Random model for 3-manifolds yielding a uniform coexact 1-form spectral gap

Develop a probabilistic model for constructing hyperbolic 3-manifolds that, with high probability, produces manifolds having a uniform positive lower bound on the smallest eigenvalue of the Laplacian on coclosed 1-forms (i.e., a uniform spectral gap on coexact 1-forms).

Background

The paper investigates spectral gaps for the Laplacian on coclosed 1-forms in hyperbolic 3-manifolds and constructs deterministic sequences exhibiting a uniform gap. In contrast to graphs and surfaces, where random models often yield expanders (function spectral gaps) with high probability, the authors note the absence of an analogous successful random model in the 3-manifold setting for coexact 1-forms.

They discuss prominent random constructions: the Dunfield–Thurston model of random Heegaard splittings and the Petri–Raimbault model built from randomly glued truncated tetrahedra. These models fail to produce uniform coexact 1-form gaps due to features like short injectivity radius or the presence of nontrivial rational homology. Thus, an appropriate random model that typically guarantees a uniform coexact 1-form spectral gap remains to be developed.