Coexact $1$-form spectral gaps of hyperbolic rational homology spheres

1. Problem Statement and Context

The paper addresses the spectral geometry of hyperbolic 3-manifolds, specifically focusing on the coexact 1-form spectrum.

• The Gap ($\lambda^*_1$): Unlike the classical spectral gap for functions ($\lambda_1$), the bottom of the $L^2$-spectrum for coexact 1-forms on hyperbolic space $\mathbb{H}^3$ is 0. The first non-zero eigenvalue, $\lambda^*_1$, is a crucial invariant.
• Motivation:
  • Torsion Homology: $\lambda^*_1$ is central to the Bergeron-Venkatesh conjecture regarding the growth of torsion homology in congruence covers.
  • Floer Theory: There is a known relationship where if $\lambda^*_1(Y) > 2$, the manifold $Y$ is an $L$-space (trivial reduced monopole Floer homology).
  • Open Question: While infinite families of hyperbolic rational homology spheres (RHS) with a uniform lower bound on $\lambda^*_1$ were previously constructed (AAG+24), the specific limit points of these gaps were not explicitly located. Furthermore, it was unknown if arithmetic families with such gaps exist.
• Goal: To construct explicit infinite families of hyperbolic RHS with uniformly bounded $\lambda^*_1$, explicitly compute the limit points of these gaps, and determine if arithmetic examples exist.

2. Methodology

A. Geometric Construction: Mapping Tori with Special Involutions

The authors construct their families using a specific geometric setup:

1. Base Surface: Start with a closed orientable surface $\Sigma$ of genus $g \geq 2$.
2. Mapping Torus: Consider a mapping torus $M_\phi$ defined by a pseudo-Anosov diffeomorphism $\phi$.
3. Special Involution ($\iota$): They introduce an orientation-reversing, fixed-point-free involution $I$ on $\Sigma$ such that $I \circ \phi \circ I^{-1} = \phi^{-1}$. This induces an involution $\iota(t, x) = (-t, I(x))$ on $M_\phi$.
4. Quotient Manifold: The manifolds in the family are defined as $Y_{\phi, n} = M_{\phi, n} / \iota$, where $M_{\phi, n}$ is the $n$-fold cyclic cover of $M_\phi$.
   • Because $\iota$ reverses the cohomology class $[dt]$, the quotient $Y_{\phi, n}$ has $b_1 = 0$, making it a rational homology sphere.
   • The construction relies on "reverse palindromic" words in the mapping class group (generated by Dehn twists) to ensure the condition $I \circ \phi \circ I^{-1} = \phi^{-1}$ holds.

B. Spectral Analysis via Twisted Hodge Laplacians

To analyze the spectrum of the quotient manifolds $Y_{\phi, n}$, the authors relate them to the base mapping torus $M_\phi$:

• Dihedral Symmetry: The cyclic covers $M_{\phi, n}$ admit an action by the dihedral group $D_n$. The spectrum of $Y_{\phi, n}$ corresponds to the $s$-invariant part of the spectrum of $M_{\phi, n}$.
• Twisted Laplacians: The authors utilize a twisting homomorphism $\psi: \pi_1(M_\phi) \to U(1)$. They study the twisted Hodge Laplacian $\Delta_\psi$ acting on coexact 1-forms.
• Key Proposition: If the eigenvalues of $\phi^*$ on $H_1(\Sigma)$ have absolute values $\neq 1$, then the spectral gap $\delta(\phi)$ of $\Delta_\psi$ on $M_\phi$ (minimized over all $\psi \in U(1)$) is strictly positive.
• Convergence: They prove that $\lim_{n \to \infty} \lambda^*_1(Y_{\phi, n}) = \delta(\phi)$. Thus, the limit of the gaps in the family is determined by the twisted spectral gap of the single base manifold $M_\phi$.

