle\leq\sup\limits_{||f||_{2}\leq
1}\int_{a}D\textquoteright\sqrt{\operatorname{vol}(M)}W_{\beta}(f)$
$\displaystyle=D\textquoteright\sqrt{\operatorname{vol}(M)}\sup\limits_{||f||_{2}\leq
1}\int_{a}W_{\beta}(f)$
$\displaystyle=D\textquoteright\sqrt{\operatorname{vol}(M)}||a||_{2}^{*}.$

Set $B=\max\\{D\sqrt{T},D\textquoteright\\}$ to obtain the claim. ∎

Recall from Section 2 that there is a polyhedral celluation $K^{*}$ dual to
$K$ that can be canonically subdivided into a triangulation $\tau(K)$.
Equipping these dual complexes with the Gromov norm, we have the following two
propositions relating these norms by the Poincaré duality and subdivision
maps.

######  Proposition 3.2.

There is a constant $D=D(\varepsilon)$ such that for any $\bullet$-cochain
$f\in C^{\bullet}(K)$ one has $||f||_{2}\leq D||f||_{G}$.

###### Proof.

Let $f=\sum a_{i}\delta_{\sigma_{i}}$ be a $\bullet$-cochain. Then
$||\omega||_{G}=\sum|a_{i}|$ and
$||f||_{2}\leq\sum|a_{i}|||\delta_{\sigma_{i}}||_{2}.$ Then, for any fixed
$\bullet$-simplex $\sigma$ that is a face of an $n$-simplex from
$\mathcal{G}_{e}$, using the $L^{2}$-change of variables formula and
Proposition 2.13 we can take $D=A\mathcal{L}^{n/2}||W(\delta_{\sigma})||_{2}$,
so that $||\delta_{\sigma_{i}}||_{2}\leq D.$

The comparison

$||f||_{2}\leq\sum|a_{i}|||\delta_{\sigma_{i}}||_{2}\leq
D\sum|a_{i}|=D||f||_{G}$

then follows. ∎

######  Proposition 3.3.

The Poincaré duality map $\Phi:C^{\bullet}(K)\to C_{n-\bullet}(K^{*})$
preserves the Gromov norm

$||f||_{G}=||\Phi(f)||_{G}.$

###### Proof.

Let $f=\sum a_{i}\delta_{\sigma_{i}}$, then $\Phi(f)=\sum
a_{i}(\sigma_{i})^{*}$. ∎

######  Proposition 3.4.

Let $N$ be the constant from Proposition 2.3, which bounds the number of
simplices in the star of a simplex in a deeply embedded triangulation. Then
the subdivision map $\tau:C_{2}(K^{*})\to C_{2}(T)$ satisfies

$||\tau(c)||_{G}\leq N||c||_{G}.$

###### Proof.

The number of sides of a $2$-cell in $K^{*}$ dual to a $(n-2)$-simplex
$\sigma$ in $K$ corresponds to the number of $n$-simplices in $K$ that contain
$\sigma$. The number of such simplices is bounded by $N$. ∎

The following estimates are essential in comparing the first eigenvalue of the
Whitney Laplacian to the genuine first eigenvalue. For this, we need to work
with various Sobolev spaces to control the orthogonal projection of a Whitney
form onto its coexact part. This discussion is the reason we use the smoothed
Whitney forms in place of the standard ones.

We will require the following version of the Gaffney inequality, which follows
from Lemma 2.4.10 in [Sch95]. To simplify the following discussion, for a
smooth manifold $Y$ possibly with boundary, we introduce an alternative
Sobolev norm on $H^{k+1}_{\nabla}\Omega^{\bullet}(Y)$:

$||\omega||_{A^{k}(Y)}:=||\omega||_{H^{k}_{\nabla}(Y)}+||d\omega||_{H^{k}_{\nabla}(Y)}+||d^{*}\omega||_{H^{k}_{\nabla}(Y)}.$

Since $d$ and $d^{*}$ are bounded operators
$H^{k+1}\Omega^{\bullet}_{\nabla}(Y)\to H^{k}\Omega^{\bullet\pm
1}_{\nabla}(Y)$, we immediately have that there is a constant $C$ such that
$||\omega||_{A^{k}(Y)}\leq C||\omega||_{H^{k+1}_{\nabla}(Y)}.$

Recall that the marking $\mathring{H}$ denotes the subspace of given Sobolev
space that is the closure of smooth functions supported in the interior.

######  Lemma 3.5.

(Gaffney inequality) Let $Y$ be a smooth manifold with boundary. Let
$\omega\in\mathring{H}^{k}_{\nabla}\Omega^{1}(Y)$. Then there is a constant
$C=C(Y)>0$ such that $||\omega||_{\mathring{H}^{k}_{\nabla}(Y)}\leq
C||\omega||_{A^{k-1}(Y)}$.

######  Lemma 3.6.

Let $B_{0}=B_{r}(p)$ and $B_{1}=B_{r+\delta}(p)$ be a pair of concentric balls
in $\mathbb H^{n}$ and let $\phi$ be a smooth bump function that is
identically 1 on $B_{0}$ and vanishes in a neighborhood of $\partial B_{1}$.
There is a constant $C=C(\phi,k)$ that depends only on the norm of the
covariant derivatives of $\phi$ up to order $k+1$ such that if
$\omega\in\Omega^{1}(\mathbb H^{n})$, then

$||\phi\omega||_{A^{k}(B_{1})}\leq C||\omega||_{A^{k}(B_{1})}.$

###### Proof.

Notice that $d\phi\omega=d\phi\wedge\omega+\phi d\omega$ and
$d^{*}\phi\omega=\phi d^{*}\omega+g(\nabla\phi,X_{\omega}),$ where
$X_{\omega}$ is the vector field dual to $\omega$. As a result, the triangle
inequality yields

$\displaystyle||\phi\omega||_{A^{k}(B_{1})}$ $\displaystyle\leq||\phi
d\omega||_{H^{k}_{\nabla}(B_{1})}+||\phi
d^{*}\omega||_{H^{k}_{\nabla}(B_{1})}+||\phi\omega||_{H^{k}_{\nabla}(B_{1})}$
$\displaystyle+||d\phi\wedge\omega||_{H^{k}_{\nabla}(B_{1})}+||g(X_{\omega},\nabla\phi)||_{H^{k}_{\nabla}(B_{1})}.$

The estimate $||\alpha\wedge\beta||_{2}\leq||\alpha||_{\infty}||\beta||_{2}$
implies

$||d\phi\wedge\omega||_{H^{k}_{\nabla}(B_{1})}\leq
C||\omega||_{H^{k}_{\nabla}(B_{1})},$

where the constant $C$ is given by the sum of the $||\cdot||_{\infty}$-norms
of the covariant derivatives of the bump function $\phi$. The same argument
gives for any form $\xi$ that $||\phi\wedge\xi||_{H^{k}_{\nabla}(B_{1})}\leq
C||\xi||_{H^{k}_{\nabla}(B_{1})}$. Applying this estimate to $\phi
d\omega,\phi d^{*}\omega$, and $\phi\omega$ handles all terms in the
comparison save for $||g(X_{\omega},\nabla\phi)||_{H^{k}_{\nabla}(B_{1})}$.
For this term, notice that $\nabla g(X_{\omega},\nabla\phi)=g(\nabla
X_{\omega},\nabla\phi)+g(X_{\omega},\nabla^{2}\phi).$ We therefore have the
pointwise estimate

$\displaystyle|\nabla g(X_{\omega},\nabla\phi)|$ $\displaystyle\leq|g(\nabla
X_{\omega},\nabla\phi)|+|g(X_{\omega},\nabla^{2}\phi)|\text{ by the triangle
inequality,}$
$\displaystyle\leq|\nabla\omega|^{2}|\nabla\phi|^{2}+|\omega|^{2}|\nabla^{2}\phi|^{2},$

by applying Cauchy-Schwarz and the musical isomorphism. Integrating then gives
the corresponding inequality for the Sobolev norm $H^{1}_{\nabla}$. Repeating
this calculation for higher order covariant derivatives completes the proof. ∎

The two previous lemmas combine to give the following statement.

######  Proposition 3.7.

Let $B_{0}=B_{r}(p)$ and $B_{1}=B_{r+\delta}(p)$ be a pair of concentric balls
in $\mathbb H^{n}$ Then there is a constant $C=C(r,\delta)$ such that for any
$\omega\in H^{k}_{\nabla}(B_{1})$ one has

$||\omega||_{H^{k}_{\nabla}(B_{0})}\leq C||\omega||_{A^{k-1}(B_{1})}.$

###### Proof.

Let $\sqrt{C}$ be the maximum of the constants from Gaffney’s inequality and
Lemma 3.6 with bump function $\phi$. Then for a form $\omega\in
H^{k}_{\nabla}(B_{1})$, one has

$||\omega||_{H^{k}_{\nabla}(B_{0})}\leq||\phi\omega||_{H^{k}_{\nabla}(B_{1})},$

since $\phi\omega|_{B_{0}}=\omega.$ Since $\phi\omega$ vanishes on $\partial
B_{1}$, Gaffney’s inequality and Lemma 3.6 give

$||\phi\omega||_{H^{k}_{\nabla}(B_{1})}=||\phi\omega||_{\mathring{H}^{k}_{\nabla}(B_{1})}\leq\sqrt{C}||\phi\omega||_{A^{k-1}(B_{1})}\leq
C||\omega||_{A^{k-1}(B_{1})}.$

