Stable Isoperimetric Ratios and the Hodge Laplacian of Hyperbolic Manifolds
SStable Isoperimetric Ratios and the Hodge Laplacian ofHyperbolic Manifolds
Cameron Gates Rudd
Abstract
We show that for a closed hyperbolic 3-manifold, the size of the first eigenvalue of the Hodge Laplacianacting on coexact 1-forms is comparable to an isoperimetric ratio relating geodesic length and stablecommutator length, with comparison constants that depend polynomially on the volume and on a lowerbound on injectivity radius. We use this estimate to show that there exist sequences of closed hyperbolic3-manifolds with injectivity radius bounded below and volume going to infinity for which the 1-formLaplacian has spectral gap vanishing exponentially fast in the volume.
The spectrum of the Hodge Laplacian is a fundamental and well studied geometric invariant of Riemannianmanifolds. The Hodge theorem partitions the positive spectrum into exact and coexact eigenvalues. Fordifferential forms of degree one, the exact eigenvalues contain exactly the data of the Laplacian acting onfunctions, 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 toother aspects of its geometry and topology. For the function Laplacian, the first eigenvalue is known to becomparable to the square of the isoperimetric Cheeger constant. In this paper, we derive a similar estimatefor the first coexact eigenvalue, building on work of Lipnowski and Stern in [LS18] motivated by torsiongrowth.Given a hyperbolic 3-manifold M , it is natural to try and extract information about M from its finite covers.A deep and interesting conjecture of Bergeron-Venketesh, Le, and Lück (see [BV13], [Lê18], and [Lüc16]) asksin part whether the volume of M can be found by studying the torsion in the homology of a family of finitecovers of M . In studying this question, Bergeron, Şengün, and Venketesh in [BŞV16] relate the growth rateof the cardinality of the torsion in the first homology of a tower of covers of a closed arithmetic hyperbolic3-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 → M of congruence covers of an arithmetic hyperbolic 3-manifold M satisfies the few small eigenvalues, small Betti numbers, and simple cycles conditions, then thelog torsion growth rate is proportional to the volume. In particular, one has lim n →∞ log | H ( M n ; Z ) torsion | vol( M n ) = 16 π . The conditions appearing in Theorem 1.1. are:1. (Few small eigenvalues) For all ε > there is a c > such that lim sup n →∞ M n ) (cid:88) <λ Theorem A. Let M be a closed hyperbolic 3-manifold with injectivity radius bounded below by ε > and let λ denote the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms. Then there is a constant A = A ( ε ) such that for any nontrivial element γ ∈ Γ (cid:48) Q , one has √ λ ≤ A vol( M ) | γ | scl ( γ ) , where | γ | denotes the geodesic length of γ . Theorem A has the following obvious corollary: Corollary. Let M be a hyperbolic -manifold with injectivity radius bounded below by ε > and let λ be thefirst eigenvalue of the Hodge Laplacian acting on coexact 1-forms. Then for the constant A = A ( ε ) fromTheorem A, √ λ ≤ A vol( M ) ρ ( M ) . The analogue of Theorem A in [LS18] studies a cochain version of the Hodge Laplacian introduced byDodziuk 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 -de Rham complex. Theorem 1.2. ( [LS18] Theorem 1.4) Let M be a closed hyperbolic n -manifold. Let K be a sufficientlyfine triangulation. Let M be a finite cover of M . Let λ W ( M ) d ∗ be the first coexact eigenvalue for theWhitney cochain Laplacian associated to the pullback of the triangulation K to M . Then if some multiple of γ ∈ π ( M ) bounds a surface, then (cid:18) scl ( γ ) | γ | (cid:19) ≤ W M vol( M ) λ W ( M ) d ∗ , for a constant W M depending on the triangulation K . Theorem 1.3. ( [LS18] Theorem 1.5) Let M be a closed hyperbolic n -manifold. Let K be a sufficientlyfine triangulation of M . Let M be a finite cover of M . Let λ W ( M ) d ∗ be the first coexact eigenvalue for theWhitney cochain Laplacian associated to the pullback of the triangulation K to M . Then, λ W ( M ) d ∗ ≤ max (cid:26) G M vol( M ) λ d ∗ ( M ) , G M C M vol( M ) (cid:27) . The constants C M and G M depend only on K . In the course of proving Theorem A, we too require a comparison of this sort. By using a smoothedversion of the Whitney embedding of cochains into the de Rham complex and triangulations with uniformlycontrolled geometry (these are called deeply embedded triangulations and are introduced in Section 2), weprove the following Whitney-Hodge eigenvalue comparison. Proposition 1.4. Let λ denote the first positive eigenvalue for the Hodge Laplacian acting on -forms andlet λ W denote the first positive eigenvalue for the smoothened Whitney Laplacian on 1-cochains associated toa deeply embedded triangulation. There is a constant G = G ( ε ) such that λ ≤ G vol( M ) λ W . Our second theorem uses the isoperimetric constant ρ ( M ) to provide a lower bound on the first coexacteigenvalue of the 1-form Laplacian. Theorem B. Let M be a closed hyperbolic n -manifold with inj( M ) > ε . Let λ be the first positive eigenvaluefor the Hodge Laplacian acting on coexact 1-forms and let H > λ . Then there is a constant P ( H, ε, n ) > such that P ρ ( M )vol( M ) / /n ≤ √ λ. Theorem B corresponds to Theorem 1.5 below from [LS18], which uses stable area in place of stablecommutator length. Note that Theorem B above remains true if one replaces ρ ( M ) with ρ Area ( M ) . Theorem 1.5. ( [LS18] Theorem 1.3 ) Let M be a closed hyperbolic n -manifold and let M be a finite coverof M . Then there are constants A and C , where A is depends only on M and C is a constant that isuniformly bounded when the injectivity radius of M is bounded below and λ ( M ) is bounded above, for which λ ( M ) d ∗ ≤ A C vol( M ) / diam( M ) (cid:0) ρ Area ( M ) − (cid:1) . Our approach to proving Theorems A and B is grounded in the following dual characterizations of scl .By Bavard duality, stable commutator length is related to the defect norm for quasimorphisms and theGersten filling norm for singular 1-chains. The Gersten filling norm for a nullhomologous loop γ is givenby the infimal (cid:96) -norm of a singular 2-chain whose boundary is a fundamental cycle for γ m , normalized by m . A quasimorphism for a group Γ is a map from Γ → R that is nearly a homomorphism in the sense thatits coboundary is a bounded map on Γ . The defect of a quasimorphism is the sup norm of its coboundary.Bavard duality says scl ( γ ) = 4 fill ( γ ) = 12 sup q q ( γ ) 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 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 thefilling norm to the spectrum of the Hodge Laplacian via the Whitney Laplacian and Poincaré duality (which4orces us to restrict to dimension 3). For Theorem B, we use de Rham quasimorphisms, which are given byintegrating coclosed forms over geodesics. Studying the de Rham quasimorphism of a coexact eigenform givesthe 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 aspecific triangulation, we use that closed hyperbolic manifolds with injectivity radius greater than some ε > can all be triangulated so that the simplices come from a compact collection. The local structure of thesetriangulations can then be compared in a uniform way, thereby allowing us to relate various combinatorialand geometric norms. By working in the smooth setting instead of the L setting, we are able to make use ofgeometric estimates that require higher regularity. This leads to the more direct Whitney-Hodge eigenvaluecomparison 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 rational homology spheres for which we have control over the stable isoperimetricconstant. Theorem C. There is a family W n of closed hyperbolic 3-manifolds with injectivity radius bounded below bysome ε > and volume growing linearly in n such that the 1-form Laplacian spectral gap vanishes exponentiallyfast in relation to volume: λ ( W n ) ≤ B vol( W n ) e − r vol( W n ) where r and B are positive constants and λ ( W n ) is the first positive eigenvalue of the 1-form Laplacian on W n . A key point in controlling the stable isoperimetric constants in this family is the ability to compare scl and length for certain curves. Proposition 1.6. Let M be a compact oriented hyperbolic 3-manifold with totally geodesic boundary ∂M .Let γ be a nullhomologous geodesic that that embeds in ∂M . Let B = { B , . . . , B n } be a collection of essentialbranched surfaces carrying all incompressible surfaces in M . Then there is a constant D > , depending onthe collection of essential branched surfaces B , such that | γ | ≤ D scl ( γ ) . If one then glues such a manifold to itself along its boundary using a suitable psuedo-Anosov mappingclass, one can then show that as the mapping class is replaced with powers of itself, certain curves in theboundary will have geodesic length that is bounded but stable commutator length growing exponentially in n . In Section 2, we show existence and study the local properties of the triangulations we use throughout thepaper. 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. InSection 4, we compare the eigenvalues of the approximation of the Laplacian to the genuine eigenvalues ofthe Laplacian, then use this comparison and the estimates from Section 3 to prove Theorem A. In Section5, we prove Theorem B. Finally, in Section 6, we use branched surfaces to compare the length and stablecommutator length of specific curves and construct the example of Theorem C. Section 6 makes use ofTheorem A, but is otherwise independent from the rest of the paper. Remark . Throughout the paper, numerous constants are used. Constants defined inside proofs have nomeaning outside the local setting of the proof. The letter C is repeatedly reused in Sections 2 and 3 to denotea constant coming from a Sobolev type estimate and can at any time be taken to be the maximum among allconstants denoted by C . Such constants depend only on injectivity radius and local choices of things likebump functions, unless specifically noted. Acknowledgments Countless thanks are due to my advisor Nathan Dunfield for suggesting the questions considered in this paperand for invaluable support. I also would like to thank Michael Lipnowski for a helpful conversation. Thiswork was partially supported by US NSF grant DMS-1811156.5 Triangulations and Whitney Forms In this section we study certain triangulations of hyperbolic manifolds with injectivity radius bounded belowthat enjoy useful combinatorial and geometric properties that will facilitate the estimates in Sections 3through 5.We begin by establishing the existence of such triangulations. For this, we use Delaunay complexes. Toobtain a Delaunay complex in a Riemannian manifold M , take a finite collection of points P ⊂ M andconsider the Voronoi celluation consisting of cells V p = { x ∈ M : d ( x, p ) ≤ d ( x, q ) for all p (cid:54) = q ∈ P } for p ∈ P. Dual to the Voronoi celluation is the Delaunay complex. The cells of the Delauney complex are theconvex 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 isa quantifiably small perturbation of the point set P whose Delaunay complex is a triangulation. The preciseconditions are as follows.Let M be a closed Riemannian manifold with distance function d . Given a pair ≥ µ > , ε > , a ( µ, ε ) -net is a collection P of points in M for which the following hold:1. ( P is ε -dense) For all x ∈ M , there is a p ∈ P such that d ( x, p ) < ε .2. ( P is µ -separated) All distinct p, q ∈ P satisfy d ( p, q ) ≥ µε .Theorem 3 of [BDG18] says that if µ and ε satisfy several inequalities relating to the injectivity radius andsectional curvatures, a ( µ, ε ) -net can be perturbed to ( µ (cid:48) , ε (cid:48) ) -net such that the resulting Delaunay complex isindeed 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 ( µ, ε ) -net