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

Francesco Lin, Michael Lipnowski

Abstract

We discuss a construction of families of hyperbolic rational homology spheres with coexact $1$-form spectral gap uniformly bounded below which is well-suited for explicit computations. Using this, we provide several disjoint intervals containing a limit point of such spectral gaps, the rightmost of which is $[0.8196,0.8277]$. Furthermore, we also exhibit a family of arithmetic examples, answering a question of Abdurrahman-Adve-Giri-Lowe-Zung.

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

Abstract

We discuss a construction of families of hyperbolic rational homology spheres with coexact -form spectral gap uniformly bounded below which is well-suited for explicit computations. Using this, we provide several disjoint intervals containing a limit point of such spectral gaps, the rightmost of which is . Furthermore, we also exhibit a family of arithmetic examples, answering a question of Abdurrahman-Adve-Giri-Lowe-Zung.
Paper Structure (3 sections, 5 theorems, 27 equations, 3 figures, 1 table)

This paper contains 3 sections, 5 theorems, 27 equations, 3 figures, 1 table.

Table of Contents

  1. Mapping tori and special involutions
  2. Uniform spectral gaps
  3. Explicit intervals

Key Result

Theorem 1

There exists an infinite family $\{Y_n\}$ of hyperbolic rational homology spheres such that $\lim_{n\rightarrow\infty}\lambda_1^*(Y_n)\in [0.8196,0.8277]$.

Figures (3)

  • Figure 1: A standard set of curves on a genus $2$ surface such that the corresponding Dehn twists generate the mapping class group FM.
  • Figure 2: An $I$-invariant collection of curves on a genus three surface; removing $x$, we obtain (after isotopy) the Lickorish generators.
  • Figure 3: The plot $J_{\psi}$ (with abscissa $\sqrt{\lambda^*}$) for a twisting parameter $\psi=e^{2\pi i\cdot 0.3}\in U(1)$, which is close to the one realizing the actual value of $\delta(\varphi)$. The very localized peak corresponds to (the square root) of an eigenvalue of the twisted Hodge Laplacian $\Delta_\psi$ of multiplicity two.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Conjecture
  • Definition 1.1
  • Lemma 1.1
  • Proposition 2.1
  • proof : Proof of Theorem \ref{['thm2']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 1 more