Existence of Selberg surfaces of arbitrarily large genus

Determine whether there exists a sequence of closed hyperbolic surfaces X_i with genera g_i tending to infinity such that the Laplacian spectral gap satisfies λ1(X_i) ≥ 1/4 for every surface in the sequence (i.e., whether Selberg surfaces exist in arbitrarily large genus).

Background

The paper studies spectral gaps on random hyperbolic surfaces and discusses the asymptotically optimal gap 1/4 as genus grows. While sequences approaching 1/4 are known (resolving a conjecture of Buser), the existence of surfaces with spectral gap at least 1/4 in arbitrarily large genus—the analogue of Ramanujan graphs for surfaces—remains unresolved and is posed as a central open problem.

This problem is framed in the context of Selberg’s conjecture for modular surfaces and recent advances showing near-optimal spectral gaps in various random models, motivating a finer understanding of extremal spectral phenomena in large-genus families.

References

A significant and well known open problem, which has been around in some form presumably since the time of Buser's conjecture and seen a resurgence in interest since the resolution of this conjecture, is whether there exists {X_i} with g_i→∞ and λ_1(X_i)≥1/4. Do there exist Selberg surfaces of arbitrarily large genus?

— Spectral gap with polynomial rate for Weil-Petersson random surfaces  (2508.14874 - Hide et al., 20 Aug 2025) in Motivation, Section 1 (Introduction); Problem 1.1