such that ε ≤ min (cid:26) inj( M )4 , Ψ( µ ) (cid:27) , where inj( M ) is the injectivity radius of M and Ψ is a function of the net parameter µ . The function Ψ isdescribed in [BDG18] and is independent of the manifold M . Then there is a point set P (cid:48) that is a ( µ (cid:48) , ε (cid:48) ) -netwith resulting Delaunay complex a triangulation. Moreover, µ (cid:48) and ε (cid:48) satisfy the following : ε ≤ ε (cid:48) ≤ ε, (1) µ ≤ µ (cid:48) ≤ µ. (2)The separation and density conditions of a ( µ (cid:48) , ε (cid:48) ) -net ensure that the resulting Delaunay triangulation hasedge lengths lying in a closed interval [ µε, ε ] . We now specialize to the case µ = 1 . Set (cid:15) = min { ε/ , Ψ(1) } and let G ε be the space of hyperbolic n -simplices with edge lengths in the interval [ (cid:15) , (cid:15) ] . Then, in viewof Proposition 2.2 below, a manifold M with injectivity radius bounded below by ε admits a triangulationwhose simplices are isometric to those in G ε . Since the space of all hyperbolic n -simplices is parametrized byedge lengths, the space G ε is compact.The following proposition guarantees the existence of the triangulations we use in the paper. To state theproposition, we define the k -star star k ( σ ) of a simplex σ to be the union of the stars of all k -faces of σ . Proposition 2.2. Let M be a closed hyperbolic n -manifold and let < ε < inj( M ) (When n = 3 , ψ (1) =2 × . × − , which is roughly 45.15). Then there is a geodesic triangulation K of M whose simplicescome from G ε for which the k -star of every simplex isometrically embeds in H n .Proof. Set (cid:15) = min { ε/ , Ψ(1) } . Take a maximal collection of points P ⊂ M such that the balls B (cid:15) / ( p ) for p ∈ P are all disjoint. By maximality, the B (cid:15) ( p ) balls then cover M . Since the B (cid:15) / ( p ) balls are all the perturbed net inequalities come from equation (2) in [BDG18] p, q ∈ P , d ( p, q ) ≥ (cid:15) = µ(cid:15) , so P is µ -separated. Since the B (cid:15) ( p ) balls cover,every point x ∈ M is (cid:15) -close to some point p ∈ P , so P is (cid:15) -dense. The collection P therefore is a ( µ, (cid:15) ) -net,with µ = 1 and (cid:15) satisfying the hypotheses of Theorem 2.1. Consequently, there is a perturbation of P thatis a ( µ (cid:48) , (cid:15) (cid:48) ) -net whose Delaunay complex is a triangulation. Since, as remarked above, the edge lengths ofsimplices in this Delaunay triangulation lies in the interval [ (cid:15) , (cid:15) ] , the simplices come from G ε . The edgelength bound along with the fact (cid:15) < inj( M ) / ensures the diameter of any k -star (which will be less than3 times the length of the longest edge of a simplex) will be less than (cid:15) . Thus, the star of every simplexembeds isometrically in H n via the local inverse of the exponential map.We call the triangulations obtained by the previous proposition deeply embedded triangulations .Throughout, when referring to deeply embedded triangulations, we suppress reference to some fixed ε .Every simplicial triangulation K of a closed Riemannian manifold admits a dual celluation K ∗ comprisedof cells σ ∗ dual to the simplices σ of the triangulation K in the following sense (for a reference, see [Bre97]chapter VI.6): Take the first barycentric subdivision T of K , then the n -cells of the dual celluation K ∗ arethe closed stars of the vertices of the original triangulation K in the barycentric subdivision. This celluationis naturally triangulated by the the barycentric subdivision triangulation T . If the triangulation K hassimplices coming from G ε , the simplices of the triangulation T built in this way have bounded edge lengthssince they can be no longer than the diameter of a star built from simplices in G ε and no shorter than thedistance from the barycenter of a simplex to the barycenter of one of its faces, again extremizing over G ε .We now record some useful properties of these deeply embedded triangulations and their dual celluations. Proposition 2.3. Let M be a closed hyperbolic n -manifold of injectivity radius inj( M ) > ε and a fixed deeplyembedded triangulation K . Then there is a positive constant N = N ( ε ) such that the number of k -simplicesin 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 vertexthat span a face via the hyperbolic law of cosines. This implies that the number of n -simplices meeting at avertex v is bounded uniformly, since for any ball around the vertex, there is a uniform lower bound on thevolume 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 simplex is less than N for k = 0 , . . . , n . Proposition 2.4. Let M be a closed hyperbolic n -manifold with injectivity radius inj( M ) > ε and a fixeddeeply embedded triangulation K . Let γ be a geodesic in M . Then there is a constant J = J ( ε ) such that thenumber v of cells in the dual cell complex K ∗ that γ intersects (counted with multiplicity) satisfies v ≤ J | γ | . Proof. Suppose γ moves from an n -cell σ to an n -cell σ (cid:48) , intersecting the ( n − -skeleton of K ∗ at a point p ∈ σ ∩ σ (cid:48) . Consider the closed radius ε -ball at the point p , V = ¯ B ε ( p ) . Let x be the point at which γ enters V and let y be the point at which it exits V . Then the geodesic subarc of γ running from x to y has length ε .Since the triangulation K has simplices from G ε , the restrained combinatorics of the dual celluation ensuresthat the ball V intersects a universally bounded number of dual cells. Let R ( ε ) denote this bound.Consider the sequence x n , y n of points such that x = x and y = y from above for the first simplexcrossing, 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 γ that intersects at most R ( ε ) simplices. Thus, ν ≤ ( | γ | ε + 1) R ( ε ) . Ittherefore follows that vR ( ε ) ε ≤ | γ | + 2 ε. Since ε ≤ | γ | , we have | γ | + 2 ε ≤ | γ | , and the stated linear boundfollows with J = R ( ε )2 ε .We also need to compare the length of closed geodesics to paths in the 1-skeleton of dual celluation K ∗ approximating them. To measure the complexity of paths in K ∗ , let || · || G be the (cid:96) -norm on chains and len ( · ) the word length of the cellular path. For a cellular path c , let || c || G be the (cid:96) -norm of the correspondingchain. Proposition 2.5. There is a constant L = L ( ε ) > such that for any geodesic curve γ in M , there is acellular path c in K ∗ homotopic to γ such that || c || G ≤ len ( c ) ≤ L | γ | . roof. Fix a base point and orientation for γ such that the base point lies on a face of a top dimensional cell.The curve γ can be replaced by a homotopic curve whose length is bounded by a constant times the geodesiclength of γ 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 γ intersects (counting these intersections withmultiplicity) that depends linearly on the length of γ and the fact the simplices of the triangulation havebounded diameter. Using the orientation and basepoint, we obtain a sequence of vertices with line segmentsbetween them that lie entirely in a cell. We can further modify γ by replacing these curve segments withcurves that lie in the 1-skeleton by traversing the 1-simplex joining the two boundary vertices. Since theedges in the celluation K ∗ have bounded length, this again adds bounded length to the curve. Let c denotethe cellular path we have constructed. By the previous considerations, there is a constant L depending onlyon ε giving the comparison len ( c ) ≤ L | γ | . The inequality || c || G ≤ len ( c ) is trivial.The combinatorial geometry of the triangulation K is related to the Riemannian geometry of M by way ofthe Whitney form map W : C • ( K ) → L Ω • ( M ) relating the cochain complex C • ( K ) (with R coefficients) tothe L -de Rham complex. It will be useful to view C • ( K ) as a subcomplex of the singular cochain complex.With this in mind, we often identify singular simplices in manifolds with their images. The Whitney map isreadily defined using the basis for C • ( K ) dual to the basis of simplices. This basis consists of the cochains δ σ that take the value 1 on the simplex σ and zero on all other simplices. The Whitney form W ( δ σ ) associatedto the cochain δ σ dual to an oriented simplex σ = [ v , . . . , v q ] is given by W ( δ σ ) = q ! q (cid:88) k =0 ( − k b k db ∧ · · · ∧ db k − ∧ db k +1 ∧ · · · ∧ db q , where b k : M → [0 , is the barycentric coordinate associated to the vertex v k . We now outline several important features of the Whitney form map, see [Dod76] for details. Later on,we will work with a smoothed version of Whitney forms. The formal properties outlined in this paragraphhold there as well. We call L -forms in the image of W Whitney forms. The support of a Whitney form W ( δ σ ) is contained in the closed star of the simplex σ ; this important property allows us to localize manyarguments to stars, which vary in a controlled way. The barycentric coordinates used to define the Whitneyforms are not smooth. They are smooth, however, in the compliment of the ( n − -skeleton of K . Onecan define the exterior derivative of a Whitney form in a weak sense, which yields a differential that is welldefined as an L -form. With this exterior derivative, the Whitney map becomes a chain map. For anycochain f and simplex σ , the restriction of the Whitney cochain ω = W ( f ) to σ , denoted ω | σ , can be uniquelyextended to a smooth form on the boundary of σ . This extension however is not unique when σ lies in theboundary of multiple simplices. In addition to the restriction of Whitney forms, we have the restriction forcochains. If f = (cid:80) a i δ σ i is a cochain and σ is a simplex, then f | σ = (cid:80) σ i ⊂ σ a i δ σ i . This cochain restrictionsatisfies ω | σ = W ( f | σ ) | σ .The geometry of geodesic simplices in hyperbolic space may be understood in a straightforward wayusing barycentric coordinates and Thurston’s straightening map. Identify H n with the upper sheet of thehyperboloid in Minkowski space R n, with quadratic form Q = x + x + · · · + x n − − x n and consider asingular simplex σ : ∆ → H n . Let b , b , . . . , b n : ∆ → [0 , be the barycentric coordinates on the standardEuclidean simplex ∆ with vertices e , . . . , e n . Then for v ∈ ∆ , write v = (cid:80) b i ( v ) e i . The straightening st ( σ ) of a simplex σ is the singular simplex defined by st ( σ )( v ) = (cid:88) b i ( v ) σ ( v i ) . If π : H n → M is the projection map and σ is a simplex in M , let st ( σ ) : ∆ → M be the composition of thestraightening of σ applied to some lift of σ and the projection map. This is well-defined and independent of This is the completion of the usual de Rham complex under the L -norm determined by the Hodge star (cid:63) , given by || ω || = (cid:82) M ω ∧ (cid:63)ω . In any simplicial complex, every point is in the interior of exactly one simplex σ with vertices v , . . . , v n . The identificationof σ with the standard simplex in R n +1 maps every point of σ to a convex combination of the vertices; p = (cid:80) i a i v i . Thebarycentric coordinate map is then given by b i ( p ) = a i . H n acts linearly on R n, preserving the quadratic form Q . For eachsimplex σ in M , let V σ : st ( σ ) → ∆ be the map from the straight simplex st ( σ ) to the standard simplex given by the barycentric coordinates: V σ (cid:16)(cid:88) b i σ ( e i ) (cid:17) = (cid:88) b i e i ∈ ∆ . This map is really just the inverse of the singular straight simplex st ( σ ) . A Whitney form associated to thesimplex σ is then the corresponding Whitney form on the standard Euclidean simplex pulled back to σ . Wecan compare two straight simplices using the composition V − σ (cid:48) ◦ V σ , which is given by V − σ (cid:48) ◦ V σ (cid:16)(cid:88) b i σ ( e i ) (cid:17) = (cid:88) b i σ (cid:48) ( e i ) . The maps V σ depend continuously on σ in the sense that if σ is a straight simplex in H n and σ (cid:48) is obtainedby perturbing the