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

TL;DR

This paper constructs hyperbolic rational homology spheres with uniformly bounded coexact 1-form spectral gaps, identifies limit points within specific intervals, and provides arithmetic examples answering a prior open question.

Contribution

It introduces explicit constructions of hyperbolic rational homology spheres with controlled spectral gaps and identifies their limit points, advancing understanding of spectral geometry in these manifolds.

Findings

Spectral gaps are uniformly bounded below in constructed families.

Identified disjoint intervals containing limit points of spectral gaps.

Provided arithmetic examples answering an open question.

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.