C. Explicit Computation via Selberg Trace Formula

To compute $\delta(\phi)$ explicitly, the authors employ the Selberg trace formula for twisted coexact 1-forms:
$$\frac{1}{2} \sum H(\sqrt{\lambda^*_{\psi, j}}) = \frac{\text{vol}(M_\phi)}{2\pi} (H(0) - H''(0)) + \sum_{\gamma} \frac{\ell(\gamma_0) \cosh(\text{hol}(\gamma)) \cos(\psi(\gamma))}{|1 - e^{\ell(\gamma)}||1 - e^{-\ell(\gamma)}|} H(\ell(\gamma))$$

• Fourier Optimization: They adapt the Fourier optimization techniques (from [BS07]) to construct test functions $J_\psi$.
• Lower Bounds: By analyzing the length spectrum of $M_\phi$ (computed via SnapPy and Twister) up to a cutoff $R$, they derive explicit intervals where eigenvalues cannot exist, thereby establishing lower bounds for $\delta(\phi)$.
• Amphicheirality: For their primary example, they utilize the fact that $M_\phi$ is amphicheiral (admits an orientation-reversing isometry), which forces eigenvalues to have even multiplicity, sharpening the bounds.

3. Key Contributions and Results

Theorem 1: Explicit Limit Points

The authors construct an infinite family $\{Y_n\}$ of hyperbolic rational homology spheres such that the sequence of spectral gaps converges to a specific limit point:
$$\lim_{n \to \infty} \lambda^*_1(Y_n) \in [0.8196, 0.8277]$$

• This is the largest limit point found among their examples.
• They provide three additional disjoint intervals containing limit points for other families (ranging from $\approx 0.17$ to $\approx 0.71$), demonstrating that limit points are not unique.

Theorem 2: Existence of Arithmetic Families

The paper answers a question posed by Abdurrahman-Adve-Giri-Lowe-Zung (AAG+24) regarding arithmeticity:

• There exists an infinite family of arithmetic hyperbolic rational homology spheres with a uniform lower bound on $\lambda^*_1$.
• These manifolds are commensurable but not congruence covers (a necessary condition for uniform gaps in arithmetic families).
• The specific example is derived from the mapping class $\phi = CeBCdcbCDaC$ in genus 2, corresponding to the manifold $s861(3,1)$ in the Hodgson-Weeks census.

Conjecture on Large Gaps

Based on the fact that their construction yields significant uniform gaps, the authors conjecture:

• Conjecture: There exist infinitely many hyperbolic rational homology spheres with $\lambda^*_1 > 2$.
• If true, this would imply the existence of hyperbolic RHS that are $L$-spaces, challenging the intuition that hyperbolic manifolds are rarely $L$-spaces (though the Ozsváth-Szabó conjecture suggests integral homology spheres are never $L$-spaces).

4. Significance

1. Explicitness: Unlike previous existence proofs, this work provides explicit constructions and numerical intervals for the limit of spectral gaps. It bridges the gap between abstract spectral theory and concrete computational geometry.
2. Arithmeticity: It resolves the open question of whether arithmetic families can exhibit uniform spectral gaps for coexact 1-forms, expanding the known landscape of arithmetic manifolds.
3. Methodological Advance: The paper demonstrates a powerful synergy between:
   • Geometric group theory (mapping class groups, Dehn twists).
   • Hyperbolic geometry (mapping tori, involutions).
   • Spectral theory (Selberg trace formula, twisted Laplacians).
   • Computational topology (SnapPy, Twister).
4. Floer Theory Implications: By identifying families with large $\lambda^*_1$, the work provides new candidates for testing the relationship between spectral gaps and Floer homological invariants ($L$-spaces), potentially leading to new counterexamples or confirmations of deep conjectures in low-dimensional topology.

In summary, Lin and Lipnowski provide a robust framework for generating and analyzing hyperbolic rational homology spheres with controlled spectral properties, successfully locating specific limit points for the coexact 1-form gap and proving the existence of arithmetic examples.