Combining these two estimates gives the proposition. ∎

######  Proposition 3.8.

Let $M$ be a hyperbolic $n$-manifold with injectivity radius greater than
$\varepsilon>0$. There is a constant $H(\varepsilon)=H>0$ depending only on
$\varepsilon$ such that if the $L^{2}$-Hodge decomposition of a smooth 1-form
$\omega\in\Omega^{1}$ has coexact part $\alpha$, then for any point $p\in M$
there is a ball $B\subset M$ centered at $p$ of radius determined by
$\varepsilon$ such that for $k\geq n$ one has

$|\nabla\alpha(p)|\leq
H\left(||\omega||_{H^{k}_{\nabla}(B)}+||\alpha||_{2,B}\right)$

and consequently

$|\nabla\alpha(p)|\leq H||\omega||_{H^{k}_{\nabla}(M)}.$

###### Proof.

The $L^{2}$-Hodge decomposition of $M$ determines an orthogonal decomposition
of $\omega=\alpha+\eta+h$, where $\alpha=d^{*}A$, $\eta=db$, and $h$ harmonic.
Cantor’s estimate (Theorem 2.15 implies at any point $p$ of $M$,

$|\nabla\alpha(p)|\leq C(r)||\alpha||_{H^{k}_{\nabla}(B_{r}(p))}$

so long as $r<\operatorname{inj}(M)$ and $k>n/2+1$. Since we assume dimension
$n>2$, any $k\geq n$ suffices. Take a concentric family of balls
$B_{i}=B_{r+i\delta}(p)$ where $r=6\varepsilon$, and
$k\delta+r<10\varepsilon<\operatorname{inj}(M)$; note that $B_{0}$ contains
the 0-star of $\sigma$. Let $\phi_{i}$ be a bump function that is identically
one on $B_{i}$ and vanishes on $\partial B_{i+1}$.

Letting $C_{i}$ be the maximum of the constant from Cantor’s estimate on the
ball $B_{i}$ and the constant from Proposition 3.7 for the balls $B_{i}\subset
B_{i+1}$, we have

$|\nabla\alpha(p)|\leq C_{i}||\alpha||_{H^{k}_{\nabla}(B_{i})}\leq
C_{i}^{2}||\alpha||_{A^{k-1}(B_{i+1})}.$

Since $\alpha$ is coexact,

$||\alpha||_{A^{k-1}(B_{i+1})}=||\alpha||_{H^{k-1}_{\nabla}(B_{i+1})}+||d\alpha||_{H^{k-1}_{\nabla}(B_{i+1})}.$

Notice that $d\alpha=d\omega$, and that $d:H^{k}_{\nabla}(B_{i})\to
H^{k-1}_{\nabla}(B_{i})$ is a bounded operator, say with operator norm $C$.
Thus, we have

$||d\alpha||_{H^{k-1}_{\nabla}(B_{i+1})}\leq C||\omega||_{H^{k}_{\nabla}}.$

As a result, we get the estimate

$||\alpha||_{A^{k-1}(B_{i+1})}\leq||\alpha||_{H^{k-1}_{\nabla}(B_{i+1})}+C||\omega||_{H^{k}_{\nabla}(B_{i+1})}$

Combining this with the estimate of $\nabla\alpha$ above gives

$|\nabla\alpha(p)|\leq
C_{i}\left(||\omega||_{H^{k}_{\nabla}(B_{i+1})}+||\alpha||_{H^{k-1}_{\nabla}(B_{i+1})}\right).$

We can repeat argument this using the family of balls $B_{i}$ to reduce the
order of the Sobolev norm of $\alpha$ on the right-hand side until we obtain

$|\nabla\alpha(p)|\leq
H\left(||\omega||_{H^{k}_{\nabla}(B_{n})}+||\alpha||_{2,B_{n}}\right),$

where $H$ is obtained by combining all the constants appearing in the iterated
calculation. Orthogonality of the Hodge decomposition implies
$||\alpha||_{2}\leq||\omega||_{2}$, and we clearly have
$||\omega||_{2,B_{n}}\leq||\omega||_{H^{k}_{\nabla}(M)},$ so we are done after
increasing $H$ by 1. ∎

When the form in the previous proposition is a Whitney form, we can make the
following refinement.

######  Proposition 3.9.

Let $M$ be a hyperbolic $n$-dimensional manifold with a deeply embedded
triangulation $K$ and let $f\in C^{1}(K)$. There is a constant
$H(\varepsilon)=H>0$ depending only on $\varepsilon$ such that if the
$L^{2}$-Hodge decomposition of the generalized Whitney form $W_{\beta}(f)$ has
coexact part $\alpha$, then

$||\nabla\alpha||_{\infty}\leq H||W_{\beta}(f)||_{2}.$

###### Proof.

Let $\omega=W_{\beta}(f)$ be a smooth Whitney form. We will apply Proposition
3.8 at a point $p$ at which $||\nabla\alpha||_{\infty}$ is realized.
Proposition 3.8 gives a ball $B$ about $p$ of radius depending only on
$\varepsilon$ such that

$|\nabla\alpha(p)|\leq
H\left(||\omega||_{H^{k}_{\nabla}(B)}+||\alpha||_{2,B}\right),$

for a constant $H$ depending only on $\varepsilon$. The ball $B$ intersects
some uniformly bounded collection of $n$-simplices $\sigma\textquoteright$
from $K$ where the constant depends only on $\varepsilon$; let
$T=T(\varepsilon)$ be this bound. We can therefore estimate the norm of the
Whitney form term by

$||\omega||_{H^{k}_{\nabla}(B)}\leq\sum\limits_{\sigma\textquoteright\cap
B\neq\emptyset}||\omega|_{\sigma\textquoteright}||_{H^{k}_{\nabla}(\sigma\textquoteright)}.$

Applying Lemma 2.13 to each summand in the previous estimate then gives

$||\omega||_{H^{k}_{\nabla}(B)}\leq\sum\limits_{\sigma\textquoteright\cap
B_{i}\neq\emptyset}||\omega|_{\sigma\textquoteright}||_{H^{k}_{\nabla}(\sigma\textquoteright)}\leq
R\sum\limits_{\sigma\textquoteright\cap
B\neq\emptyset}||\omega|_{\sigma\textquoteright}||_{2}\leq
R\sqrt{T}||\omega||_{2}.$

Combining the above then gives that

$|\nabla\alpha(p)|\leq HR\sqrt{T}||\omega||_{2}+H||\alpha||_{2,B}.$

Clearly

$H||\alpha||_{2,B}\leq H||\alpha||_{2}\leq H||\omega||_{2},$

so that after increasing $H$ to absorb the $R\sqrt{T}$ term we are done. ∎

######  Proposition 3.10.

Let $M$ be a hyperbolic $n$-manifold. Let $\omega\in\Omega^{1}(M).$ Assume
there exists a constant $H$ such that $|\nabla\omega|\leq H$. Then there is a
constant $C(H,\varepsilon)$ such that $||\omega||_{\infty}\leq
C(H,\varepsilon)||\omega||_{2}.$

###### Proof.

Assume $||\omega||_{\infty}=1$ and is realized at the point $p$. By Kato’s
inequality and the hypothesis, away from the zeros of $\omega$ one has
$|\nabla|\omega||\leq|\nabla\omega|\leq H$. Fix a normal coordinate frame
$x_{0},\dots,x_{n-1}$ at $p$ of radius $2\varepsilon$. Define the function
$\phi$ on this normal coordinate neighborhood by $\phi(x)=1-Hd(x,p)$ for
$d(x,p)<1/H$ and extend by zero. Then $||\phi||_{\infty}=||\omega||_{\infty}$
and $||\phi||_{2}\leq||\omega||_{2}$. The claim then holds for
$C(H,\varepsilon)=1/||\phi||_{2}$ by scaling the unit norm case. ∎

######  Proposition 3.11.

There is a constant $C=C(\varepsilon)$ such that if $f\in C^{1}(K)$ and
$\omega=W_{\beta}(f)=\alpha+\eta$ where $\alpha$ is $L^{2}$-coexact and $\eta$
is closed, then

$||\alpha||_{\infty}\leq C||\alpha||_{2}.$

###### Proof.

Assume that $||f||_{2}=1$. By Proposition 3.9, $||\nabla\alpha||_{\infty}\leq
H||f||_{2}=H$. Proposition 3.10 gives a constant $C=C(H(\varepsilon))$ (so
this really just depends on $\varepsilon$) such that $||\alpha||_{\infty}\leq
C||\alpha||_{2}$. If $f$ does not have unit $L^{2}$-norm, then either $f=0$,
in which case the result is trivial, or else $f=\lambda f\textquoteright$ for
some unit $L^{2}$-norm cochain $f\textquoteright$ and positive number
$\lambda$. The coexact part of $W_{\beta}(f\textquoteright)$ is
$\alpha\textquoteleft=\alpha/\lambda$. Hence,
$||\alpha\textquoteright||_{\infty}\leq C||\alpha\textquoteright||_{2}$, and
the result follows. ∎

##  4 The upper bound

In this section we prove Theorem A, which states that in a closed hyperbolic
3-manifold $M$ the first positive eigenvalue of the Hodge Laplacian acting on
coexact 1-forms is bounded above by a multiple of the stable isoperimetric
ratio $\rho(M)$. The background results of this section all hold in any
dimension greater than 2, however the proof of Theorem A makes use of Poincaré
duality to relate 1-forms and surfaces, this forces us to restrict Theorem A
to the 3-dimensional setting.

