Typical hyperbolic surfaces have a spectral gap greater than $2/9 - \epsilon$

TL;DR

This paper proves that most hyperbolic surfaces have a spectral gap exceeding 2/9 minus epsilon, advancing towards the optimal gap of 1/4 minus epsilon.

Contribution

It establishes a lower bound on the spectral gap for typical hyperbolic surfaces sampled via Weil-Petersson measure, using explicit combinatorial exclusion techniques.

Findings

Spectral gap exceeds 2/9 - epsilon for typical surfaces.

Uses inclusion-exclusion to exclude tangles at precision 1/g.

Progress towards proving the optimal spectral gap 1/4 - epsilon.

Abstract

In this article, we prove that typical hyperbolic surfaces, sampled with the Weil-Petersson probability measure, have a spectral gap at least $2/9 - \epsilon$. This is an intermediate result on the way to our proof of the optimal spectral gap $1/4 - \epsilon$, building on the results of the first part of this series. A significant part of the proof is an explicit inclusion-exclusion argument to exclude tangles at the level of precision $1/g$.