vertices of σ , then the composition map V − σ (cid:48) ◦ V σ is almost an isometry, where the failureto be an isometry is controlled by the size of the vertex perturbation. Indeed, we have the following: Proposition 2.6. Let σ and σ (cid:48) be straight simplices from G ε embedded in H n . Then the map V − σ (cid:48) ◦ V σ is κ -biLipschitz for some κ = κ ( ε ) > that does not depend on σ and σ (cid:48) .Proof. The biLipschitz constant for the comparison map between two simplices depends continuously on thesimplices. Take the supremum over all pairs in G ε × G ε , this is finite by compactness.If K is a deeply embedded triangulation in M , by Proposition 2.3, any vertex of K is contained in atmost N = N ( ε ) simplices. As a result, there are finitely many possible finite simplicial complexes that appearas the star of a simplex in a deeply embedded triangulation. Recall that the k -star star k ( σ ) of a simplex σ is the union of the stars of all k -faces of σ . Then there are finitely many combinatorial types of k -stars of an n -simplex. Of course, if σ is a k -simplex, then the k -star of simplex is its usual star. Let A be the finite setof possible k -star complexes in a deeply embedded triangulation For any complex a ∈ A , say with | a | many n -cells, a geometric structure on a is given by identifying each n -cell in a with a model simplex in G ε in sucha way that the face gluing maps are isometries. Each star complex a has geometric structures parametrizedby a subspace of G | a | ε . By taking the disjoint union over all shapes a ∈ A , we obtain a precompact space S ε ( k ) that parametrizes the geometry of stars in deeply embedded triangulations. We have uniform controlover the geometry of these complexes in the same way that we do over the simplices. To see this, define mapsfrom the complex to a Euclidean model complex of the combinatorial type of the complex by gluing themaps V σ defined above together according to the combinatorics of the complex. Since the gluing maps areisometries, this is well defined. Since the map restricted to each simplex is uniformly biLipschitz equivalentto the model simplex, the same holds for the stars. We encode this in the following proposition. Proposition 2.7. There is a constant L = L ( ε ) such that if s and s (cid:48) are two complex of the same combinatorialtype in A , then s and s (cid:48) are L -biLipschitz equivalent. Using this comparison, we are able to compare locally the geometric norms on cochains determined by theWhitney map with combinatorial norms. This is done using the L p -change of variables formula for k -formsintroduced in [Ste13]. Proposition 2.8. Let s ∈ S ε ∪ G ε . Let W ( f ) be the Whitney form associated to a cochain f ∈ C k ( s ; R ) . Let || · || be some fixed norm on the real vector space C k ( s ; R ) and let || · || p,s be the p -norm associated to s on Ω k ( s ) , where p = ∞ or p = 2 . Then there is a constant A = A ( ε, || · || ) > such that A − || W ( f ) || p,s ≤ || f || ≤ A|| W ( f ) || p,s . Proof. By Proposition 2.6, there is a constant L (note κ ≤ L , so that if we’re working with simplices, L worksas Lipschitz constant) such that any pair s, s (cid:48) ∈ S ε ∪ G ε in the same combinatorial type are L -biLipschitzequivalent. For each combinatorial type, fix a model s a ∈ S ε ∪ G ε and let µ : s → s a be a L -biLipschitz9omparison map. Then, applying the L -change of variables formula for k -forms and using the biLipschitzcomparison, we get || W ( f ) || ,s || f || ≤ L n/ || W ( f ) || ,µ ( s ) || f || , and applying the L ∞ -change of variables formula for k -forms, we get || W ( f ) || ∞ ,s || f || ≤ L k || W ( f ) || ∞ ,s a || f || . For p = 2 , set A = L n/ max a sup g ∈ C k ( s a ) || W ( g ) || ,s a || g || and for p = ∞ , set A ∞ = L n max a sup g ∈ C k ( s a ) || W ( g ) || ,s a || g || , where the maximum runs over all combinatorial types a . Then, A = max {A , A ∞ } gives the first inequality.The second inequality is obtained from an identical argument via the lower bound in biLipschitz comparison.Let A be the maximum of these two constants.We now turn to the smooth Whitney forms mentioned above, which are defined by replacing the barycentriccoordinates with a smooth partition of unity indexed by the vertices of a triangulation. The existence of asuitable smooth partition of unity, which we will call a smooth barycentric partition of unity, was proved byDodziuk. The estimates throughout rely on the covariant derivative bounds for the partition of unity, not theparticular partition of unity constructed. Proposition 2.9. ( [Dod81], Lemma 2.11) If M admits a deeply embedded triangulation K , then there existsa C ∞ partition of unity β i indexed by the vertices of K and subordinate to the covering of M by open starsof vertices of K (indeed, compactly supported in each open star). Moreover, each β i has covariant derivativessatisfying the pointwise bound |∇ k β i | < C for some constant C = C ( ε ) , for k ≤ n . For completeness, and to identify what the constant C depends on, we recall Dodziuk’s construction. Proof. Let s ∈ S ε be the 0-star of a simplex. Denote the vertices of s by v , . . . , v n and let b i be the standardbarycentric coordinates associated to the vertex v i . Define ¯ b i ( x ) = (cid:40) b i ( x ) ≤ / ( n + 2) , ( n +2) b i ( x ) − n +1 b i ( x ) ≥ / ( n + 2) . Observe that (cid:80) i ¯ b i ( y ) ≥ n +2)( n +1) . Define δ ( s ) = inf x ∈ supp(¯ b i ) y ∈ ∂ star ( v i ) d ( x, y ) , where d is the distance function induced by the Riemannian metric. Set δ = inf s ∈S ε δ ( s ) and notice δ > . Forany point x / ∈ star ( v i ) , the ball B δ ( x ) is disjoint from the support of ¯ b i . Let η be a smooth cutoff functionsuch that η ( r ) = 1 when | r | < δ/ and η ( r ) = 0 when | r | > / δ . The function η ( d ( x, y )) is smooth on theopen star of a vertex, so the operator given by integrating against η ( d ( x, y )) is smoothing. Therefore, if wedefine ˜ b i ( x ) = (cid:90) star ( v i ) η ( d ( x, y ))¯ b i ( y ) dy, the result is a smooth function. Notice ˜ b i is supported on the star of v by virtue of our choice of δ . We nowdefine smoothed barycentric partitions of unity for a smooth manifold with deeply embedded triangulation K 10y normalizing the functions ˜ b i associated to the vertices of K : β i ( x ) = (cid:88) j ˜ b j ( x ) − ˜ b i ( x ) . Notice that if ˜ b i ( x ) (cid:54) = 0 , then this normalizing sum really just runs over the vertices of star ( v i ) . Thisnormalizing constant can be bounded from below: (cid:88) j ˜ b j ( x ) ≥ (cid:88) j (cid:90) B δ/ ( x ) ¯ b j ( y ) dy = (cid:90) B δ/ ( x ) (cid:88) j ¯ b j ( y ) dy ≥ vol( B δ/ ( x )) 1( n + 2)( n + 1) . The covariant derivative bound follows from repeated application of the quotient rule and the correspondingbounds for ¯ b i , which depends only on the derivatives of cutoff function η and the covariant derivatives of themetric. The choice of δ ensures each function β i is compactly supported in the star of v i .We now study the local properties of these smooth barycentric partitions of unity, which will replacethe barycentric coordinates in the Whitney cochain map. Our aim in particular is to establish a version ofProposition 2.8 for these partitions of unity that will allow us to relate the geometric norms induced by theWhitney map to combinatorial norms. Our analysis of these partitions of unity relies on showing they live ina suitable compact function space. Dodziuk’s construction then ensures that this space is nonempty.Let σ be an n -simplex simplex from G ε with a fixed 0-star s = star ( σ ) ∈ S ε . Let v , . . . , v n be thevertices of σ . Let H d ( star ( v i )) be the completion of the space of smooth functions on star ( v i ) with respectto the norm || f || H d = || f || + || df || . This norm is equivalent to the usual Sobolev norm defined by thecovariant derivative, which we use when working with higher order derivatives (we denote that Sobolevspace by H k ∇ ( star ( v i )) with norm || · || H k ∇ ( star ( v i )) = k (cid:80) i =0 ||∇ i f || ). We also will use Sobolev spaces of formson manifolds. For a manifold Y , possibly with boundary, we denote these spaces H d Ω • ( Y ) and H k ∇ Ω • ( Y ) respectively and their norms by || · || H d ( Y ) and || · || H k ∇ ( Y ) . When • = 0 , we drop Ω from this notation.When Y has boundary, the marking ˚ H k in the above Sobolev spaces denotes the subspace of forms that areapproximated by smooth forms supported in the interior of Y .The Rellich-Kondrachov embedding theorem (see [Ada03], Theorem 6.3 Part II and Part IV) ensuresthe functions that are in sufficiently high order Sobolev spaces (for dimension n > , taking n derivativessuffices) embed compactly in ˚ H d ( star ( v i )) . Thus if we topologize V = (cid:81) i ˚ H n ∇ ( star ( v i )) as a subspace of (cid:81) i ˚ H d ( star ( v i )) , it is compact. Define the subset B ( σ, s ) ⊂ V of smooth barycentric partitions of unity tobe the subspace of functions that satisfy the following:1. |∇ k β i | ≤ C where C is the constant from Proposition 2.9 above and k ∈ { , . . . , n } . ≤ β i ( x ) ≤ for all x .3. For all x ∈ σ , (cid:80) β i ( x ) = 1 .Note that each β i ∈ ˚ H d ( star ( v i )) , so in particular each β i is supported on the interior of the star of thecorresponding vertex. Additionally, observe that if σ is an n -simplex in a deeply embedded triangulation K of M , β i the corresponding partition of unity constructed in Proposition 2.9, and if σ has vertices v , . . . , v n and s = star ( σ ) , then ( β , . . . , β n ) ∈ B ( σ, s ) . Lemma 2.10. The space B ( σ, s ) of smooth barycentric partitions of unity on an n -simplex σ with -star s is compact.Proof. Since B ( σ, s ) is a subspace of a compact space, we need only verify that it is closed. Conditions 2and 3 are defined by bounded operators, so define closed sets. For Condition 1, we use that the embedding ˚ H n ∇ → ˚ H d is a compact operator and the fact that unform pointwise bounds hold in the weak limit. Thus, ifa sequence of functions f n in ˚ H n ∇ weakly converge to f and |∇ k f n ( x ) | ≤ C for every x , then |∇ k f ( x ) | ≤ C .This gives that the set of functions satisfying Condition 1 is weakly closed in ˚ H n ∇ . Since compact operators11ake weakly convergent sequences to norm convergent sequences, it follows that the set of functions thatsatisfy Condition 1 in ˚ H d is closed. The claim then follows by identifying B ( σ, s ) with the intersection of theclosed sets defined by these conditions.To a partition of unity indexed by the vertices of a triangulation and subordinate to the covering by openstars of vertices, one can define a generalized Whitney mapping, given by the same formula as the standardWhitney map but with the smooth barycentric partitions in place of the standard barycentric coordinates.Like the standard Whitney map, these generalized Whitney maps satisfy:1. For a chain a and cochain f of the same degree, (cid:82) a W β ( f ) = f ( a ) . 