Sunada’s no-gap conjecture for abelian covers
Let $X$ be a closed connected orientable surface of genus $g \ge 2$ with a metric of constant curvature $-1$, let $\widetilde X$ be its universal cover and let $$Y = \widetilde X / [\pi_1(X), \pi_1(X)]$$ with the lifted metric. This is the maximal abelian covering of $X$: the deck group of $Y \to X$ is $H_1(X; \mathbb{Z}) \cong \mathbb{Z}^{2g}$. Let $\Delta_Y$ be the Laplace–Beltrami operator of $Y$, self-adjoint on $L^2(Y)$. For a Riemannian covering with amenable deck group the spectrum of the Laplacian of the base is contained in that of the cover (Sunada; see Kuchment), so among the abelian coverings of $X$ the maximal one has the largest spectrum. The analogous no-gap statement for maximal abelian coverings of regular graphs was proved by Higuchi and Shirai for every even degree.
Problem. Prove or disprove that $$\sigma(-\Delta_Y) = [0, \infty).$$
Source: Sunada (1990; 2008), as recorded by Kuchment (2016), Conjecture 6.4, for maximal abelian coverings of compact manifolds of constant negative curvature. Stated for closed hyperbolic surfaces by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DG
- Keywords
- abelian coverings, AIM Open Applied Problems, hyperbolic surfaces, spectral gaps
References
@article{sunada1990,
author = {Sunada, Toshikazu},
title = {A periodic {S}chr{\"o}dinger operator on an abelian cover},
journal = {J. Fac. Sci. Univ. Tokyo Sect. IA Math.},
volume = {37},
number = {3},
year = {1990},
pages = {575--583},
}
@incollection{sunada2008,
author = {Sunada, Toshikazu},
title = {Discrete geometric analysis},
booktitle = {Analysis on Graphs and Its Applications},
series = {Proc. Sympos. Pure Math.},
volume = {77},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2008},
pages = {51--83},
doi = {10.1090/pspum/077/2459864},
}
@article{kuchment2016,
author = {Kuchment, Peter},
title = {An overview of periodic elliptic operators},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {53},
number = {3},
year = {2016},
pages = {343--414},
doi = {10.1090/bull/1528},
eprint = {1510.00971},
archivePrefix = {arXiv},
}
@incollection{higuchi2004,
author = {Higuchi, Yusuke and Shirai, Tomoyuki},
title = {Some spectral and geometric properties for infinite graphs},
booktitle = {Discrete Geometric Analysis},
series = {Contemp. Math.},
volume = {347},
publisher = {Amer. Math. Soc.},
address = {Providence, RI},
year = {2004},
pages = {29--56},
doi = {10.1090/conm/347/06265},
}
@misc{aim-sunada-no-gap-cover,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Sunada’s no-gap conjecture for abelian covers}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/096-sunada-no-gap-cover.md},
note = {Entry 096 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00393, "Sunada’s no-gap conjecture for abelian covers", clopen.solutions.