The cochain results of the previous section are connected to spectral geometry
via the inner product induced by the Whitney map associated to a triangulation
and barycentric coordinate:

$\langle f,g\rangle=\int_{M}W_{\beta}(f)\wedge\star W_{\beta}(g),$

which along with the corresponding norm $||\cdot||_{2}$, determine a Hodge
theory for the cochain complex $C^{\bullet}(K)$. This inner product determines
a codifferential

$d^{*}_{W}:C^{\bullet}(K)\to C^{\bullet-1}(K)$

which, as the adjoint of the standard differential, satisfies $\langle
df,g\rangle=\langle f,d_{W}^{*}g\rangle.$ The corresponding Whitney Laplacian
$\Delta_{W}:C^{\bullet}(K)\to C^{\bullet}(K)$ is then given by the standard
formula $\Delta_{W}=dd_{W}^{*}+d_{W}^{*}d.$ This inner product was introduced
using the standard barycentric coordinates in [Dod76].

This Laplacian decomposes the space $C^{\bullet}(K)$ into harmonic, exact, and
coexact components:

$C^{\bullet}(K)\cong H^{\bullet}(M)\oplus dC^{\bullet-1}(K)\oplus
d_{W}^{*}C^{\bullet+1}(K).$

This combinatorial Hodge decomposition serves as a good approximation of the
$L^{2}$-Hodge decomposition of $M$, though it does not capture the
$L^{2}$-Hodge decomposition exactly. In particular, the Whitney coexact chains
may not be $L^{2}$-coexact.

We begin by relating the Whitney and the Riemannian coexact eigenvalues.

######  Lemma 4.1.

Let $M$ be a closed Riemannian $n$-manifold with triangulation $K$ and an
associated barycentric partition of unity $\beta$. Give the cochain complex
the Whitney $L^{2}$-norm induced by the Whitney map determined by $\beta.$
Likewise, give the chain complex the dual norm $||\cdot||_{2}^{*}$ determined
by the integration pairing. Then for every coexact cochain $f\in
d_{W}^{*}C^{2}(K)$, there is an exact chain $a\in\partial C_{2}(K)$ of unit
norm such that $||f||_{2}=\int_{a}W_{\beta}(f).$

###### Proof.

The cochain Hodge decomposition from the Whitney inner product gives the
orthogonal decomposition

$C^{1}(K)=H^{1}(M)\oplus d_{W}^{*}C^{2}(K)\oplus dC_{0}(K).$

Let $Z^{1}(K)=H^{1}(M)\oplus dC_{0}(K)$. Identify $C_{1}(K)$ with
$C^{1}(K)^{*}$ via the integration pairing. The composition of the inclusion
and quotient map determines an isomorphism $d_{W}^{*}C^{2}(K)\to
C^{1}(K)/Z^{1}(K)$ that allows us to identify these spaces. If Ann assigns to
a subspace its annihalator, then there is also an isomorphism
$(C^{1}(K)/Z^{1}(K))^{*}\to\text{Ann}(Z^{1}(K)).$ By Stokes’ theorem and
dimension counting, $\text{Ann}(Z^{1}(K))=\partial C_{2}(K)$. Thus, the dual
of $d_{W}^{*}C^{2}(K)$ is exactly $\partial C_{2}(K)$. The dual norm of an
element $a\in\partial C_{2}(K)$ is given by

$||a||_{2}^{*}=\sup\limits_{\begin{subarray}{c}f\in C^{1}(K)\\\ ||f||_{2}\leq
1\end{subarray}}\int_{a}W_{\beta}(f).$

If $f$ has unit $L^{2}$-norm and $f=g+h$ where $g\in d_{W}^{*}C^{2}(K)$ and
$h\in Z^{1}(K)$, then orthogonality implies $||g||_{2}\leq 1.$ Whence,

$||a||_{2}^{*}=\sup\limits_{\begin{subarray}{c}f=g+h\in C^{1}(K)\\\
||f||_{2}\leq
1\end{subarray}}\int_{a}W_{\beta}(g)=\sup\limits_{\begin{subarray}{c}g\in
d^{*}_{W}C^{2}(K)\\\ ||g||_{2}\leq 1\end{subarray}}\int_{a}W_{\beta}(g).$

The isometric identification of $(d^{*}_{W}C^{2}(K),||\cdot||_{2})$ with its
double dual therefore implies we can compute the norm of an element $f\in
d^{*}_{W}C^{2}(K)$ via the integration pairing integrating only against chains
in $\partial C_{2}(K)$:

$||f||_{2}=\sup\limits_{\begin{subarray}{c}a\in\partial C_{2}(K)\\\
||a||_{2}^{*}=1\end{subarray}}\int_{a}W_{\beta}(f).$

In particular, for any coexact cochain $f\in d^{*}_{W}C^{1}(K)$, there exists
an exact chain $a$ with $||a||^{*}_{2}=1$ and

$\int_{a}W_{\beta}(f)=||f||_{2}.$

∎

######  Proposition 4.2.

Let $\lambda$ denote the first eigenvalue for the Hodge Laplacian acting on
coexact $1$-forms and let $\lambda_{W}$ denote the first eigenvalue for the
Whitney Laplacian acting on coexact 1-cochains associated to a deeply embedded
triangulation $K$ and barycentric partition of unity $\beta$. There is a
constant $G=G(\varepsilon)$ such that

$\lambda\leq G\operatorname{vol}(M)\lambda_{W}.$

###### Proof.

The main issue here is that a Whitney coexact cochain will not generally map
to an $L^{2}$-coexact form. This potentially adds a closed term to the
denominator in the Whitney Rayleigh quotient, causing the Whitney Rayleigh
quotient to be smaller than the Riemannian Rayleigh quotient. However, this
failure can be controlled.

Let $f$ be a coexact eigen-cochain with eigenvalue $\lambda_{W}$. Set
$\omega=W_{\beta}(f)\in\Omega^{1}(M)$, so that
$\frac{||d\omega||_{2}^{2}}{||\omega||_{2}^{2}}=\lambda_{W}$. Let
$p:\Omega^{1}(M)\to\Omega^{1}(M)$ be the $L^{2}$-orthogonal projection onto
coexact forms. Let $a\in C_{1}(K)$ be the unit norm exact chain that realizes
the norm of $f$ by integration given by Lemma 4.1. Then using that
$d\omega=d(p(\omega))$ and the fact $a$ is exact, we obtain

$||f||_{2}=||\omega||_{2}=\int_{a}\omega=\int_{a}p(\omega).$

Using 3.11, we have $||p(\omega)||_{\infty}\leq C||f||_{2}.$ Hence,
$||f||_{2}\leq C\text{len}(a)||p(\omega)||_{2}.$ We can therefore obtain a
lower bound on $||p(\omega)||_{2}$ by bounding $\text{len}(a)$ from above.
Applying Proposition 3.1 to the chain $a$ above gives

$\mathcal{|}|a||_{G}\leq
B\sqrt{\operatorname{vol}(M)}||a||_{2}^{*}=B\sqrt{\operatorname{vol}(M)}.$

Since the lengths of the edges in the triangulation are bounded, we conclude
the length of the support of $a$ is bounded. Take $E$ to be the length of the
largest edge possible in a deeply embedded triangulation, so that
$\text{len}(a)\leq BE\sqrt{\operatorname{vol}(M)}$ Then we have obtained

$||\omega||_{2}=\int_{a}\omega=\int_{a}p(\omega)\leq||p(\omega)||_{2}BCE\sqrt{\operatorname{vol}(M)}.$

Setting $G=(BCE)^{2}$ and using that $\omega$ is a Whitney eigenform, we
obtain the result by the following short computation:

$\lambda\leq\frac{||d\omega||_{2}^{2}}{||p(\omega)||_{2}^{2}}\leq
G\operatorname{vol}(M)\frac{||d\omega||_{2}^{2}}{||\omega||_{2}^{2}}=G\operatorname{vol}(M)\lambda_{W}.$

∎

######  Remark 2.

Note that the above estimate in fact holds for the first positive eigenvalue
since the first positive eigenvalue $\lambda$ is the minimium of the first
eigenvalue of the Laplacian acting on functions and the first eigenvalue of
the Laplacian acting on coexact 1-forms. The first eigenvalue $\lambda_{f}$
for the Laplacian acting on functions automatically satisfies the comparison
$\lambda_{f}\leq\lambda_{W}$, as can be seen by studying the Rayleigh quotient
and noticing that the estimate above controlling the projection in the
denominator is immaterial in the function case.

We are now ready to introduce stable commutator length, a thorough reference
for which is [Cal09]. For a group $\Gamma$, let $\Gamma\textquoteright$ denote
the commutator subgroup and define the rational commutator subgroup to be

$\Gamma_{{\mathbb
Q}}\textquoteright=\text{Ker}(\Gamma\to\Gamma^{ab}\otimes{\mathbb Q}).$

Note that when $\Gamma$ is the fundamental group of a manifold, these
subgroups correspond to the integrally nullhomologous and rationally
nullhomologous loops respectively. The commutator length of an element
$\gamma\in\Gamma\textquoteright$, denoted ${\tt{cl}}(\gamma)$ is the shortest
word length of $\gamma$ with respect to the generating set of all commutators.
The stable commutator length for $\gamma\in\Gamma\textquoteright_{{\mathbb
Q}}$ is then defined to be

$\operatorname{{\tt{scl}}}(\gamma)=\inf\limits_{m\geq
1}\frac{{\tt{cl}}(\gamma^{m})}{m}.$

Topologically, stable commutator length corresponds to the stable complexity
of a surface bounding a nullhomologous curve. In particular, for
$\gamma\in\Gamma\textquoteright_{{\mathbb Q}}$, one has

$\operatorname{{\tt{scl}}}(\gamma)=\inf\left\\{\frac{\chi_{-}(S)}{2m}~{}:~{}S\text{
with }\partial S=\gamma^{m}\text{ and $S$ with no closed
components}\right\\},$

where for a connected surface $S$ we define
$\chi_{-}(S)=\max\\{0,-\chi(S)\\}$, and extend this additively to disconnected
surfaces. There is another natural complexity measure for loops in
$\Gamma\textquoteright_{\mathbb Q}$, the Gersten filling norm. For a loop
$\gamma\in\Gamma\textquoteright_{\mathbb Q}$,
$\operatorname{{\tt{fill}}}(\gamma)$ is the infimum of the Gromov norm
$\frac{||A||_{G}}{m}$ for all singular 2-chains $A$ bounding a 1-cycle
representing a singular fundamental class of $\gamma^{m}$. A fundamental
theorem of Bavard relates the filling norm to the stable commutator length.