2. For any cochain f , dW β ( f ) = W β ( df ) .3. If p is contained in the interior of an n -simplex σ , and any cochain f , W β ( f ) p = W β ( f | σ ) p .For smooth barycentric partitions of unity, we have the following. Lemma 2.11. The Whitney map W β : B ( σ, s ) × C • ( σ ) → L Ω • ( σ ) varies continuously with β. Proof. It suffices to show that exterior products of dβ i vary continuously in the L -norm. The degree 0 anddegree 1 cases are immediate by our choice of topology. We treat only the degree 2 case. Assume || β − β (cid:48) || < (cid:15) in the product norm on B ( σ, s ) . Notice || dβ ∧ dβ − dβ (cid:48) ∧ dβ || = || dβ ∧ dβ − dβ (cid:48) ∧ dβ + dβ ∧ dβ (cid:48) − dβ ∧ dβ (cid:48) || ≤ || dβ ∧ dβ − dβ ∧ dβ (cid:48) || + || dβ ∧ dβ (cid:48) − dβ (cid:48) ∧ dβ || = || dβ ∧ ( dβ − dβ (cid:48) ) || + || dβ ∧ ( dβ (cid:48) − dβ ) || ≤ || dβ || ( || dβ − dβ (cid:48) || + || dβ (cid:48) − dβ || ) ≤ C(cid:15). Proposition 2.12. There is a constant A = A ( ε ) > such that for any β ∈ B ( σ, s ) and any cochain f ∈ C • ( σ ) , there is a comparison: A − || W ( f ) || ,σ ≤ || W β ( f ) || ,σ ≤ A || W ( f ) || ,σ . Proof. Since the L -norm induced by β is continuous on the vector space of cochains C • ( σ ) and for anycochain f , W β ( f ) varies continuously with β in the L -norm on Ω • , it follows that || W β ( f ) || is continuousas a function on B ( σ, s ) × C • ( σ ) . Since each W β ( f ) sends nonzero cochains to nonzero L -forms, for any f (cid:54) = 0 , one has < || W β ( f ) || . This along with the continuity and compactness of Lemma 2.10 and Lemma2.11 implies the constants A • = inf β ∈ B ( σ,s ) inf f ∈ C • ( σ ) || W ( f ) || =1 || W β ( f ) || , and B • = sup β ∈ B ( σ,s ) sup f ∈ C • ( σ ) || W ( f ) || =1 || W β ( f ) || , are strictly positive real numbers. Take A = max { A − • , B • } .The upshot of this is that any smooth barycentric partition of unity induces a norm on the cochaincomplex of a simplex that locally is uniformly comparable to the L -norm induced by the standard barycentriccoordinate. Because the L -norm associated to the standard Whitney forms on a simplex are all uniformlycomparable by Proposition 2.8, this comparison implies for each geometric structure from S ε on the star ofsome simplex, the cochain norms induced by smooth barycentric coordinates on the simplex are all comparable.Moreover, this shows that the one can even compare the norms determined by different combinatorial types12nd geometric structures on the 0-star of the simplex. This gives an analogue of Proposition 2.8 for the L -norm. We also need such a comparison for the L ∞ -norm. The upgraded version of Proposition 2.8 appearsbelow as Proposition 2.15. For this, we require the following lemma. Lemma 2.13. There is a constant R ( ε ) > such that for any n -simplex σ ∈ G ε with 0-star s ∈ S ε andbarycentric partition of unity β ∈ B ( σ, s ) , the map W β : C • ( σ ) → H n ∇ Ω • ( σ ) satisfies || W β ( f ) || H n ∇ ( σ ) ≤ R || W β ( f ) || ,σ . Proof. We first observe that the covariant derivative bounds for a smooth barycentric partition of unity implythat || W β ( f ) || H n ∇ ( σ ) is bound by some constant times || f || G,σ , where || · || G,σ is the (cid:96) -norm on C • ( σ ) . For acochain f = (cid:80) a i δ σ i , let ω i = W β ( δ σ i ) so that W β ( f ) = (cid:80) a i ω i . We can therefore compute, || W β ( f ) || H k ∇ ( σ ) = || (cid:88) a i ω i || H k ∇ ( σ ) ≤ (cid:88) j ||∇ j (cid:88) i a i ω i || ,σ ≤ (cid:88) j (cid:88) i | a i |||∇ j ω i || ,σ . Each summand above satisfies ||∇ j ω i || < C for a constant C depending on the covariant derivative boundsof the barycentric partition of unity. There is a constant T such that the number of • -faces of an n -simplex isless than T . Thus, || W β ( f ) || H k ∇ ( σ ) ≤ (cid:88) j (cid:88) i | a i |||∇ j ω i || ,σ ≤ (cid:88) j C (cid:88) i | a i |≤ T C (cid:88) i | a i | = T C || f || G,σ . This combinatorial (cid:96) -norm is comparable to the β -induced L -norm by Proposition 2.8 and Proposition 2.12.Thus, || W β ( f ) || H n ∇ ( σ ) ≤ R || W β ( f ) || ,σ .The following result, a consequence of Theorem 1 in [Can75], ensures that control over certain Sobolevnorms implies pointwise norm control. Let | · | be the pointwise norm induced by the Riemannian metric. Theorem 2.14. (Cantor) Suppose M is a hyperbolic n -manifold with injectivity radius bounded below by ε .Let r < ε , and l ≥ , k ≥ be such that l + n/ < k . Then if ω is in the Sobolev space H k ∇ ( M ) , there is aconstant C = C ( r ) such that for every p ∈ M , |∇ l ω ( p ) | ≤ C || ω || H k ∇ ( B r ( p )) . Cantor’s theorem along with Lemma 2.13 implies the smooth barycentric partition of unity induced L -norm (cid:107)·(cid:107) ,σ and L ∞ -norm (cid:107)·(cid:107) ∞ ,σ are comparable for any simplex σ ∈ G ε with star s ∈ S ε and any smoothbarycentric partition of unity β ∈ B ( σ, s ) .This discussion provides us with the following upgraded version of Proposition 2.8. Proposition 2.15. Let s ∈ S ε be the star of an n -simplex σ . Let β be a smooth barycentric partition ofunity in B ( σ, s ) , or the standard barycentric coordinate on s . Let W β ( f ) be the resulting generalized Whitneyform associated to a cochain f ∈ C • ( σ ; R ) . Let || · || be some fixed norm on the real vector space C • ( σ ; R ) and let || · || p,σ be the p -norm associated to σ on Ω • ( σ ) , where p = ∞ or p = 2 . Then there is a constant B = B ( ε, || · || ) > (independent of σ and s ) such that B − || W β ( f ) || p,σ ≤ || f || ≤ B|| W β ( f ) || p,σ . Norm Estimates In this section, we use deeply embedded triangulations and the Whitney maps described in the previoussection to compare various geometric and combinatorial norms on forms and cochains. Throughout, let be M a closed hyperbolic manifold of dimension n > with injectivity radius bounded below by ε > and a fixeddeeply embedded triangulation K . Let β be a smooth barycentric partition of unity for K . For concreteness,one can always assume we are working with barycentric partition of unity given by Dodziuk’s construction(see Proposition 2.9).We require various comparisons of the following norms on cochain and chain complexes associated to M and K . The relevant norms are:1. The combinatorial Gromov norm || · || G on any chain or cochain complex given by || (cid:80) a i σ i || G = (cid:80) | a i | and || (cid:80) a i δ σ i || G = (cid:80) | a i | .2. The combinatorial max norm ||·|| max on any chain or cochain complex given by || (cid:80) a i σ i || max = max | a i | .3. The Whitney induced L -norm || · || on the cochain complex C • ( K ) , given by || f || = (cid:113)(cid:82) M W β ( f ) ∧ (cid:63)W β ( f ) .4. The Whitney induced L ∞ -norm || · || ∞ on the cochain complex C • ( K ) , given by taking the essentialsupremum of the pointwise Riemannian metric operator norms || f || ∞ = ess sup p ∈ M || W β ( f ) p || ∞ . Given a norm || · || on the cochain complex C • ( K ) , let || · || ∗ denote the dual norm on the linear dualchain complex C • ( K ) induced by the integration pairing: || a || ∗ = sup || f || =1 f ∈ C • ( K ) (cid:90) a W β ( f ) . Proposition 3.1. There is a constant B = B ( ε ) > such that the norms || · || G and || · || on C • ( K ) satisfy || · || G ≤ B (cid:112) vol( M ) || · || , and the norms || · || G and || · || ∗ on C • ( K ) satisfy || · || G ≤ B (cid:112) vol( M ) || · || ∗ . Proof. Let f = (cid:80) F a F δ F be a cochain. Then for any n -simplex σ , f | σ = (cid:80) F ⊂ σ a F δ F and || f || = (cid:88) σ ∈ K ( n ) || W β ( f | σ ) | σ || = (cid:88) σ ∈ K ( n ) || W β ( f ) | σ || . Apply Proposition 2.15 to obtain D = B ( || · || , || · || G ) . This gives || f | σ || G ≤ D || f | σ || ,σ , where || · || ,σ isthe L -norm on the simplex σ associated to the smooth barycentric coordinate β . Then, by applying theEuclidean (cid:96) - (cid:96) -comparison to the cochain complex and using the fact there is a constant T such that thenumber of n -simplices in K is less than T vol( M ) , we find || f || G ≤ (cid:88) σ ∈ K ( n ) || f | σ || G ≤ (cid:88) σ ∈ K ( n ) D || f | σ || ,σ ≤ D (cid:115) T vol( M ) (cid:88) σ ∈ K ( n ) || W β ( f ) | σ || ≤ D (cid:112) T vol( M ) || f || . || · || G is the usual (cid:96) -norm on a finite dimensional vector space, so its dualnorm is the max norm || · || max .Apply Proposition 2.15 and set D (cid:48) = B ( ε, ∞ , || · || max ) , so that, || f || ∞ ≤ D (cid:48) || f || max . Then, || f || ≤ (cid:112) vol( M ) || f || ∞ ≤ D (cid:48) (cid:112) vol( M ) || f || max . Dualizing gives || · || G = || · || ∗ max ≤ D (cid:48) (cid:112) vol( M ) || · || ∗ , since || a || G = || a || ∗ max = sup || f || max ≤ (cid:90) a W β ( f )= sup || D (cid:48) √ vol( M ) f || max ≤ (cid:90) a D (cid:48) (cid:112) vol( M ) W β ( f ) ≤ sup || f || ≤ (cid:90) a D (cid:48) (cid:112) vol( M ) W β ( f )= D (cid:48) (cid:112) vol( M ) sup || f || ≤ (cid:90) a W β ( f )= D (cid:48) (cid:112) vol( M ) || a || ∗ . Set B = max { D √ T , D (cid:48) } to obtain the claim.Recall from Section 2 that there is a polyhedral celluation K ∗ dual to K that can be canonically subdividedinto a triangulation T . Equipping these dual complexes with the Gromov norm, we have the following twopropositions relating these norms by the Poincaré duality and subdivision maps. Proposition 3.2. There is a constant D = D ( ε ) such that for any • -cochain f ∈ C • ( K ) one has || f || ≤ D || f || G .Proof. Let f = (cid:80) a i δ σ i be a • -cochain. Then || ω || G = (cid:80) | a i | and || f || ≤ (cid:80) | a i ||| δ σ i || . Then, for any fixed • -simplex σ that is a face of an n -simplex from G e , using the L -change of variables formula and Proposition2.12 we can take D = A L n/ || W ( δ σ ) || , so that || δ σ i || ≤ D. The comparison || f || ≤ (cid:88) | a i ||| δ σ i || ≤ D (cid:88) | a i | = D || f || G then follows. Proposition 3.3. The Poincaré duality map Φ : C • ( K ) → C n −• ( K ∗ ) preserves the Gromov norm || f || G = || Φ( f ) || G . Proof. Let f = (cid:80) a i δ σ i , then Φ( f ) = (cid:80) a i ( σ i ) ∗ . Proposition 3.4. Let N be the constant from Proposition 2.3, which bounds the number of simplices in thestar of a simplex in a deeply embedded triangulation. Then the subdivision map τ : C ( K ∗ ) → C ( T ) satisfies || c || G ≤ N || τ ( c ) || G . Proof. The number of sides of a -cell in K ∗ dual to a ( n − -simplex σ in K corresponds to the number of n -simplices in K that contain σ . The number of such simplices is bounded by N .The following estimates are essential in comparing the first eigenvalue of the Whitney Laplacian to thegenuine first eigenvalue. For this, we need to work with various Sobolev spaces to control the orthogonal15rojection of a Whitney form onto its coexact part. This discussion is the reason we use the smoothedWhitney 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 analternative Sobolev norm on H k +1 ∇ Ω • ( Y ) : || ω || A k ( Y ) := || ω || H k ∇ ( Y ) + || dω || H k ∇ ( Y ) + || d ∗ ω || H k ∇ ( Y ) . Since d and d ∗ are bounded operators H k +1 Ω •∇ ( Y ) → H k Ω •± ∇ ( Y ) , we immediately have that there is aconstant C such that || ω || A k ( Y ) ≤ C || ω || H k +1 ∇ ( Y ) . Recall that the marking ˚ H denotes the subspace of given Sobolev space that is the closure of smoothfunctions supported in the interior. Lemma 3.5. (Gaffney inequality) Let Y be a smooth manifold with boundary. Let ω ∈ ˚ H k ∇ Ω ( Y ) . Thenthere is a constant C = C ( Y ) > such that || ω || ˚ H k ∇ ( Y ) ≤ C || ω || A k − ( Y ) . The following proposition shows the norm || · || A k ( Y ) behaves as expected when a form is multiplied by abump function. Lemma 3.6. Let B = B r ( p ) and B = B r + δ ( p ) be a pair of concentric balls in H n and let φ be a smoothbump function that is identically 1 on B and vanishes in a neighborhood of ∂B . There is a constant C = C ( φ, k ) that depends only on the norm of the covariant derivatives of φ up to order k + 1 such that if ω ∈ Ω ( H n ) , then || φω || A k ( B ) ≤ C || ω || A k ( B ) . Proof. Notice that dφω = dφ ∧ ω + φdω and d ∗ φω = φd ∗ ω + g ( ∇ φ, X ω ) , where X ω is the vector field dual to ω . As a result, the triangle inequality yields || φω || A k ( B ) ≤ || φdω || H k ∇ ( B ) + || φd ∗ ω || H k ∇ ( B ) + || φω || H k ∇ ( B ) + || dφ ∧ ω || H k ∇ ( B ) + || g ( X ω , ∇ φ ) || H k ∇ ( B ) . The estimate || α ∧ β || ≤ || α || ∞ || β || implies || dφ ∧ ω || H k ∇ ( B ) ≤ C || ω || H k ∇ ( B ) , where the constant C is given by the sum of the || · || ∞ -norms of the covariant derivatives of the bump function φ . The same argument gives for any form ξ that || φ ∧ ξ || H k ∇ ( B ) ≤ C || ξ || H k ∇ ( B ) . Applying this estimate to φdω, φd ∗ ω , and φω handles all terms in the comparison save for || g ( X ω , ∇ φ ) || H k ∇ ( B ) . For this term, noticethat ∇ g ( X ω , ∇ φ ) = g ( ∇ X ω , ∇ φ ) + g ( X ω , ∇ φ ) . We therefore have the pointwise estimate |∇ g ( X ω , ∇ φ ) | ≤ | g ( ∇ X ω , ∇ φ ) | + | g ( X ω , ∇ φ ) | by the triangle inequality, ≤ |∇ ω | |∇ φ | + | ω | |∇ φ | by Cauchy-Schwarz and the musical isomorphism.Integrating then gives the corresponding inequality for the Sobolev norm H ∇ . Repeating this calculation forhigher order covariant derivatives completes the proof.The two previous lemmas combine to give the following statement. Proposition 3.7. Let B = B r ( p ) and B = B r + δ ( p ) be a pair of concentric balls in H n Then there is aconstant C = C ( r, δ ) such that for any ω ∈ H k ∇ ( B ) one has || ω || H k ∇ ( B ) ≤ C || ω || A k − ( B ) . Proof. Let √ C be the maximum of the constants from Gaffney’s inequality and Lemma 3.6 with bumpfunction φ . Then for a form ω ∈ H k ∇ ( B ) , one has || ω || H k ∇ ( B ) ≤ || φω || H k ∇ ( B ) , φω | B = ω. Since φω vanishes on ∂B , Gaffney’s inequality and Lemma 3.6 give || φω || H k ∇ ( B ) = || φω || ˚ H k ∇ ( B ) ≤ √ C || φω || A k − ( B ) ≤ C || ω || A k − ( B ) . Combining these two estimates gives the proposition. Proposition 3.8. Let M be a hyperbolic n -manifold with injectivity radius greater than ε > . There is aconstant H ( ε ) = H > depending only on ε such that if the L -Hodge decomposition of a smooth 1-form ω ∈ Ω has coexact part α , then for any point p ∈ M there is a ball B ⊂ M centered at p of radius determinedby ε such that for k > n one has |∇ α ( p ) | ≤ H (cid:16) || ω || H k ∇ ( B ) + || α || ,B (cid:17) and consequently |∇ α ( p ) | ≤ H || ω || H k ∇ ( M ) . Proof. The L -Hodge decomposition of M determines an orthogonal decomposition of ω = α + η + h , where α = d ∗ A , η = db , and h harmonic. Cantor’s estimate (Theorem 2.14) implies at any point p of M , |∇ α ( p ) | ≤ C ( r ) || α || H k ∇ ( B r ( p )) so long as r < inj( M ) and k > n/ . Since we assume dimension n > , any k ≥ n suffices. Take aconcentric family of balls B i = B r + iδ ( p ) where r = 6 ε , and kδ + r < ε < inj( M ) ; note that B contains the0-star of σ . Let φ i be a bump function that is identically one on B i and vanishes on ∂B i +1 .Letting C i be the maximum of the constant from Cantor’s estimate on the ball B i and the constant fromProposition 3.7 for the balls B i ⊂ B i +1 , we have |∇ α ( p ) | ≤ C i || α || H k ∇ ( B i ) ≤ C i || α || A k − ( B i +1 ) . Since α is coexact, || α || A k − ( B i +1 ) = || α || H k − ∇ ( B i +1 ) + || dα || H k − ∇ ( B i +1 ) . Notice that dα = dω , and that d : H k ∇ ( B i ) → H k − ∇ ( B i ) is a bounded operator, say with operator norm C .Thus, we have || dα || H k − ∇ ( B i +1 ) ≤ C || ω || H k ∇ . As a result, we get the estimate || α || A k − ( B i +1 ) ≤ || α || H k − ∇ ( B i +1 ) + C || ω || H k ∇ ( B i +1 ) Combining this with the estimate of ∇ α above gives |∇ α ( p ) | ≤ C i (cid:16) || ω || H k ∇ ( B i +1 ) + || α || H k − ∇ ( B i +1 ) (cid:17) . We can repeat argument this using the family of balls B i to reduce the order of the Sobolev norm of α on theright-hand side until we obtain |∇ α ( p ) | ≤ H (cid:16) || ω || H k ∇ ( B n ) + || α || ,B n (cid:17) , where H is obtained by combining all the constants appearing in the iterated calculation. Orthogonality ofthe Hodge decomposition implies || α || ≤ || ω || , and we clearly have || ω || ,B n ≤ || ω || H k ∇ ( M ) , so we are doneafter 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 ∈ C ( K ) . There is a constant H ( ε ) = H > depending only on ε such that if the L -Hodge ecomposition of the generalized Whitney form W β ( f ) has coexact part α , then ||∇ α || ∞ ≤ H || W β ( f ) || . Proof. Let ω = W β ( f ) be a smooth Whitney form. We will apply Proposition 3.8 at a point p in the interiorof an n -simplex. Since such a point can be chosen so that |∇ α ( p ) | is arbitrarily close to ||∇ α || ∞ , there is noloss in doing this. Proposition 3.8 gives a ball B about p of radius depending only on ε such that |∇ α ( p ) | ≤ H (cid:16) || ω || H k ∇ ( B ) + || α || ,B (cid:17) , for a constant H depending only on ε . The ball B intersects some uniformly bounded collection of n -simplices σ (cid:48) from K where the constant depends only on ε ; let T = T ( ε ) be this bound. We can therefore estimate thenorm of the Whitney form term by || ω || H k ∇ ( B ) ≤ (cid:88) σ (cid:48) ∩ B (cid:54) = ∅ || ω | σ (cid:48) || H k ∇ ( σ (cid:48) ) . Applying Lemma 2.13 to each summand in the previous estimate then gives || ω || H k ∇ ( B ) ≤ (cid:88) σ (cid:48) ∩ B i (cid:54) = ∅ || ω | σ (cid:48) || H k ∇ ( σ (cid:48) ) ≤ R (cid:88) σ (cid:48) ∩ B (cid:54) = ∅ || ω | σ (cid:48) || ≤ R √ T || ω || . Combining the above then gives that |∇ α ( p ) | ≤ HR √ T || ω || + H || α || ,B . Clearly H || α || ,B ≤ H || α || ≤ H || ω || , so that after increasing H to absorb the R √ T term we are done. Proposition 3.10. Let M be a hyperbolic n -manifold with a deeply embedded triangulation K . Let ω ∈ Ω ( M ) . Assume there exists a constant H such that |∇ ω | ≤ H . Then there is a constant C ( H, ε ) such that || ω || ∞ ≤ C ( H, ε ) || ω || . Proof. Assume || ω || ∞ = 1 and is realized at the point p . By Kato’s inequality and the hypothesis, away fromthe zeros of ω one has |∇| ω || ≤ |∇ ω | ≤ H . Fix a normal coordinate frame x , . . . , x n − at p of radius ε .Define the function φ on this normal coordinate neighborhood by φ ( x ) = 1 − Hd ( x, p ) for d ( x, p ) < /H and extend by zero. Then || φ || ∞ = || ω || ∞ and || φ || ≤ || ω || . Set C ( H, ε ) = 1 / || φ || . This comparison givesthe general result by scaling: That is, for an arbitrary nonzero 1-form ω we can write ω = || ω || ∞ ω (cid:48) , where || ω (cid:48) || ∞ = 1 . Then, C ( H, ε ) || ω (cid:48) || ≥ . Multiplying through by || ω || ∞ gives the proposition since C ( H, ε ) || ω || ∞ || ω (cid:48) || = C ( H, ε ) || ω || ≥ || ω || ∞ . Proposition 3.11. There is a constant C = C ( ε ) such that if f ∈ C ( K ) and ω = W β ( f ) = α + η where α is L -coexact and η is closed, then || α || ∞ ≤ C || α || . Proof. Assume that || f || = 1 . By Proposition 3.9, ||∇ α || ∞ ≤ H || f || = H . Proposition 3.10 gives a constant C = C ( H ( ε )) (so this really just depends on ε ) such that || α || ∞ ≤ C || α || . If f does not have unit L -norm,then either f = 0 , in which case the result is trivial, or else f = λf (cid:48) for some unit L -norm cochain f (cid:48) andpositive number λ . The coexact part of W β ( f (cid:48) ) is α (cid:48) = α/λ . Applying the result for the unit norm case gives || α (cid:48) || ∞ ≤ C || α (cid:48) || , and multiplying through by λ gives the general result.18 The Upper Bound In this section we prove Theorem A, which states that in a closed hyperbolic 3-manifold M the first positiveeigenvalue of the Hodge Laplacian acting on coexact 1-forms is bounded above by a multiple of the stableisoperimetric ratio ρ ( 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 usto restrict Theorem A to the 3-dimensional setting.The cochain results of the previous section are connected to spectral geometry via the inner productinduced by the Whitney map associated to a triangulation and barycentric coordinate: (cid:104) f, g (cid:105) = (cid:90) M W β ( f ) ∧ (cid:63)W β ( g ) , which along with the corresponding norm || · || , determine a Hodge theory for the cochain complex C • ( K ) .This inner product determines a codifferential d ∗ W : C • ( K ) → C •− ( K ) which, as the adjoint of the standarddifferential, satisfies (cid:104) df, g (cid:105) = (cid:104) f, d ∗ W g (cid:105) . The corresponding Whitney Laplacian ∆ W : C • ( K ) → C • ( K ) is thengiven by the standard formula ∆ W = dd ∗ W + d ∗ W d. This inner product was introduced using the standardbarycentric coordinates in [Dod76].This Laplacian decomposes the space C • ( K ) into harmonic, exact, and coexact components: C • ∼ = H • ( M ) ⊕ dC •− ( K ) ⊕ d ∗ W C • +1 ( K ) . This combinatorial Hodge decomposition serves as a good approximationof the L -Hodge decomposition of M , though it does not capture the L -Hodge decomposition exactly. Inparticular, the Whitney coexact chains may not be L -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 barycentricpartition of unity β . Give the cochain complex the Whitney L -norm induced by the Whitney map determinedby β. Likewise, give the chain complex the dual norm || · || ∗ determined by the integration pairing. Thenfor every coexact cochain f ∈ d ∗ W C ( K ) , there is an exact chain a ∈ ∂C ( K ) of unit norm such that || f || = (cid:82) a W β ( f ) . Proof. The cochain Hodge decomposition from the Whitney inner product gives the orthogonal decomposition C ( K ) = H ( M ) ⊕ d ∗ W C ( K ) ⊕ dC ( K ) . Let Z ( K ) = H ( M ) ⊕ dC ( K ) . Identify C ( K ) with C ( K ) ∗ via the integration pairing. The compositionof the inclusion and quotient map determines an isomorphism d ∗ W C ( K ) → C ( K ) /Z ( K ) that allows usto identify these spaces. If Ann assigns to a subspace its annihalator, then there is also an isomorphism ( C ( K ) /Z ( K )) ∗ → Ann ( Z ( K )) . By Stokes’ theorem and dimension counting, Ann ( Z ( K )) = ∂C ( K ) .Thus, the dual of d ∗ W C ( K ) is exactly ∂C ( K ) . The dual norm of an element a ∈ ∂C ( K ) is given by || a || ∗ = sup f ∈ C ( K ) || f || ≤ (cid:90) a W β ( f ) . If f has unit L -norm and f = g + h where g ∈ d ∗ W C ( K ) and h ∈ Z ( K ) , then orthogonality implies || g || ≤ . Whence, || a || ∗ = sup f = g + h ∈ C ( K ) || f || ≤ (cid:90) a W β ( g ) = sup g ∈ d ∗ W C ( K ) || g || ≤ (cid:90) a W β ( g ) . The isometric identification of ( d ∗ W C ( K ) , || · || ) with its double dual therefore implies we can compute thenorm of an element f ∈ d ∗ W C ( K ) via the integration pairing integrating only against chains in ∂C ( K ) : || f || = sup a ∈ ∂C ( K ) || a || ∗ =1 (cid:90) a W β ( f ) . In particular, for any coexact cochain f ∈ d ∗ W C ( K ) , there exists an exact chain a with || a || ∗ = 1 and19 a W β ( f ) = || f || . Proposition 4.2. Let λ denote the first eigenvalue for the Hodge Laplacian acting on coexact -forms andlet λ W denote the first eigenvalue for the Whitney Laplacian acting on coexact 1-cochains associated to adeeply embedded triangulation K . There is a constant G = G ( ε ) such that λ ≤ G vol( M ) λ W . Proof. The main issue here is that a Whitney coexact cochain will not generally map to an L -coexactform. This potentially adds a closed term to the denominator in the Whitney Rayleigh quotient, causing theWhitney Rayleigh quotient smaller than the Riemannian Rayleigh quotient. However, this failure can becontrolled.Let f be a coexact eigen-cochain with eigenvalue λ W . Set ω = W β ( f ) ∈ Ω ( M ) , so that || dω || || ω || = λ W .Let p : Ω ( H n ) → Ω ( H n ) be the orthogonal projection onto coexact forms. Let a ∈ C ( K ) be the unit normexact chain that realizes the norm of f by integration given by Lemma 4.1. Then using that dω = d ( p ( ω )) and the fact a is exact, we obtain || f || = || ω || = (cid:90) a ω = (cid:90) a p ( ω ) . The only way a sequence of unit norm coexact Whitney forms ω n could have L -coexact part that vanishesin the limit is if the lengths of the supports of the corresponding test chains a n that realize the norm go toinfinity. Applying Proposition 3.1 to the chain a above gives || a || G ≤ B (cid:112) vol( M ) || a || ∗ = B (cid:112) vol( M ) . Since the lengths of the edges in the triangulation are bounded, the length of the support of a is bounded.Indeed, we have || ω || = (cid:90) a ω = (cid:90) a p ( ω ) ≤ || p ( ω ) || ∞ BE (cid:112) vol( M ) , where E is the length of the largest edge possible in a deeply embedded triangulation. Apply Proposition3.11 to obtain || ω || = (cid:90) a ω = (cid:90) a p ( ω ) ≤ || p ( ω ) || BCE (cid:112) vol( M ) . Setting G = ( BCE ) and using that ω is a Whitney eigenform, we obtain