Jakobson–Naud’s essential spectral-gap conjecture
Let $X = \Gamma \backslash \mathbb{H}^2$ be a hyperbolic surface of infinite area, where $\Gamma \subset \operatorname{PSL}_2(\mathbb{R})$ is a discrete, torsion-free, non-elementary convex co-compact group, and let $\delta \in (0,1)$ be the Hausdorff dimension of its limit set. Let $\Delta_X$ be the nonnegative Laplacian. The resolvent $(\Delta_X - s(1-s))^{-1}$, defined on $L^2(X)$ for $\operatorname{Re} s > 1/2$ apart from finitely many poles, continues meromorphically to $s \in \mathbb{C}$ as an operator $C_c^\infty(X) \to C^\infty(X)$; let $\mathcal{R}_X$ be the set of its poles, the resonances of $X$, and let $$G_X = \inf\{a \in \mathbb{R} : \#\{s \in \mathcal{R}_X : \operatorname{Re} s \ge a\} < \infty\}.$$ By the theory of Lax and Phillips, $G_X \le 1/2$ when $\delta > 1/2$, and Naud proved $G_X < \delta$ when $\delta \le 1/2$. Bourgain and Dyatlov proved that every such surface has an essential spectral gap, $G_X < 1/2$, with no condition on $\delta$. Jakobson and Naud proved $G_X \ge \delta(1 - 2\delta)/2$, and $G_X \ge \delta/2 - 1/4$ when $\delta > 1/2$ and $\Gamma$ is a subgroup of an arithmetic group. For surfaces of finite area, where $\delta = 1$, a result of Selberg gives the corresponding value $1/2$.
Problem. Is $G_X = \delta/2$ for every such surface? Equivalently, for every $\varepsilon > 0$, are there only finitely many resonances with $\operatorname{Re} s > \delta/2 + \varepsilon$ and infinitely many with $\operatorname{Re} s > \delta/2 - \varepsilon$?
Source: Jakobson and Naud (2012), Conjecture 1.1 (arXiv:1011.6264, 2010).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- DG, DS
- Keywords
- AIM Open Applied Problems, hyperbolic surfaces, scattering resonances, spectral gaps
References
@article{jakobson2012,
author = {Jakobson, Dmitry and Naud, Fr{\'e}d{\'e}ric},
title = {On the critical line of convex co-compact hyperbolic surfaces},
journal = {Geom. Funct. Anal.},
volume = {22},
number = {2},
year = {2012},
pages = {352--368},
doi = {10.1007/s00039-012-0154-y},
eprint = {1011.6264},
archivePrefix = {arXiv},
}
@article{naud2005,
author = {Naud, Fr{\'e}d{\'e}ric},
title = {Expanding maps on {C}antor sets and analytic continuation of zeta functions},
journal = {Ann. Sci. {\'E}cole Norm. Sup. (4)},
volume = {38},
number = {1},
year = {2005},
pages = {116--153},
doi = {10.1016/j.ansens.2004.11.002},
}
@article{bourgain2018,
author = {Bourgain, Jean and Dyatlov, Semyon},
title = {Spectral gaps without the pressure condition},
journal = {Ann. of Math. (2)},
volume = {187},
number = {3},
year = {2018},
pages = {825--867},
doi = {10.4007/annals.2018.187.3.5},
eprint = {1612.09040},
archivePrefix = {arXiv},
}
@misc{aim-jakobson-naud-essential-gap,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Jakobson–Naud’s essential spectral-gap conjecture}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/115-jakobson-naud-essential-gap.md},
note = {Entry 115 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-00356, "Jakobson–Naud’s essential spectral-gap conjecture", clopen.solutions.