######  Theorem 4.3.

([Bav91]) For any group element $\gamma$, there is an equality:

$\operatorname{{\tt{scl}}}(\gamma)=4\operatorname{{\tt{fill}}}(\gamma).$

For proof, see for instance Lemma 2.69 in [Cal09].

######  Remark 3.

Let $B(\Gamma)$ be the $\mathbb R$-vector space of 1-boundaries. Then stable
commutator length can be extended to a psuedo-norm on $B(\Gamma)$. After
identifying chains with vanishing psuedo-norm, Bavard duality, which relates
the filling norm to quasimorphisms and their defect norm, becomes a genuine
functional analytic duality theorem. One could define the stable isoperimetric
ratio in this chain setting, and the results of this paper would go through
for that (smaller) ratio as well.

We can now prove the main theorem of this section.

######  Theorem A.

Let $M$ be a closed hyperbolic 3-manifold with injectivity radius bound below
by $\varepsilon$. There is a constant $A=A(\varepsilon)$ that only depends on
$\varepsilon$ such that for any nontrivial boundary
$\gamma\in\Gamma\textquoteright_{{\mathbb Q}}$, one has

$\sqrt{\lambda}\leq
A\operatorname{vol}(M)\frac{|\gamma|}{\operatorname{{\tt{scl}}}(\gamma)},$

where $\lambda$ is the first coexact eigenvalue of the Hodge Laplacian on
$\Omega^{1}(M)$.

###### Proof.

First note that since stable commutator length and geodesic length are both
multiplicative under powers, it suffices to show the claim for an integrally
nullhomologous loop $\gamma$.

Fix a deeply embedded triangulation $K$ of $M$ and denote by $\lambda_{W}$ the
first eigenvalue of the Whitney Laplacian $\Delta_{W}$ acting on
$d^{*}_{W}C^{2}(K)$ associated to a smooth barycentric partition of unity.
Notice that the Hodge decomposition ensures that zero is not an eigenvalue of
this operator. Let $c:S^{1}\to M$ be a cellular path in the 1-skeleton of
$K^{*}$ representing the loop $\gamma$, constructed as in Proposition 2.5. Let
$T$ be a triangulation of $K^{*}$. Let $a\in C_{1}(K^{*})$ be the fundamental
cycle for $\gamma$ corresponding to the path $c$ in $C_{1}(K^{*})\subset
C_{1}(T)\subset C_{1}^{\text{sing}}(M)$. If $\Phi:C^{2}(K)\to C_{1}(K^{*})$ is
the Poincaré duality map, then $\Phi^{-1}(a)$ is an exact 2-cochain. We can
therefore choose $\omega\in d^{*}_{W}C^{2}(K)$ with $d\omega=\Phi^{-1}(a)$.
Setting $A=\Phi(\omega)$ in $C_{2}(K^{*})$, we have $\partial A=a$ and
$||A||_{G}=||\omega||_{G}.$ Since $\lambda_{W}$ is nonzero, we have
$||\omega||_{2}\leq\frac{||d\omega||_{2}}{\sqrt{\lambda_{W}}}$. Proposition
4.2 implies

$||\omega||_{2}\leq\frac{\sqrt{G}\sqrt{\operatorname{vol}(M)}||d\omega||_{2}}{\sqrt{\lambda}}.$

By Bavard’s theorem relating the filling norm to stable commutator length
(Theorem 4.3), our choice of $A$, and Proposition 3.4 we find that

$\operatorname{{\tt{scl}}}(\gamma)=4\operatorname{{\tt{fill}}}(\gamma)\leq
4||\tau(A)||_{G}\leq 4N||A||_{G}=4N||\omega||_{G},$

where, as in Proposition 3.4, $\tau$ is the triangulation map relating the
cellular chain $A$ to the subdivided simplicial chain in $C_{2}(T)$.
Consequently,

$\displaystyle\operatorname{{\tt{scl}}}(\gamma)$ $\displaystyle\leq
4N||\omega||_{G}$ $\displaystyle\leq
4NB\sqrt{\operatorname{vol}(M)}||\omega||_{2}~{}\text{by Proposition
\ref{prop: 3.1},}$ $\displaystyle\leq
4NB\sqrt{G}\operatorname{vol}(M)\frac{||d\omega||_{2}}{\sqrt{\lambda}}~{}\text{by
above computation,}$ $\displaystyle\leq
4NB\sqrt{G}\operatorname{vol}(M)\frac{D||d\omega||_{G}}{\sqrt{\lambda}}~{}\text{by
Proposition \ref{prop: 3.2},}$
$\displaystyle=4NB\sqrt{G}\operatorname{vol}(M)\frac{D||\partial
A||_{G}}{\sqrt{\lambda}}~{}\text{by Proposition \ref{prop: 3.3},}$
$\displaystyle=4NB\sqrt{G}\operatorname{vol}(M)\frac{D||c||_{G}}{\sqrt{\lambda}}\text{
by construction of $\partial A$,}$ $\displaystyle\leq
4NB\sqrt{G}\operatorname{vol}(M)\frac{DL|\gamma|}{\sqrt{\lambda}}~{}\text{by
Proposition \ref{prop: length comparison},}$
$\displaystyle=4NB\sqrt{G}DL\operatorname{vol}(M)\frac{|\gamma|}{\sqrt{\lambda}}.$

Setting $A=4NB\sqrt{G}DL$ and rearranging, we are done. ∎

##  5 The lower bound

We now turn to proving the lower bound on the first coexact eigenvalue of the
1-form Laplacian that constitutes Theorem B. Unlike Theorem A, we prove this
eigenvalue comparison without a dimension constraint. The line of proof
follows that of Theorem 1.3 in [LS18].

In [LS18], the authors obtain the following estimate controlling the
$L^{2}$-norm of coclosed forms. Note that this estimate does not depend on the
fundamental domain coming from a deeply embedded triangulation.

######  Proposition 5.1.

(Proposition 5.4 in [LS18]) Let $\eta$ be a 1-form on $M$ and
$\mathcal{D}\subset\mathbb H^{n}$ any fundamental domain. Then,

$||\eta||_{2}^{2}\leq\emph{Area}(\partial\mathcal{D})||\eta||_{\infty}\left(3\pi||d\eta||_{\infty}+\max_{i}\left|\int_{\gamma_{i}}\eta\right|\right)+\frac{1}{2}||d\eta||_{\infty}||\eta||_{2}\sqrt{\operatorname{vol}(M)},$

where the $\gamma_{i}$ are the geodesics in the homotopy class of the loops
representing the side pairing transformations of the fundamental domain
$\mathcal{D}$.

Studying the terms in the estimate of Proposition 5.1 for a coexact
$\lambda$-eigenform provides a lower bound on $\lambda$ given later as Theorem
B. The essential idea is that after applying an $L^{2}$-$L^{\infty}$ norm
comparison, all but one summand on the right-hand side (the integral term),
has a $||d\eta||_{2}$ term. In particular, if $\eta$ is a unit norm eigenform,
the right-hand side almost has a $\sqrt{\lambda}$ term in every summand. Our
aim, then, is to replace the integral term with something that looks like
$||d\eta||_{\infty}(\rho(M)^{-1}+\text{stuff})$, where the stuff is polynomial
in the volume of $M$ with constants that depend only on the lower bound on
injectivity radius.

######  Lemma 5.2.

Let $a$ be the lift to $\mathbb H^{n}$ of the cellular approximation of a
geodesic loop and let $\gamma$ be the (oriented) lift of the same geodesic
loop. If $\tilde{\eta}$ is the pullback to $\mathbb H^{n}$ of a 1-form
$\eta\in\Omega^{1}(M)$, then

$\left|\int_{a}\tilde{\eta}-\int_{\gamma}\tilde{\eta}\right|\leq\pi||d\eta||_{\infty}||a||_{G}.$

###### Proof.

Let $x$ be the starting point of $\gamma$, let $a_{i}$ be the geodesic arcs of
$a$ (so that the word corresponding to the cellular path $a$ is the word
$a_{1}\cdots a_{||a||_{G}}$), and let $y$ be the end point of $a_{||a||_{G}}$.
Let $Q$ be the piecewise geodesic $(||a||_{G}+3)$-gon obtained by taking the
union of the triangles $\text{convex hull}(a_{i},x)$ and the triangle
$\text{convex hull}(\gamma,y)$. Since $Q$ is the union of $(||a||_{G}+1)$
geodesic triangles of area bounded by $\pi$, we have the upper bound of
$(||a||_{G}+1)\pi$ for the area of $Q$. Then, since

$\left|\int_{a}\tilde{\eta}-\int_{\gamma}\tilde{\eta}\right|=\left|\int_{\partial
Q}\tilde{\eta}\right|\leq\pi||d\eta||_{\infty}(||a||_{G}+1),$

where the first equality follows from the deck transformation invariance of
$\tilde{\eta}$, which makes the integral over $\partial
Q\setminus(a\cup\gamma)$ vanish. If one then considers the cellular s $ma$ and
the geodesic $\gamma^{m}$ integrated over the form $\frac{1}{m}\tilde{\eta}$,
one gets

$\left|\int_{a}\tilde{\eta}-\int_{\gamma}\tilde{\eta}\right|=\frac{1}{m}\left|\int_{ma}\tilde{\eta}-\int_{\gamma^{m}}\tilde{\eta}\right|\leq\frac{1}{m}\pi||d\eta||_{\infty}(||ma||_{G}+1).$

Taking the limit as $m\to\infty$ then results in

$\left|\int_{a}\tilde{\eta}-\int_{\gamma}\tilde{\eta}\right|\leq\pi||d\eta||_{\infty}||a||_{G},$

proving the lemma. ∎

######  Lemma 5.3.

There is a constant $B_{0}=B_{0}(\varepsilon)$ such that
$\operatorname{diam}(M)\leq B_{0}\operatorname{vol}(M)$.