the result by the following shortcomputation: λ ≤ || dω || || p ( ω ) || ≤ G vol( M ) || dω || || ω || = G vol( M ) λ W . Remark . Note that the above estimate in fact holds for the first positive eigenvalue since the first positiveeigenvalue λ is the minimium of the first eigenvalue of the Laplacian acting on functions and the firsteigenvalue of the Laplacian acting on coexact 1-forms. The first eigenvalue λ f for the Laplacian acting onfunctions automatically satisfies the comparison λ f ≤ λ W , as can be seen by studying the Rayleigh quotientand noticing that the estimate above controlling the projection in the denominator is immaterial in thefunction case.We are now ready to introduce stable commutator length, a thorough reference for which is [Cal09]. For agroup Γ , let Γ (cid:48) denote the commutator subgroup and define the rational commutator subgroup to be Γ (cid:48) Q = Ker (Γ → Γ ab ⊗ Q ) . Note that when Γ is the fundamental group of a manifold, these subgroups correspond to the integrallynullhomologous and rationally nullhomologous loops respectively. The commutator length of an element γ ∈ Γ (cid:48) , denoted cl ( γ ) is the shortest word length of γ with respect to the generating set of all commutators.20he stable commutator length for γ ∈ Γ (cid:48) Q is then defined to be scl ( γ ) = inf m ≥ cl ( γ m ) m . Topologically, stable commutator length corresponds to the stable complexity of a surface bounding anullhomologous curve. In particular, for γ ∈ Γ (cid:48) Q , one has scl ( γ ) = inf (cid:26) χ − ( S )2 m : S with ∂S = γ m and S having no closed components (cid:27) , where for a connected surface S we define χ − ( S ) = max { , − χ ( S ) } , and extend this additively to disconnectedsurfaces. There is another natural complexity measure for loops in Γ (cid:48) Q , the Gersten filling norm. For a loop γ ∈ Γ (cid:48) Q , fill ( γ ) is the infimum of the Gromov norm || A || G m for all singular 2-chains A bounding a 1-cyclerepresenting a singular fundamental class of γ m . A fundamental theorem of Bavard relates the filling norm tothe stable commutator length. Theorem 4.3. ( [Bav91]) For any group element γ , there is an equality: scl ( γ ) = 4 fill ( γ ) . For proof, see for instance Lemma 2.69 in [Cal09]. Remark . Let B (Γ) be the R -vector space of 1-boundaries. Then stable commutator length can be extendedto a psuedo-norm on B (Γ) . After identifying chains with vanishing psuedo-norm, Bavard duality, whichrelates the filling norm to quasimorphisms and their defect norm, becomes a genuine functional analyticduality theorem. One could define the stable isoperimetric ratio in this chain setting, and the results of thispaper would go through for that (smaller) ratio as well.We can now prove the main theorem. Theorem A. Let M be a closed hyperbolic 3-manifold with injectivity radius bound below by ε . There is aconstant A = A ( ε ) that only depends on ε such that for any nontrivial boundary γ ∈ Γ (cid:48) Q , one has √ λ ≤ A vol( M ) | γ | scl ( γ ) , where λ the first coexact eigenvalue of the Hodge Laplacian on Ω ( M ) .Proof. First note that since stable commutator length and geodesic length are both multiplicative underpowers, it suffices to show the claim for an integrally nullhomologous loop γ .Fix a deeply embedded triangulation K of M and denote by λ W the first eigenvalue of the WhitneyLaplacian ∆ W acting on d ∗ W C ( K ) associated to a smooth barycentric partition of unity. Notice thatthe Hodge decomposition ensures that zero is not an eigenvalue of this operator. Let c : S → M be acellular path in the 1-skeleton of K ∗ representing the loop γ , constructed as in Proposition 2.5. Let T be a triangulation of K ∗ . Let a ∈ C ( K ∗ ) be the fundamental cycle for γ corresponding to the path c in C ( K ∗ ) ⊂ C ( T ) ⊂ C sing ( M ) . If Φ : C ( K ) → C ( K ∗ ) is the Poincaré duality map, then Φ − ( a ) is an exact2-cochain. We can therefore choose ω ∈ d ∗ W C ( K ) with dω = Φ − ( a ) . Setting A = Φ( ω ) in C ( K ∗ ) , we have ∂A = a and || A || G = || ω || G . A short computation shows (cid:104) dω, dω (cid:105) = (cid:104) ω, d ∗ W dω (cid:105) = (cid:104) ω, ∆ W ω (cid:105) by coexactness, ≥ λ W (cid:104) ω, ω (cid:105) by Courant-Fischer, = λ W || ω || . 21e can rewrite this as || ω || ≤ || dω || √ λ W . Proposition 4.1 implies || ω || ≤ √ G (cid:112) vol( M ) || dω || √ λ . By Bavard’s theorem relating the filling norm to stable commutator length (Theorem 4.2), our choice of A ,and Proposition 3.4, we find that scl ( γ ) = 4 fill ( γ ) ≤ || τ ( A ) || G ≤ N || A || G = 4 N || ω || G , where, as in Proposition 3.4, τ is the triangulation map relating the cellular chain A to the subdividedsimplicial chain in C ( T ) . Consequently, scl ( γ ) ≤ N || ω || G ≤ N B (cid:112) vol( M ) || ω || by Proposition 3.1, ≤ N B √ G vol( M ) || dω || √ λ by above computation, ≤ N B √ G vol( M ) D || dω || G √ λ by Proposition 3.2, = 4 N B √ G vol( M ) D || ∂A || G √ λ by Proposition 3.3, ≤ N B √ G vol( M ) D || c || G √ λ by construction of ∂A , ≤ N B √ G vol( M ) DL | γ |√ λ by Proposition 2.5, = 4 N B √ GDL vol( M ) | γ |√ λ . Setting A = 4 N B √ GDL and rearranging, we are done. We now turn to proving the lower bound on the first coexact eigenvalue of the 1-form Laplacian that constitutesTheorem B. Unlike Theorem A, we prove this eigenvalue comparison without a dimension constraint. Theline of proof follows that of Theorem 1.3 in [LS18].In [LS18], the authors obtain the following estimate controlling the L -norm of coclosed forms. Note thatthis estimate does not depend on the fundamental domain coming from a deeply embedded triangulation. Proposition 5.1. (Proposition 5.4 in [LS18]) Let η be a 1-form on M and D ⊂ H n any fundamental domain.Then, || η || ≤ Area ( ∂ D ) || η || ∞ (cid:18) π || dη || ∞ + max i (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) γ i η (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + 12 || dη || ∞ || η || (cid:112) vol( M ) , where the γ i are the geodesics in the homotopy class of the loops representing the side pairing transformationsof the fundamental domain D . Studying the terms in the estimate of Proposition 5.1 for a coexact λ -eigenform provides a lower boundon λ given later as Theorem B. The essential idea is that after applying an L - L ∞ norm comparison, all butone summand on the right-hand side (the integral term), has a || dη || term. In particular, if η is a unit normeigenform, the right-hand side almost has a √ λ term in every summand. Our aim, then, is to replace theintegral term with something that looks like || dη || ∞ ( ρ ( M ) − + stuff ) , where the stuff is polynomial in thevolume of M with constants that depend only on the lower bound on injectivity radius.22 emma 5.2. Let a be the lift to H n of the cellular approximation of a geodesic loop and let γ be the (oriented)lift of the same geodesic loop. If ˜ η is the pullback to H n of a 1-form η ∈ Ω ( M ) , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) a ˜ η − (cid:90) γ ˜ η (cid:12)(cid:12)(cid:12)(cid:12) ≤ π || dη || ∞ || a || G . Proof. Let x be the starting point of γ , let a i be the geodesic arcs of a (so that the word corresponding tothe cellular path a is the word a · · · a || a || G ), and let y be the end point of a || a || G . Let Q be the piecewisegeodesic ( || a || G + 3) -gon obtained by taking the union of the triangles convex hull ( a i , x ) and the triangleconvex hull ( γ, y ) . Since Q is the union of ( || a || G + 1) geodesic triangles of area bounded by π , we have theupper bound of ( || a || G + 1) π for the area of Q . Then, since (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) a ˜ η − (cid:90) γ ˜ η (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂Q ˜ η (cid:12)(cid:12)(cid:12)(cid:12) ≤ π || dη || ∞ ( || a || G + 1) , where the first equality follows from the deck transformation invariance of ˜ η , which makes the integral over ∂Q \ ( a ∪ γ ) vanish. If one then considers the cellular paths ma and the geodesic γ m integrated over the form m ˜ η , one gets (cid:12)(cid:12)(cid:12)(cid:82) a ˜ η − (cid:82) γ ˜ η (cid:12)(cid:12)(cid:12) = m (cid:12)(cid:12)(cid:12)(cid:82) ma ˜ η − (cid:82) γ m ˜ η (cid:12)(cid:12)(cid:12) ≤ m π || dη || ∞ ( || ma || G + 1) . Taking the limit as m → ∞ thenresults in (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) a ˜ η − (cid:90) γ ˜ η (cid:12)(cid:12)(cid:12)(cid:12) ≤ π || dη || ∞ || a || G , proving the lemma. Lemma 5.3. There is a constant B = B ( ε ) such that diam( M ) ≤ B vol( M ) .Proof. First note that there is a constant T = T ( ε ) such that the number of simplices in M is bounded by T vol( M ) and that each simplex from a deeply embedded triangulation has bounded diameter, say boundedby C . With B = C T , one has that B vol( M ) bounds the diameter of M , as desired. Lemma 5.4. Let η ∈ Ω ( M ) be a 1-form and γ a rationally nullhomologous loop in M . Then integratingover the geodesic in the free homotopy class satisfies (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) γ η (cid:12)(cid:12)(cid:12)(cid:12) ≤ π || dη || ∞ scl ( γ ) . Proof. This follows from Bavard duality and the fact that (cid:82) γ η , where the integral is over the geodesic in thefree homotopy class of γ , is a quasimorphism with defect bounded by π || dη || ∞ (see [Cal09], pages 21 and39).The key estimate allowing us to replace the integral term with one involving the stable isoperimetricconstant ρ ( M ) is the following proposition; compare with Proposition 5.24 in [LS18]. Proposition 5.5. Let η be a 1-form on M . Then there is a harmonic form h and a constant L = L ( ε ) > such that for every closed geodesic α in M , one has (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α ( η − h ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | α | L vol( M ) / || dη || ∞ (cid:0) ρ ( M ) − + 1 (cid:1) . Proof. If M is a Q -homology sphere, α is rationally nullhomologous and h can only be 0. Lemma 5.4 gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α η (cid:12)(cid:12)(cid:12)(cid:12) ≤ π scl ( α ) || dη || ∞ . By multiplying the right-hand side by | α | / | α | , this becomes (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α η (cid:12)(cid:12)(cid:12)(cid:12) ≤ π | α | scl ( α ) | α | || dη || ∞ ≤ π | α | ρ ( M ) − || dη || ∞ . 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 ≥ that the set of hyperbolic volumes is discrete in R ,see [Bel14]), the claim follows with L = π vol( W ) / .Thus, we assume M has nontrivial real homology classes. Fix a basepoint x ∈ M . Take a basis c , . . . , c n of harmonic 1-chains for C ( K ∗ ) by taking harmonic 2-cochains in C ( K ) and using the Poincaré dualitymap Φ . The Poincaré duality map endows C ( K ∗ ) with the Whitney L -norm and for this norm harmonicchains are norm minimizing in their homology classes. Additionally, note that since Φ preserves the Gromovnorm, the Gromov- L norm comparison of Proposition 3.2 holds for cellular chains in C ( K ∗ ) . Let h be the(unique) harmonic form that satisfies (cid:82) c i η − h = 0 for each i . Let a be a cellular path in K ∗ approximating α , as in Proposition 2.5, so || a || G ≤ L | α | and identify the cellular path a with the chain it represents.Then, a = a h + ∂S where a h is harmonic and S is some rational 2-chain. A short computation shows, || ∂S || G = || a − a h || G ≤ || a || G + || a h || G ≤ B (cid:112) vol( M )( || a || + || a h || ) , by Proposition 3.1, ≤ B (cid:112) vol( M ) || a || , since harmonic chains are L -norm minimizing , ≤ BC (cid:112) vol( M ) || a || G , by Proposition 3.2 . Additionally, since there is a universal upper bound on the length of an edge in K ∗ , there is a constant E > , such that | α | ≤ E len ( c ) for any path cellular path c homotopic to α .Since ∂S is a rational cycle, take N > to be an integer so that N ∂S is integral. Then one can gluetogether oriented copies of the edges on which ∂S is supported along their boundaries to obtain a (nonunique) collection of closed cellular loops b , . . . , b m whose union represents the cycle N ∂S . Fix a vertex v i in each loop b i . Note that by construction, len ( b i ) = || b i || G for each i . Let τ i be the geodesic arc connectingthe basepoint x to v i and τ − i the oppositely oriented geodesic arc. Define the curve b to be the path τ b τ − τ b τ − · · · τ m b m τ − m . Let β be the geodesic loop through x homotopic to b . Notice (cid:80) i || b i || G = || b || G ,where || b || G is meant in the sense of the norm on singular chains, where the τ ± terms cancel. This gives apossibly trivial element of Γ (cid:48) Q whose length is bounded as follows: | β | ≤ | b | = (cid:88) i (2 | τ i | + | b i | ) ≤ M ) m + E || b || G ≤ (2 diam( M ) + E ) || b || G ≤ (2 B vol( M ) + E ) || b || G , where we use that m ≤ || b || G , the diameter bound of Proposition 5.3, along with the remarks in the abovediscussion.Since N || b || G = || ∂S || G ≤ BC (cid:112) vol( M ) || a || G , and || a || G ≤ L | α | , we obtain || b || G N ≤ BCL (cid:112) vol( M ) | α | . As a result, | β | N ≤ BCL (2 B vol( M ) + E ) (cid:112) vol( M ) | α | . 