Jakobson–Naud essential spectral gap conjecture

Establish that every convex co-compact hyperbolic surface has an essential spectral gap of size \(\beta=\tfrac12-\tfrac12\delta\), where \(\delta\) is the exponent of convergence of its Poincaré series.

Background

For a convex co-compact hyperbolic surface $M=\Gamma\backslash\mathbb H^2$, the essential spectral gap is defined through the Selberg zeta function: a gap of size $\beta$ means that $Z_M(s)$ has only finitely many zeros in the half-plane $\{\operatorname{Re}s>\tfrac12-\beta\}$. The parameter $\delta$ denotes the exponent of convergence of the Poincaré series of $\Gamma$.

The paper proves a new universal spectral-gap bound, including an essential spectral gap of any size $\beta<\tfrac1{64}$ when $\delta=\tfrac12$. The conjectured value $\beta=\tfrac12-\tfrac12\delta$ is presented as a stronger general target that remains unresolved by the results established in the paper.