###### Proof.

First note that there is a constant $T=T(\varepsilon)$ such that the number of
simplices in $M$ is bounded by $T\operatorname{vol}(M)$ and that each simplex
from a deeply embedded triangulation has bounded diameter, say bounded by
$C_{0}$. With $B_{0}=C_{0}T$, one has that $B_{0}\operatorname{vol}(M)$ bounds
the diameter of $M$, as desired. ∎

######  Lemma 5.4.

Let $\eta\in\Omega^{1}(M)$ be a 1-form and $\gamma$ a rationally
nullhomologous loop in $M$. Then integrating over the geodesic in the free
homotopy class satisfies

$\left|\int_{\gamma}\eta\right|\leq
2\pi||d\eta||_{\infty}\operatorname{{\tt{scl}}}(\gamma).$

###### Proof.

This follows from Bavard duality and the fact that $\int_{\gamma}\eta$, where
the integral is over the geodesic in the free homotopy class of $\gamma$, is a
quasimorphism with defect bounded by $\pi||d\eta||_{\infty}$ (see [Cal09],
page 21). ∎

The key estimate allowing us to replace the integral term with one involving
the stable isoperimetric constant $\rho(M)$ is the following proposition;
compare with Proposition 5.24 in [LS18].

######  Proposition 5.5.

Let $\eta$ be a 1-form on $M$. Then there is a harmonic form $h$ and a
constant $L_{0}=L_{0}(\varepsilon)>0$ such that for every closed geodesic
$\alpha$ in $M$, one has

$\left|\int_{\alpha}(\eta-h)\right|\leq|\alpha|L_{0}\operatorname{vol}(M)^{3/2}||d\eta||_{\infty}\left(\rho(M)^{-1}+1\right).$

###### Proof.

If $M$ is a ${\mathbb Q}$-homology sphere, $\alpha$ is rationally
nullhomologous and $h$ can only be 0. Lemma 5.4 gives

$\left|\int_{\alpha}\eta\right|\leq
2\pi\operatorname{{\tt{scl}}}(\alpha)||d\eta||_{\infty}.$

By multiplying the right-hand side by $|\alpha|/|\alpha|$, this becomes

$\left|\int_{\alpha}\eta\right|\leq
2\pi|\alpha|\frac{\operatorname{{\tt{scl}}}(\alpha)}{|\alpha|}||d\eta||_{\infty}\leq
2\pi|\alpha|\rho(M)^{-1}||d\eta||_{\infty}.$

If $W$ is the minimal volume hyperbolic $n$-manifold (in dimension 3, this is
the Weeks manifold, see[GMM09], more generally it is know that in dimension
$n\geq 4$ that the set of hyperbolic volumes is discrete in $\mathbb R$, see
[Bel14]), the claim follows with
$L_{0}=\frac{2\pi}{\operatorname{vol}(W)^{3/2}}$.

Thus, we assume $M$ has nontrivial real homology classes. Fix a basepoint
$x_{0}\in M$. Take a basis $c_{1},\dots,c_{n}$ of harmonic 1-chains for
$C_{1}(K^{*})$ using the Euclidean inner product on $C_{1}(K^{*})$. Harmonic
chains are norm minimizing for the induced $\ell^{2}$-norm. Let
$||\cdot||_{E}$ denote this norm. Let $h$ be the (unique) harmonic form that
satisfies $\int_{c_{i}}\eta-h=0$ for each $i$. Let $a$ be a cellular path in
$K^{*}$ approximating $\alpha$, as in Proposition 2.5, so $||a||_{G}\leq
L|\alpha|$ and identify the cellular path $a$ with the chain it represents.

Then, using the Hodge decomposition induced by the Euclidean inner product, we
get $a=a_{h}^{E}+\partial S,$ where $a_{h}^{E}$ is harmonic with respect to
the Euclidean inner product on the chain complex $C_{1}(K^{*})$ and $S$ is
some 2-chain. Since $\partial$ and the Euclidean adjoint $\partial^{*}_{E}$
have integral bases, and since $a$ is integral, $\partial S$ is a rational
2-chain. Recall that there is a constant $T=T(\varepsilon)$ such that the
number of 2-simplices in $K^{*}$ is bounded by $T\operatorname{vol}(M)$. A
short computation then shows,

$\displaystyle||\partial S||_{G}$ $\displaystyle=||a-a_{h}^{E}||_{G}$
$\displaystyle\leq||a||_{G}+||a_{h}^{E}||_{G}$
$\displaystyle\leq||a||_{G}+\sqrt{T\operatorname{vol}(M)}||a_{h}^{E}||_{E},\text{~{}by
the Euclidean $\ell^{1}$-$\ell^{2}$ norm comparison,}$
$\displaystyle\leq||a||_{G}+\sqrt{T\operatorname{vol}(M)}||a||_{E},\text{~{}
as $a_{h}^{E}$ is $\ell^{2}$-norm minimizing in its class},$
$\displaystyle\leq||a||_{G}+\sqrt{T\operatorname{vol}(M)}||a||_{G},\text{~{}by
the Euclidean $\ell^{1}$-$\ell^{2}$ norm comparison},$
$\displaystyle=(\sqrt{T\operatorname{vol}(M)}+1)||a||_{G}.$

Because there is a minimal volume hyperbolic 3-manifold, we can increase $T$
so that we can write the above as $||\partial S||_{G}\leq
T\sqrt{\operatorname{vol}(M)}||a||_{G}$. Additionally, since there is a
universal upper bound on the length of an edge in $K^{*}$, there is a constant
$E>0$, such that the geodesic length $|\gamma|$ of a loop $\gamma$ satisfies
$|\gamma|\leq E\operatorname{{\tt{len}}}(c)$ for any cellular path $c$ in
$K^{*}$ homotopic to $\gamma$.

Since $\partial S$ is a rational cycle, take $N>0$ to be an integer so that
$N\partial S$ is integral. Then one can glue together oriented copies of the
edges on which $\partial S$ is supported along their boundaries to obtain a
(non unique) collection of closed cellular loops $b_{1},\dots,b_{m}$ whose
union represents the cycle $N\partial S$. Fix a vertex $v_{i}$ in each loop
$b_{i}$. Note that by construction,
$\operatorname{{\tt{len}}}(b_{i})=||b_{i}||_{G}$ for each $i$. Let $\tau_{i}$
be the geodesic arc connecting the basepoint $x_{0}$ to $v_{i}$ and
$\tau_{i}^{-1}$ the oppositely oriented geodesic arc. Define the curve $b$ to
be the path

$\tau_{1}b_{1}\tau_{1}^{-1}\tau_{2}b_{2}\tau_{2}^{-1}\cdots\tau_{m}b_{m}\tau_{m}^{-1}.$

Let $\beta$ be the geodesic loop through $x_{0}$ homotopic to $b$. Notice
$\sum_{i}||b_{i}||_{G}=||b||_{G}$, where $||b||_{G}$ is meant in the sense of
the norm on singular chains, where the $\tau^{\pm 1}$ terms cancel. This gives
a possibly trivial element of $\Gamma_{\mathbb Q}\textquoteright$ whose length
is bounded as follows:

$\displaystyle|\beta|\leq|b|$ $\displaystyle=\sum_{i}(2|\tau_{i}|+|b_{i}|)$
$\displaystyle\leq 2\operatorname{diam}(M)m+E||b||_{G}$
$\displaystyle\leq(2\operatorname{diam}(M)+E)||b||_{G}$
$\displaystyle\leq(2B_{0}\operatorname{vol}(M)+E)||b||_{G},$

where we use that $m\leq||b||_{G}$, the diameter bound of Lemma 5.3, along
with the remarks in the above discussion.