24e compute, (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α η − h (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α − a h η − h (cid:12)(cid:12)(cid:12)(cid:12) since a h is a linear combination of the c i , and (cid:90) c i η − h = 0, ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂S η − h (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:18)(cid:90) α η − h (cid:19) − (cid:18)(cid:90) a η − h (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∂S η − h (cid:12)(cid:12)(cid:12)(cid:12) + π || dη || ∞ || a || G , by Lemma 5.2, = 1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) N∂S η − h (cid:12)(cid:12)(cid:12)(cid:12) + π || dη || ∞ || a || G = 1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) b η − h (cid:12)(cid:12)(cid:12)(cid:12) + π || dη || ∞ || a || G , since b abelianizes to N ∂S , ≤ N (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) β η − h (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) b ( η − h ) − (cid:90) β ( η − h ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) + π || dη || ∞ || a || G ≤ N (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) β η − h (cid:12)(cid:12)(cid:12)(cid:12) + 1 N π || dη || ∞ || b || G + π || dη || ∞ || a || G , by Lemma 5.4 . If β is trivial, then the integral term | (cid:82) β η − h | vanishes, and we can replace that term with BCLπ | α ||| dη || ∞ (cid:112) vol( M ) ρ ( M ) − to obtain (after using our estimate for || b || G /N and || a || G ≤ L | α | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α η − h (cid:12)(cid:12)(cid:12)(cid:12) ≤ BCLπ | α ||| dη || ∞ (cid:112) vol( M ) (cid:0) ρ ( M ) − + 1 (cid:1) + πL || dη || ∞ | α |≤ BCLπ | α ||| dη || ∞ (cid:112) vol( M ) (cid:0) ρ ( M ) − + 1 (cid:1) + vol( M ) / vol( W ) / πL || dη || ∞ || α |≤ BCLπ | α ||| dη || ∞ vol( M ) / vol( W ) (cid:0) ρ ( M ) − + 1 (cid:1) + vol( M ) / vol( W ) / πL || dη || ∞ | α | Setting L = 2 max { πBCL vol( W ) , πL vol( W ) / } and factoring gives the result.Assume now that β is nontrivial. Combining the above estimates yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α η − h (cid:12)(cid:12)(cid:12)(cid:12) ≤ | β | N | β | (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) β η − h (cid:12)(cid:12)(cid:12)(cid:12) + 1 N π || dη || ∞ || b || G + πL || dη || ∞ | α |≤ BCL (2 B vol( M ) + E ) (cid:112) vol( M ) | α | | β | (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) β η − h (cid:12)(cid:12)(cid:12)(cid:12) + π || dη || ∞ BCL (cid:112) vol( M ) | α | + πL || dη || ∞ | α | = 2 BCL (cid:112) vol( M ) | α | (cid:18) (2 B vol( M ) + E ) 1 | β | (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) β η − h (cid:12)(cid:12)(cid:12)(cid:12) + π || dη || ∞ (cid:19) + πL || dη || ∞ | α | . Since the geodesic β is nullhomologous, Lemma 5.4 implies (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) β ( η − h ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ π || dη || ∞ scl ( β ) . Replacing the integral term with this estimate and using that scl ( β ) | β | ≤ ρ ( M ) − gives (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α η − h (cid:12)(cid:12)(cid:12)(cid:12) ≤ | α | BCL (cid:112) vol( M ) || dη || ∞ (cid:0) π ( B vol( M ) + E ) ρ ( M ) − + π (cid:1) + πL || dη || ∞ | α | . Again using the existence of a minimal volume hyperbolic n -manifold, one can replace B with theconstant B = 2 B + E/ vol( W ) since B vol( M ) > B vol( M ) + E . Then, after combining constants in the25rst summand (and using that π > π to pull out the terms containing π ) into a single constant L , oneobtains: (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) α ( η − h ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | α | L vol( M ) / || dη || ∞ (cid:0) ρ ( M ) − + 1 (cid:1) + πL || dη || ∞ | α | . Set L = 2 max { L , πL vol( W ) / } and multiply the second summand by vol( M ) / to obtain the claim. Lemma 5.6. Let M have deeply embedded triangulation K and let ˜ K be the pullback of this triangulation to H n . Then there is a fundamental domain D ⊂ H n for M that is a union of simplices from ˜ K such that thediameter of D satisfies diam( D ) ≤ M ) . Proof. Fix a top dimensional simplex σ ∈ K ( n ) and let ˜ σ be a lifted copy in H n . Let ˜ x be the barycenterof ˜ σ . For every other top dimensional simplex σ in K ( n ) there is a lift ˜ σ whose barycenter ˜ x σ is within diam( M ) of ˜ x . Choose one such lift for every σ in such a way the resulting fundamental domain D isconnected. Then the diameter of the fundamental domain satisfies diam( D ) ≤ diam( M ) + 2 e , where e is themaximum distance from the barycenter of a simplex in a deeply embedded triangulation to its boundary.Clearly e < diam( M ) , so the lemma immediately follows.Now, assume M has a fixed deeply embedded triangulation and let D be a fundamental domain as inLemma 5.6. Let γ i be the geodesics in the free homotopy class of the side pairing transformations of thefundamental domain D , and notice by construction | γ i | ≤ M ) . With this, we modify the estimate inProposition 5.1 to obtain the following. Proposition 5.7. Let η be a coclosed 1-form on M . Let h be the harmonic form of Proposition 5.5 associatedto η . Then for a constant A = A ( ε ) > , the following holds: || η − h || ≤ A vol( M ) || η − h || ∞ (cid:16) π || dη || ∞ + 3 L B vol( M ) / || dη || ∞ (cid:0) ρ ( M ) − + 1 (cid:1)(cid:17) + 12 || dη || ∞ || η − h || (cid:112) vol( M ) . Proof. Let γ i realize the maximum among the integrals (cid:82) γ i η . Substitute the estimate of Proposition 5.5 forthe integral term in Proposition 5.1 applied to the coclosed form η − h and the fundamental domain D toobtain || η − h || ≤ Area ( ∂ D ) || η − h || ∞ (cid:16) π || dη || ∞ + L vol( M ) / || dη || ∞ | γ i | (cid:0) ρ ( M ) − + 1 (cid:1)(cid:17) + 12 || dη || ∞ || η − h || (cid:112) vol( M ) . Then, replace | γ i | with B vol( M ) , using Lemma 5.6 and Lemma 5.3. Lastly, since there is an upperbound on the area of a face of any simplex in G ε (a consequence of the bounds on the dihedral angles), thetotal area of the boundary of a complex made from no more than T vol( M ) simplices from G ε is bounded by A vol( M ) for a constant A depending on ε . Substituting this estimate for the Area ( ∂ D ) term completesthe proof.The last tool needed is an estimate comparing the L and L ∞ -norms. We do this using a Sobolev estimate.To apply this estimate to a form that is the linear combination of eigenforms, one needs an upper bound onthe eigenvalues of the eigenforms. To avoid this, one might hope to use an estimate in the combinatorialsetting of Whitney forms. To do this, one needs to relate an arbitrary smooth form to approximating Whitneyforms. This can be done by assigning to a smooth form ω the cochain f ω defined by f ω ( c ) = (cid:82) c ω , thentake the form ¯ ω = W β ( f ω ( c )) . Dodziuk observed that as one takes finer and finer triangulations, this mapconverges to the identity in a suitable norm. However, without a bound on the derivatives of the smoothforms, one could choose forms whose L -norm is large, but with support in the complement of the 1-skeletonof the triangulation, making ¯ ω = 0 . For this reason, we stick to the following estimate from [LS18], specializedto the case of coexact eigenforms. Proposition 5.8. (Proposition 2.2 of [LS18]) Let M be a closed hyperbolic n-manifold with inj( M ) > ε .Assume the first positive eigenvalue λ of the Laplacian acting on coexact 1-forms is less than some fixed onstant H > . Then there is a constant C ( H, ε ) > such that for a coexact λ -eigenform ω , one has || ω || ∞ ≤ C ( H, ε ) || ω || . Proposition 5.9. Let M be a closed hyperbolic n-manifold with inj( M ) > ε . Let λ < H be the first positiveeigenvalue for the Hodge Laplacian acting on coexact 1-cochains. Then the following holds: √ λ ≤ A vol( M ) C ( H, ε ) (cid:16) π + 3 L B vol( M ) / (cid:0) ρ ( M ) − + 1 (cid:1)(cid:17) + C ( H, ε )2 (cid:112) vol( M ) . Proof. Let η be a λ coexact eigenform. Applying the Sobolev type estimate of Proposition 5.8 to each instanceof the sup norm in Proposition 5.7 and using that || dη || = √ λ || η || ≤ √ λ || η − h || , where the inequalityfollows from the orthogonality of the Hodge decomposition, gives || η − h || ≤ A vol( M ) C ( H, ε ) √ λ || η − h || (cid:16) π + 3 L B vol( M ) / (cid:0) ρ ( M ) − + 1 (cid:1)(cid:17) + C ( H, ε )2 √ λ || η − h || (cid:112) vol( M ) . Dividing both sides by √ λ || η − h || then gives √ λ ≤ A vol( M ) C ( H, ε ) (cid:16) π + 3 L B vol( M ) / (cid:0) ρ ( M ) − + 1 (cid:1)(cid:17) + C ( H, ε )2 (cid:112) vol( M ) . Rearanging the terms and combining constants (which again requires the existence of a minimal volumehyperbolic n -manifold) in the previous proposition and applying a geometric estimate of Calegari and asystolic inequality due to Sabourau leads to the main theorem of this section. Theorem B. Let M be a closed hyperbolic n -manifold with inj( M ) > ε . Let λ be the first positive eigenvaluefor the Laplacian acting on coexact 1-forms and let H > λ . Then there is a constant P ( H, ε ) > such that P ρ ( M )vol( M ) / /n ≤ √ λ. Proof. First we rearrange the previous proposition and combine constants into one constant P to get theestimate P ρ ( M )(1 + ρ ( M )) vol( M ) / ≤ √ λ. We need an estimate of Calegari’s (see the proof of Theorem 3.9 in [Cal09], in particular, the estimate atthe bottom of page 58) which gives that for a genus g surface S with boundary ∂S that wraps around anullhomologous geodesic γ n -times, one has n | γ | g − ≤ µ + 2 π µ + 2 | γ | , where µ depends on the n -dimensional Margulis constant. Since χ − ( S ) = 2 g − , we get n | γ | χ − ( S ) ≤ (cid:18) µ + 2 π µ + 2 | γ | (cid:19) . Since this is true for any surface S bounding a power of γ , we obtain | γ | scl ( γ ) ≤ (cid:18) µ + 2 π µ + 2 | γ | (cid:19) . We also have the commutator systolic inequality of Sabourau from Theorem 1.4 in [Sab17], which bounds the27hortest nontrivial integrally nullhomologous loop γ ∈ Γ (cid:48) by | γ | ≤ c vol( M ) /n , for a dimensional constant c .Both the inequality of Calegari and the systolic inequality involve a dimensional constant; let µ be themaximum of these constants in dimension n and write Calegari’s inequality as | γ | scl ( γ ) ≤ µ (1 + | γ | ) . Then weget ρ ( M ) ≤ | γ | scl ( γ ) ≤ µ (1 + | γ | ) ≤ µ (1 + µ vol( M ) /n ) . Inserting this upper bound into the denominator of the above rearranged estimate above gives P ρ ( M )(1 + µ (1 + µ vol( M ) /n ) vol( M ) / ≤ P ρ ( M )(1 + ρ ( M )) vol( M ) / ≤ √ λ. We can then increase P to allow us to pull out the volume term and absorb µ , thereby obtaining the desiredestimate. The aim of this section is to show that the first positive eigenvalue of the 1-form Laplacian can vanishexponentially fast. This contrasts the behaviour of the first positive eigenvalue of the Laplacian on functions,whose rate of vanishing is controlled by the Cheeger-Buser estimates.Our construction is similar to that in [BD17]. Essentially, we choose a hyperbolic 3-manifold with totallygeodesic boundary and glue 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 simplemodel family. Using this model family, we show that one can find curves with uniformly bounded lengthwhose stable commutator length grows exponentially in the volume. We then use the spectral gap upperbound in Theorem A to show the first positive eigenvalue vanishes exponentially fast. An essential tool inthis section is the theory of branched surfaces which we quickly review.A branched surface in a 3-manifold M is an embedded finite, smooth 2-complex obtained from a finitecollection of smooth surfaces by identifying compact subsurfaces. A branched surface B is a smooth manifoldaway from its branch locus L , and the components of B \ L are called the sectors of B . The branched surface B has a normal fibered neighborhood N ( B ) ⊂ M , which admits a vertical foliation.A surface S in M is said to be carried by the branched surface B if there is a surface S (cid:48) embedded in N ( B ) transverse to the fibers that is isotopic to S . A surface carried by a branched surface is encoded byassigning integer weights to the sectors of B corresponding to the number of components of S (cid:48) intersectingthe fibers of a given sector. A set of weights determines a surface if it satisfies the branching equations. Ataut surface is a χ − -minimizing surface all of whose components are essential and a branched surface thatcarries only taut surfaces is called a taut branched surface. There is also a one dimensional analogue of abranched surface, called a train track. The components of a train track minus its branch locus are callededges rather than sectors. The theory of branched surfaces extends to 3-manifolds with boundary. One allowsthe branched surfaces to intersect the boundary in a train track; For such a branched surface B , let ∂B denote the train track B ∩ ∂M .If W + ( B ) and W + ( ∂B ) are the spaces of nonnegative weights for the branched surface B and train track ∂B respectively, one has a boundary projection map p ∂ : W + ( B ) → W + ( ∂B ) given by weighting each edgeof ∂B by the corresponding sector weight.Floyd and Oertel in [FO84] (Theorem 1) prove that if M is an irreducible, orientable, boundary-irreducible3-manifold, there exist finitely many branched surfaces that carry with positive weights all incompressible,boundary incompressible surfaces in M . Moreover, every surface carried with positive weights by thesebranched surfaces is incompressible. Such branched surfaces are called essential. Proposition 6.1. Let M be a compact oriented hyperbolic 3-manifold with totally geodesic boundary ∂M .Let γ be a geodesic 1-boundary that that embeds in ∂M . Let B = { B , . . . , B n } be a collection of essential ranched surfaces carrying all of incompressible surfaces in M . Then there is a constant D > , dependingon the collection of essential branched surfaces B , such that | γ | ≤ D scl ( γ ) .Proof. Denote by M A the double of M along an annular neighborhood A of γ in ∂M ; let ∂ : H ( M A ) → H ( A ) be the boundary map from Mayer-Vietoris, if ∂ [ S ] = a , say S homologically bounds a . Write M A = M + A ∪ M − A ,where M is identified with M + A under the inclusion. Set V = ∂ − ([ γ ]) . The proof of Proposition 4.4 in [Cal09]implies that scl ( γ ) = 14 inf [Σ] ∈ V || [Σ] || T h . For any class in V , we can find a rational class arbitrarily close to it, and then by clearing denominatorsobtain an integral class that homologically bounds a multiple of γ . Let Σ be a taut surface representing sucha class [Σ] ∈ V . Assume Σ homologically bounds m [ γ ] . Let Σ + = Σ ∩ M + . Then, χ − (Σ + ) ≤ || [Σ] || T h . Remark . While Σ is taut in M , it may happen that Σ + is not taut in M , since a taut surface could be asimpler surface representing the same class but with different boundary. The point of doubling along theannular region is to ensure this does not happen.We now compare χ − (Σ + ) to the length of γ . Let W + ( B ) be the positive weight space for B . Let V + be the set of weights w ∈ W + ( B ) corresponding to surfaces that represent classes in V ∩ H ( M ; ∂M ) . Themaps x i : W + ( B i ) → R taking a weight vector w to χ − of the surface with corresponding weight vector w isa rational linear map represented by the row vector − ( χ ( σ ) − c ( σ )4 ) , where σ is a sector of B i . The sector σ is a manifold with corners, and c ( σ ) denotes the number of corners (see [Cal09] page 93). Since a weightvector represents a surface if it satisfies the branching equations, whose solution set is a closed subset S + ( B i ) of W + ( B i ) , the set S +1 ( B i ) = S + ( B i ) ∩ { w ∈ W + ( B i ) : || w || = 1 } is compact. The maps x i is positive for all rational points in S +1 ( B i ) , because some multiple of any suchpoint represents a surface of nonzero Euler characteristic since the branching equations are homogeneous andthe manifold M has no essential tori. It therefore follows that on S +1 ( B i ) , the map x i is positive. As a result,if S has weight vector w , we have the comparison c i || w || ≤ χ − ( S ) , where, c i = min w ∈ S +1 ( B i ) x i ( w ) . To obtainthis comparison for all branched surfaces B i , set c = min { c i } . Next, let (cid:96) be the length of the longest edgeamong all the train tracks ∂B i . Since ∂S is carried by the train track ∂B with projected weight p ∂ ( w ) , weclearly have | ∂S | ≤ (cid:96) || p ∂ ( w ) || ≤ (cid:96) || w || . Set D = 4 (cid:96)c − and apply the inequalities above to obtain | ∂S | ≤ (cid:96) || w || ≤ Dχ − ( S ) . This then implies that for every rational class [Σ] ∈ V , | γ | ≤ D χ − (Σ + )2 m ≤ D || [Σ] || T h . Taking the infimum among all such classes [Σ] gives the claim.For any compact Riemannian manifold M one can define the stable norm on the first homology of M .For a class a ∈ H ( M ) , the stable norm of a is given by || a || s,M = inf (cid:26) | α | n : for α ∈ π M with [ α ] = na (cid:27) . Corollary. Let M be a compact oriented hyperbolic 3-manifold with totally geodesic boundary ∂M . Let γ bea geodesic 1-boundary embedded in ∂M . Then || [ γ ] || s,∂M ≤ D scl ( γ ) . Theorem C. There is a family W n of closed hyperbolic 3-manifolds with injectivity radius bounded below bysome ε > and volume growing linearly in n such that the 1-form Laplacian spectral gap vanishes exponentiallyfast in relation to volume: (cid:112) λ ( W n ) ≤ B vol( W n ) e − r vol( W n ) , where r and B are positive positive constants and λ ( W n ) is the first positive eigenvalue of the 1-form Laplacianon W n .Proof. Let W be Thurston’s tripus manifold (see [Thu97]), a compact hyperbolic 3-manifold with totallygeodesic boundary a genus 2 surface for which the inclusion map H ( ∂W ; Z ) → H ( W ; Z ) is onto. Thehomology of the boundary ∂W decomposes as the direct sum of rank 2 submodules U and V , where V ⊂ H ( ∂W ) is the image of the boundary map ∂ : H ( W, ∂W ; Z ) → H ( ∂W ; Z ) (which is also the kernel ofthe inclusion H ( ∂W ) → H ( W ) ) and U is a compliment of V (note that the inclusion map H ( ∂W ) → H ( W ) restricted to U is an isomorphism). Let S be a genus 2 surface, which we will use to mark the boundaries of W + and W − . Assume H ( S ; Z ) is generated by e , e , e , e . Choose a marking S → ∂W + so in W + onehas U = (cid:104) e , e (cid:105) and V = (cid:104) e , e (cid:105) . Similarly, choose a marking S → ∂W − so that in W − one has V = (cid:104) e , e (cid:105) and U = (cid:104) e , e (cid:105) . We then define W n = W + ∪ f n W − where f : S → S is a pseudo-Anosov that acts on H ( S ) by the symplectic matrix F = − 10 0 − (for existence of such a pseudo-Anosov mapping class, see the proof Lemma 7.1 in [BD17]). This matrixpreserves the subspace decomposition above, and so ensures that every curve in ∂W ± bounds on exactly oneside in W n (and since the boundary of the tripus carries its homology, each W n is an integer homology sphere).Moreover, it acts as an Anosov matrix on U and V . This ensures the standard Euclidean (cid:96) -norm || F n ( a ) || E of an element a ∈ H ( S ) grows exponentially in n (indeed, for our choice of F , it grows like ( √ ) n ). Sincenorms on finite dimensional real vector spaces are comparable, there is a constant comparing the stable normto the standard Euclidean (cid:96) -norm on H ( S ) . This means that the shortest curve in S , for any metric on S ,homologous to F n ( a ) has length growing exponentially in n .The hyperbolic manifolds W n admit a K -biLipschitz (where K does not depend on n ) diffeomorphism µ from a model manifold M n which is a degree n cyclic cover of the mapping torus M f cut open along afiber with W + and W − glued to the two boundary components with the corresponding orientations (thisis a consequence of [BMNS16]). This decomposes W n into three pieces, a product region S × [0 , n ] andthe ends W + and W − in a metrically controlled way. It will be convenient to set M + = W + ⊂ M n and M − = W − ⊂ M n when talking about the ends 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 . We denote the geodesic length ofa curve in W n by | · | W n and likewise let | · | M n denote the geodesic length of the curve in the model manifold.To keep track of in which manifold we are computing stable commutator length, let scl ( · , M ) denote thestable commutator length in a manifold M .That the W n have injectivity radius bounded below and volume growing linearly can be seen by theargument of Lemma 7.3 in [BD17], which uses the results of [BMNS16]. In particular, since ∂W ± areincompressible and W ± are acylindrical, the only restriction on the map f that ensures the W n have thedesired properties is that it be pseudo-Anosov. Remark . Note though that if the tripus manifold were replaced by a handlebody (as in [BD15]), one wouldneed to carefully control how the associated laminations of f interact with the disk set.30igure 1: A schematic picture of the model manifold and a surface bounding γ Remark . Using the model manifold, one can easily estimate the Cheeger constant of W n , which will decaylike /n .Fix now some n and consider the model manifold M n . Take γ in ∂M + ⊂ M n to be an embedded geodesicrepresenting the class e and let η = f n ( γ ) ⊂ ∂M − ⊂ M n . Note that η and γ are isotopic in M n .Let Σ m be a surface bounding η m in W n that realizes the commutator length of γ m and which intersects ∂M − transversely and minimally (the curves in the intersection will be essential in both ∂M − and Σ m ).Observe that since [ γ ] (cid:54) = 0 in W + ∪ S × [0 , n ] , Σ m must meet ∂W − . In case Σ m passes back into the productregion of W + , consider the decomposition of Σ m into Σ + m = Σ m ∩ M + and Σ − m = Σ ∩ M − . The boundary of Σ − m is a multicurve representing a class in V that is homologous to η m . Since χ − (Σ − m ) ≤ χ − (Σ m ) , we can usethe sequence of surfaces Σ − m to obtain a lower bound on the stable commutator length of γ in π M n . Thecorollary to Proposition 4.1 implies that || [ η m ] || s,∂W − = m || [ η ] || s,∂W − ≤ Dχ − (Σ − m ) , where || · || s,∂W − is the stable norm on homology for H ( ∂W − ) and D is the constant in Proposition 4.1.31aking the infimum over all such m we obtain: || [ η ] || s,∂W − ≤ D inf m χ − (Σ − m ) m ≤ D inf m χ − (Σ m ) m = scl ( γ, M n )= scl ( γ, W n ) . Our choice of f and the definition of η cause || [ η m ] || s,M n to grow exponentially in n . In particular, for someconstant B > , and with r = √ , Be rn ≤ scl ( γ, W n ) . 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