Since

$\frac{1}{N}||b||_{G}=||\partial S||_{G}\leq
T\sqrt{\operatorname{vol}(M)}||a||_{G},$

and $||a||_{G}\leq L|\alpha|$, we obtain

$\frac{||b||_{G}}{N}\leq TL\sqrt{\operatorname{vol}(M)}|\alpha|.$

As a result,

$\frac{|\beta|}{N}\leq
TL(2B_{0}\operatorname{vol}(M)+E)\sqrt{\operatorname{vol}(M)}|\alpha|.$

We compute,

$\displaystyle\left|\int_{\alpha}\eta-h\right|$
$\displaystyle=\left|\int_{\alpha-a_{h}}\eta-h\right|\text{~{}since $a_{h}$ is
in the span of the $c_{i}$, and $\int_{c_{i}}\eta-h$ = 0,}$
$\displaystyle\leq\left|\int_{\partial
S}\eta-h\right|+\left|\left(\int_{\alpha}\eta-h\right)-\left(\int_{a}\eta-h\right)\right|$
$\displaystyle\leq\left|\int_{\partial
S}\eta-h\right|+\pi||d\eta||_{\infty}||a||_{G},\text{~{}by Lemma
\ref{lem:5.2},}$ $\displaystyle=\frac{1}{N}\left|\int_{N\partial
S}\eta-h\right|+\pi||d\eta||_{\infty}||a||_{G}$
$\displaystyle=\frac{1}{N}\left|\int_{b}\eta-h\right|+\pi||d\eta||_{\infty}||a||_{G},\text{~{}since
$b$ abelianizes to $N\partial S$,}$
$\displaystyle\leq\frac{1}{N}\left(\left|\int_{\beta}\eta-h\right|+\left|\int_{b}(\eta-h)-\int_{\beta}(\eta-h)\right|\right)+\pi||d\eta||_{\infty}||a||_{G}$
$\displaystyle\leq\frac{1}{N}\left|\int_{\beta}\eta-h\right|+\frac{1}{N}\pi||d\eta||_{\infty}||b||_{G}+\pi||d\eta||_{\infty}||a||_{G},\text{~{}by
Lemma \ref{lem:5.2}}.$

If $\beta$ is trivial, then the integral term $|\int_{\beta}\eta-h|$ vanishes,
and we can replace that term with

$TL\pi|\alpha|||d\eta||_{\infty}\sqrt{\operatorname{vol}(M)}\rho(M)^{-1}$

to obtain (after using our estimate for $||b||_{G}/N$ and $||a||_{G}\leq
L|\alpha|$)

$\displaystyle\left|\int_{\alpha}\eta-h\right|$ $\displaystyle\leq
TL\pi|\alpha|||d\eta||_{\infty}\sqrt{\operatorname{vol}(M)}\left(\rho(M)^{-1}+1\right)+\pi
L||d\eta||_{\infty}|\alpha|$ $\displaystyle\leq
TL\pi|\alpha|||d\eta||_{\infty}\sqrt{\operatorname{vol}(M)}\left(\rho(M)^{-1}+1\right)+\frac{\operatorname{vol}(M)^{3/2}}{\operatorname{vol}(W)^{3/2}}\pi
L||d\eta||_{\infty}||\alpha|$ $\displaystyle\leq
TL\pi|\alpha|||d\eta||_{\infty}\frac{\operatorname{vol}(M)^{3/2}}{\operatorname{vol}(W)}\left(\rho(M)^{-1}+1\right)+\frac{\operatorname{vol}(M)^{3/2}}{\operatorname{vol}(W)^{3/2}}\pi
L||d\eta||_{\infty}|\alpha|,$

where in the last line we again use the minimal volume hyperbolic 3-manifold
$W$ to replace $\sqrt{\operatorname{vol}(M)}$ with
$\operatorname{vol}(M)^{3/2}.$ Setting

$L_{0}=2\max\\{\frac{2\pi BDL}{\operatorname{vol}(W)},\frac{\pi
L}{\operatorname{vol}(W)^{3/2}}\\}$

and factoring gives the result.

Assume now that $\beta$ is nontrivial. Combining the above estimates yields

$\displaystyle\left|\int_{\alpha}\eta-h\right|$
$\displaystyle\leq\frac{|\beta|}{N}\frac{1}{|\beta|}\left|\int_{\beta}\eta-h\right|+\frac{1}{N}\pi||d\eta||_{\infty}||b||_{G}+\pi
L||d\eta||_{\infty}|\alpha|$ $\displaystyle\leq
TL(2B_{0}\operatorname{vol}(M)+E)\sqrt{\operatorname{vol}(M)}|\alpha|\frac{1}{|\beta|}\left|\int_{\beta}\eta-h\right|$
$\displaystyle~{}~{}~{}~{}+\pi||d\eta||_{\infty}TL\sqrt{\operatorname{vol}(M)}|\alpha|+\pi
L||d\eta||_{\infty}|\alpha|$
$\displaystyle=TL\sqrt{\operatorname{vol}(M)}|\alpha|\left((2B_{0}\operatorname{vol}(M)+E)\frac{1}{|\beta|}\left|\int_{\beta}\eta-h\right|+\pi||d\eta||_{\infty}\right)$
$\displaystyle~{}~{}~{}~{}+\pi L||d\eta||_{\infty}|\alpha|.$

Since the geodesic $\beta$ is nullhomologous, Lemma 5.4 implies

$\left|\int_{\beta}(\eta-h)\right|\leq
2\pi||d\eta||_{\infty}\operatorname{{\tt{scl}}}(\beta).$

Replacing the integral term with this estimate and using that
$\frac{\operatorname{{\tt{scl}}}(\beta)}{|\beta|}\leq\rho(M)^{-1}$ gives

$\displaystyle\left|\int_{\alpha}\eta-h\right|\leq|\alpha|TL\sqrt{\operatorname{vol}(M)}||d\eta||_{\infty}\left(2\pi(B_{0}\operatorname{vol}(M)+E)\rho(M)^{-1}\right)+\pi)$
$\displaystyle+\pi L||d\eta||_{\infty}|\alpha|.$

Again using the existence of a minimal volume hyperbolic $n$-manifold, one can
replace $B_{0}$ with the constant $B_{1}=2B_{0}+E/\operatorname{vol}(W)$ since
$B_{1}\operatorname{vol}(M)>2B_{0}\operatorname{vol}(M)+E$. Then, after
combining constants in the first summand (and using that $2\pi>\pi$ to pull
out the terms containing $\pi$) into a single constant $L_{1}$, one obtains:

$\left|\int_{\alpha}(\eta-h)\right|\leq|\alpha|L_{1}\operatorname{vol}(M)^{3/2}||d\eta||_{\infty}\left(\rho(M)^{-1}+1\right)+\pi
L_{1}||d\eta||_{\infty}|\alpha|.$

Set $L_{0}=2\max\\{L_{1},\frac{\pi L}{\operatorname{vol}(W)^{3/2}}\\}$ and
multiply the second summand by $\operatorname{vol}(M)^{3/2}$ to obtain the
claim.

∎

######  Lemma 5.6.

Let $M$ have deeply embedded triangulation $K$ and let $\tilde{K}$ be the
pullback of this triangulation to $\mathbb H^{n}$. Then there is a fundamental
domain $\mathcal{D}\subset\mathbb H^{n}$ for $M$ that is a union of simplices
from $\tilde{K}$ such that the diameter of $\mathcal{D}$ satisfies
$\operatorname{diam}(\mathcal{D})\leq 3\operatorname{diam}(M).$

###### Proof.

Fix a top dimensional simplex $\sigma_{0}\in K^{(n)}$ and let
$\tilde{\sigma}_{0}$ be a lifted copy in $\mathbb H^{n}$. Let $\tilde{x}_{0}$
be the barycenter of $\tilde{\sigma}_{0}$. For every other top dimensional
simplex $\sigma$ in $K^{(n)}$ there is a lift $\tilde{\sigma}$ whose
barycenter $\tilde{x}_{\sigma}$ is within $\operatorname{diam}(M)$ of
$\tilde{x}_{0}$. Choose one such lift for every $\sigma$ in such a way the
resulting fundamental domain $\mathcal{D}$ is connected. Then the diameter of
the fundamental domain satisfies
$\operatorname{diam}(\mathcal{D})\leq\operatorname{diam}(M)+2e$, where $e$ is
the maximum distance from the barycenter of a simplex in a deeply embedded
triangulation to its boundary. Clearly $e<\operatorname{diam}(M)$, so the
lemma immediately follows. ∎

Now, assume $M$ has a fixed deeply embedded triangulation and let
$\mathcal{D}$ be a fundamental domain as in Lemma 5.6. Let $\gamma_{i}$ be the
geodesics in the free homotopy class of the side pairing transformations of
the fundamental domain $\mathcal{D}$, and notice by construction
$|\gamma_{i}|\leq 3\operatorname{diam}(M)$. With this, we modify the estimate
in Proposition 5.1to obtain the following.

######  Proposition 5.7.

Let $\eta$ be a coclosed 1-form on $M$. Let $h$ be the harmonic form of
Proposition 5.5 associated to $\eta$. Then for a constant
$A_{0}=A_{0}(\varepsilon)>0$, the following holds:

$\displaystyle||\eta-h||_{2}^{2}\leq
A_{0}\operatorname{vol}(M)||\eta-h||_{\infty}\left(3\pi||d\eta||_{\infty}+3L_{0}B_{0}\operatorname{vol}(M)^{5/2}||d\eta||_{\infty}\left(\rho(M)^{-1}+1\right)\right)$
$\displaystyle+\frac{1}{2}||d\eta||_{\infty}||\eta-h||_{2}\sqrt{\operatorname{vol}(M)}.$

###### Proof.

Let $\gamma_{i}$ realize the maximum among the integrals
$\int_{\gamma_{i}}\eta$. Substitute the estimate of Proposition 5.5 for the
integral term in Proposition 5.1 applied to the coclosed form $\eta-h$ and the
fundamental domain $\mathcal{D}$ to obtain

$\displaystyle||\eta-h||_{2}^{2}\leq\text{Area}(\partial\mathcal{D})||\eta-h||_{\infty}\left(3\pi||d\eta||_{\infty}+L_{0}\operatorname{vol}(M)^{3/2}||d\eta||_{\infty}|\gamma_{i}|\left(\rho(M)^{-1}+1\right)\right)$
$\displaystyle+\frac{1}{2}||d\eta||_{\infty}||\eta-h||_{2}\sqrt{\operatorname{vol}(M)}.$

Then, replace $|\gamma_{i}|$ with $3B_{0}\operatorname{vol}(M)$, using Lemma
5.6 and Lemma 5.3. Lastly, since there is an upper bound on the area of a face
of any simplex in $\mathcal{G}_{\varepsilon}$ (a consequence of the bounds on
the dihedral angles), the total area of the boundary of a complex made from no
more than $T\operatorname{vol}(M)$ simplices from $\mathcal{G}_{\varepsilon}$
is bounded by $A_{0}\operatorname{vol}(M)$ for a constant $A_{0}$ depending on
$\varepsilon$. Substituting this estimate for the
$\text{Area}(\partial\mathcal{D})$ term completes the proof. ∎

######  Proposition 5.8.

(Proposition 2.2 of [LS18]) Let $M$ be a closed hyperbolic n-manifold with
$\operatorname{inj}(M)>\varepsilon$. Assume the first positive eigenvalue
$\lambda$ of the Laplacian acting on coexact 1-forms is less than some fixed
constant $H>0$. Then there is a constant $C(H,\varepsilon)>0$ such that for a
coexact $\lambda$-eigenform $\omega$, one has

$||\omega||_{\infty}\leq C(H,\varepsilon)||\omega||_{2}.$

######  Proposition 5.9.

Let $M$ be a closed hyperbolic n-manifold with
$\operatorname{inj}(M)>\varepsilon$. Let $\lambda<H$ be the first positive
eigenvalue for the Hodge Laplacian acting on coexact 1-cochains. Then the
following holds:

$\displaystyle\frac{1}{\sqrt{\lambda}}\leq
A_{0}\operatorname{vol}(M)C(H,\varepsilon)^{2}\left(3\pi+3L_{0}B_{0}\operatorname{vol}(M)^{5/2}\left(\rho(M)^{-1}+1\right)\right)$
$\displaystyle+\frac{C(H,\varepsilon)}{2}\sqrt{\operatorname{vol}(M)}.$

###### Proof.

Let $\eta$ be a $\lambda$ coexact eigenform. Applying the Sobolev type
estimate of Proposition 5.8 to each instance of the sup norm in Proposition
5.7 and using that
$||d\eta||_{2}=\sqrt{\lambda}||\eta||_{2}\leq\sqrt{\lambda}||\eta-h||_{2}$,
where the inequality follows from the orthogonality of the Hodge
decomposition, gives

$\displaystyle||\eta-h||_{2}^{2}\leq
A_{0}\operatorname{vol}(M)C(H,\varepsilon)^{2}\sqrt{\lambda}||\eta-h||_{2}^{2}\left(3\pi+3L_{0}B_{0}\operatorname{vol}(M)^{5/2}\left(\rho(M)^{-1}+1\right)\right)$
$\displaystyle+\frac{C(H,\varepsilon)}{2}\sqrt{\lambda}||\eta-h||_{2}^{2}\sqrt{\operatorname{vol}(M)}.$

Dividing both sides by $\sqrt{\lambda}||\eta-h||_{2}^{2}$ then gives

$\displaystyle\frac{1}{\sqrt{\lambda}}\leq
A_{0}\operatorname{vol}(M)C(H,\varepsilon)^{2}\left(3\pi+3L_{0}B_{0}\operatorname{vol}(M)^{5/2}\left(\rho(M)^{-1}+1\right)\right)$
$\displaystyle+\frac{C(H,\varepsilon)}{2}\sqrt{\operatorname{vol}(M)}.$

∎

Rearanging the terms and combining constants (which again requires the
existence of a minimal volume hyperbolic $n$-manifold) in the previous
proposition and applying a geometric estimate of Calegari and a systolic
inequality due to Sabourau leads to the main theorem of this section.

######  Theorem B.

Let $M$ be a closed hyperbolic $n$-manifold with
$\operatorname{inj}(M)>\varepsilon$. Let $\lambda$ be the first positive
eigenvalue for the Laplacian acting on coexact 1-forms and let $H>\lambda$.
Then there is a constant $P(H,\varepsilon)>0$ such that

$\frac{P\rho(M)}{\operatorname{vol}(M)^{7/2+1/n}}\leq\sqrt{\lambda}.$

###### Proof.

First we rearrange the previous proposition and combine constants into one
constant $P$ to get the estimate

$\frac{P\rho(M)}{(1+\rho(M))\operatorname{vol}(M)^{7/2}}\leq\sqrt{\lambda}.$

We need an estimate of Calegari’s (see the proof of Theorem 3.9 in [Cal09],
the estimate at the bottom of page 58) which gives that for a genus $g$
surface $S$ with boundary $\partial S=\gamma^{m}$, one has

$\frac{m|\gamma|}{12g-6}\leq 4\mu+\frac{2\pi}{3\mu}+2|\gamma|,$

where $\mu$ depends only on the dimension $n$. Since $\chi_{-}(S)\geq 2g-1,$
we get

$\frac{2m|\gamma|}{\chi_{-}(S)}\leq
24\left(4\mu+\frac{2\pi}{3\mu}+2|\gamma|\right).$

Since this is true for any surface $S$ bounding a power of $\gamma$, we obtain

$\frac{|\gamma|}{\operatorname{{\tt{scl}}}(\gamma)}\leq
24\left(4\mu+\frac{2\pi}{3\mu}+2|\gamma|\right).$

We also have the commutator systolic inequality of Sabourau from Theorem 1.4
in [Sab17], which bounds the shortest nontrivial integrally nullhomologous
loop $\gamma\in\Gamma\textquoteright$ by

$|\gamma|\leq c\operatorname{vol}(M)^{1/n},$

for a dimensional constant $c$.

Both the inequality of Calegari and the systolic inequality involve a
dimensional constant; let $\mu$ be the maximum of these constants in dimension
$n$ and write Calegari’s inequality as
$\frac{|\gamma|}{\operatorname{{\tt{scl}}}(\gamma)}\leq\mu(1+|\gamma|)$. Then
we get

$\rho(M)\leq\frac{|\gamma|}{\operatorname{{\tt{scl}}}(\gamma)}\leq\mu(1+|\gamma|)\leq\mu(1+\mu\operatorname{vol}(M)^{1/n}).$

Inserting this upper bound into the denominator of the above rearranged
estimate above gives

$\frac{P\rho(M)}{(1+\mu(1+\mu\operatorname{vol}(M)^{1/n})\operatorname{vol}(M)^{7/2}}\leq\frac{P\rho(M)}{(1+\rho(M))\operatorname{vol}(M)^{7/2}}\leq\sqrt{\lambda}.$

We can then increase $P$ to allow us to pull out the volume term and absorb
$\mu$, thereby obtaining the desired estimate. ∎

##  6 An example

Figure 1:  A sketch of the example constructed in this section. The manifold
decomposes into three pieces, left and right caps and a middle region. The
loop drawn on the boundary of the left cap only bounds surfaces of high
topological complexity that are at least partly contained in the right cap.
This ensures the isoperimetric ratio of that loop is very small.

The aim of this section is to show that the first positive eigenvalue of the
1-form Laplacian can vanish exponentially fast in relation to volume. This
contrasts the behaviour of the first positive eigenvalue of the Laplacian on
functions.

Our construction is similar to that in [BD17]. Essentially, we choose a
hyperbolic 3-manifold with totally geodesic boundary and glue it to itself
using a particular psuedoAnosov with several useful properties. By [BMNS16],
this family has geometry that up to bounded error can be understood in terms
of a simple model family. Using this model family, we show that one can find
curves with uniformly bounded length whose stable commutator length grows
exponentially in the volume. We then use the spectral gap upper bound in
Theorem A to conclude the first positive eigenvalue vanishes exponentially
fast.

Throughout this section, we need to compare geodesic lengths in different
submanifolds of a given manifold $M$. Let $|\cdot|_{X}$ denote the geodesic
length of a homotopy class of curves relative endpoints in a manifold $X$ and
$\text{length}(\cdot)$ be the length in $M$ of the curve. Similarly, when we
compute stable commutator length for the fundamental group of a manifold $X$,
which may or may not be a submanifold of $M$, we denote it
$\operatorname{{\tt{scl}}}_{X}$.

We will need that for certain curves, $\operatorname{{\tt{scl}}}$ is
comparable to length. We begin with a simple but essential technical lemma.

2pt $a_{0}$ [ ] at 830 1100 $a_{1}$ [ ] at 1300 1100 $b_{0}$ [ ] at 830 1250
$b_{1}$ [ ] at 1300 1250 $t_{1}$ [ ] at 380 1290 $t_{0}$ [ ] at 380 1120

Figure 2:  Illustrating Lemma 6.1 , the rectangular base of the figure is part
of the totally geodesic surface $S$ and the box is the corresponding part of
the tubular neighborhood $N_{\varepsilon}(S)$ foliated by surfaces $S_{t}$
parallel to $S$. Drawn in the box is the surface $\Sigma$, which is transverse
the foliation except at isolated points. The multicurve $a_{0}\cup a_{1}$ is
part of a single level set $S_{t_{0}}\cap\Sigma$, but only $a_{0}$ is part of
the curve $c_{t_{0}}$ described in the lemma, whereas the multicurve
$b_{0}\cup b_{1}$ forms the multicurve $c_{t_{1}}$ in the lemma.

######  Lemma 6.1.

Let $M$ be a compact hyperbolic 3-manifold with totally geodesic boundary
$\partial M=S$. Let $\varepsilon$ be smaller than the injectivity radius of
$M$ and such that $N_{\varepsilon}(S)$ is an embedded tubular neighborhood.
Let $\\{S_{t}\\}$ be the leaves of the foliation of $N_{\varepsilon}(S)$ by
surfaces equidistant from $S$. Let $\Sigma$ be a smooth incompressible proper
not necessarily immersed surface in $M$ that is transverse to the foliation
$\\{S_{t}\\}$ except at isolated points. Let $c=\partial\Sigma$. By
transversality, for generic $t$ the multicurve $c_{t}$ given by the part of
$S_{t}\cap\Sigma$ that cobounds a subsurface of $\Sigma$ with $c=c_{0}$ is a
smooth multicurve. Let $T$ be the set (of full measure) of all
$t\in[0,\varepsilon)$ such that $c_{t}$ is a smooth multicurve. Since $S$ is
totally geodesic, each multicurve $c_{t}$ is homotopic to a possibly
degenerate geodesic multicurve $\gamma_{t}$ in $S$. Let $\Sigma_{\varepsilon}$
be the part of $\Sigma$ contained in $N_{\varepsilon}(S)$. Then for
$C=1/\varepsilon$, we have that

$\inf\limits_{t\in T}|\gamma_{t}|_{S}\leq C\emph{Area}(\Sigma_{\varepsilon}).$

###### Proof.

The coarea formula implies the inequality $\inf\limits_{t\in
T}|c_{t}|_{S_{t}}\leq C\text{Area}(\Sigma_{\varepsilon})$. Since $S$ is
totally geodesic, for all $t\in T$, we have
$|\gamma_{t}|_{S}\leq|c_{t}|_{S_{t}}.$ ∎

Note that in the previous lemma, when $\inf\limits_{t\in T}|\gamma_{t}|_{S}$
is zero, because $\Sigma$ is incompressible and any loop with length less than
$\operatorname{inj}(M)$ bounds a disk, every component of $\Sigma$ can either
be homotoped to be disjoint from $N_{\varepsilon}(S)$ or be contained in $S$.

The next proposition requires a notion of geometric complexity for homology
classes. For any compact Riemannian manifold $M$ one can define the stable
norm on the first homology of $M$ (see [Gro07] Section 4C). The mass of a
Lipschitz 1-chain $\alpha=\sum_{i}t_{i}\alpha_{i}$ in $M$ is defined to be
$\text{mass}(\alpha)=\sum_{i}|t_{i}|\text{length}(\alpha_{i}).$ The mass of a
class $a\in H_{1}(M)$ is then the infimal value of the mass of a chain
$\alpha$ representing $a$. For a class $a\in H_{1}(M)$, the stable norm of $a$
is then given by

$||a||_{s,M}=\inf\limits_{m>0}\frac{\text{mass}(ma)}{m}.$

Stable commutator length can also be generalized to geodesic multicurves (see
Section 2.6 of [Cal09]), which can naturally be viewed as Lipschitz chains.
Suppose $\gamma_{i}\in\pi_{1}M$ and $\sum_{i}n[\gamma_{i}]=0$ in $H_{1}(M)$.
Let $\gamma$ be the geodesic multicurve, which is not necessarily simple,
consisting of the geodesic loops determined by $\gamma_{i}$. Say a surface
$f:S\to M$ is admissible of degree $n(S)$ if it has no closed components and
$\partial S$ is a union of circles $S^{1}_{i}$ with $f|_{S^{1}_{i}}$ a degree
$n(S)$ cover of $\gamma_{i}$. Then we define stable commutator length of
$\gamma$ to be

$\operatorname{{\tt{scl}}}(\gamma)=\inf\limits_{S\text{
admissible}}\frac{\chi_{-}(S)}{2n(S)}.$

When $\gamma$ is a single loop, this definition agrees with the usual
definition of stable commutator length.

######  Proposition 6.2.

Let $M$ be a compact oriented hyperbolic 3-manifold with totally geodesic
boundary $\partial M=S$. Let $\gamma$ be a geodesic multicurve in $S$ that is
rationally nullhomologous in $M$. Then there is a constant $D>0$ depending
only on $M$ such that

$||[\gamma]||_{s,S}\leq D\operatorname{{\tt{scl}}}_{M}(\gamma).$

###### Proof.

If $\gamma$ is nullhomologous in $S$, then the left hand side is zero and the
inequality holds. Assume now that $[\gamma]\neq 0\in H_{1}(S)$. Fix
$\delta>0$. Let $\Sigma_{m}$ be an incompressible admissible surface for
$\gamma$ of degree $m=n(S)$ such that
$\chi_{-}(\Sigma_{m})/2m-\operatorname{{\tt{scl}}}(\gamma)<\delta$. We can
triangulate $\Sigma_{m}$ so that there is a single vertex on each boundary
component. This triangulation has $4g+3b-4$ faces, where $g$ is the genus of
$\Sigma_{m}$ and $b$ the number of boundary components. We can then straighten
this triangulation to obtain a piecewise totally geodesic triangulated
surface. Replace $\Sigma_{m}$ with this surface. Since every face of this
triangulation of $\Sigma_{m}$ is geodesic, every face has area at most $\pi$.
Since there are $4g+3b-4$ faces and $\chi_{-}(\Sigma_{m})=2g-2+b$, we can
estimate

$\text{Area}(\Sigma_{m})\leq 3\pi\chi_{-}(\Sigma_{m}).$

We can perturb $\Sigma_{m}$ to obtain a smooth surface
$\Sigma\textquoteright_{m}$ that it is transverse the foliation of
$N_{\varepsilon}(S)$ except at isolated points and in doing so increase the
area by less than $\delta$. Let $\gamma_{t}$ be the family of multicurves in
Lemma 6.1 applied to $\Sigma\textquoteright_{m}$. Since each curve
$\gamma_{t}$ cobounds a surface in $S$ with $\partial\Sigma_{m}$, they are
homologous, thus $||[\gamma_{t}]||_{s,S}=m||[\gamma]||_{s,S}$. Since
$||[\gamma_{t}]||_{s,S}\leq|\gamma_{t}|_{S}$, Lemma 6.1 implies that

$m||[\gamma]||_{s,S}\leq C\text{Area}(\Sigma_{m}\textquoteright)\leq
C\text{Area}(\Sigma_{m})+C\delta\leq 3C\pi\chi_{-}(\Sigma_{m})+C\delta.$

From this we get

$||[\gamma]||_{s,S}\leq 6C\pi\chi_{-}(\Sigma_{m})/2m+C\delta/m\leq
6C\pi\operatorname{{\tt{scl}}}_{M}(\gamma)+6C\pi\delta+C\delta/m.$

Since the stable commutator length of a nontrivial rational commutator is
bounded away from zero by a constant only depending on $M$, by Theorem 3.9 in
[Cal09], we can replace $C$ with a larger constant $D$ such that

$||[\gamma]||_{s,S}\leq D\operatorname{{\tt{scl}}}_{M}(\gamma),$

as desired. ∎

We now introduce the family of manifolds that we use in our construction. The
family $\\{W_{n}\\}$ of manifolds we study are easily understood using the
model manifold theory of [BMNS16]. In particular, there is a $K$-biLipschitz
map between $W_{n}$ and a model manifold $M_{n}$, where $K$ is independent of
$n$. The base of the construction is Thurston’s tripus manifold $W$ (see
[Thu97], Section 3.3.12), a hyperbolic manifold with totally geodesic
boundary, and a psuedoAnosov homeomorphism $f$ of the boundary surface
$\partial W$. The model manifold $M_{n}$ is a degree $n$ cyclic cover of the
mapping torus $M_{f}$ cut open along a fiber with two oppositely oriented
copies of $W$, denoted $W^{+}$ and $W^{-}$ glued as described in [BMNS16]
Section 2.15 to the two boundary components of the cut open mapping torus.
This decomposes $W_{n}$ into three pieces, a product region $S\times[0,n]$ and
the caps $W^{+}$ and $W^{-}$ in a metrically controlled way. It will be
convenient to set $M^{+}=W^{+}\subset M_{n}$ and $M^{-}=W^{-}\subset M_{n}$
when talking about the caps of the model manifold $M_{n}$ for fixed $n$, and
to let $W^{+}$ and $W^{-}$ denote the images of these spaces under the natural
inclusion into $W_{n}$.

2pt $M^{+}$ [ ] at 950 1100 $M^{-}$ [ ] at 2000 1100 $S\times[0,n]$ [ ] at
1500 1600

Figure 3:  A schematic picture of the model manifold $M_{n}$ with caps $M^{+}$
and $M^{-}$ two oppositely oriented copies of the tripus manifold.

Given a multicurve $c$ in $M^{\pm}$, we say $c$ bounds on both sides if there
are incompressible surfaces $S^{+}$ in $M^{+}$ and $S^{-}$ in $M^{-}$ both
with boundary homotopic to $c$.

We encode the construction and its essential properties in the following
proposition.

######  Proposition 6.3.

There is a family $\\{W_{n}\\}$ of closed hyperbolic 3-manifolds with
injectivity radius uniformly bounded below and volume growing linearly in $n$
constructed from the tripus and a pseudoAnosov $f$ as described above. Each
manifold $W_{n}$ is $K$-biLipschitz equivalent to the model manifold $M_{n}$
for some constant $K$ independent of $n$. Any homologically nontrivial loop in
$H_{1}(\partial W^{\pm})$ that bounds a surface in $M^{\pm}$ cannot bound on
both sides. The pseudoAnosov $f$ is such that for any nonzero class $a\in
H_{1}(\partial W^{+})$, the stable norm of $f_{*}^{n}(a)$ grows exponentially.

###### Proof.

Let $W$ be Thurston’s tripus manifold, a compact hyperbolic 3-manifold with
totally geodesic boundary a genus 2 surface for which the inclusion map
$H_{1}(\partial W;{\mathbb Z})\to H_{1}(W;{\mathbb Z})$ is onto. The homology
of the boundary $\partial W$ decomposes as the direct sum of rank 2 submodules
$U$ and $V$, where $V\subset H_{1}(\partial W)$ is the image of the boundary
map $\partial:H_{2}(W,\partial W;{\mathbb Z})\to H_{1}(\partial W;{\mathbb
Z})$ (which is also the kernel of the inclusion $H_{1}(\partial W)\to
H_{1}(W)$) and $U$ is a compliment of $V$ (note that the inclusion map