A Moebius inversion formula to discard tangled hyperbolic surfaces

Abstract

Recent literature on Weil-Petersson random hyperbolic surfaces has met a consistent obstacle: the necessity to condition the model, prohibiting certain rare geometric patterns (which we call tangles), such as short closed geodesics or embedded surfaces of short boundary length. The main result of this article is a Moebius inversion formula, allowing to integrate the indicator function of the set of tangle-free surfaces in a systematic, tractable way. It is inspired by a key step of Friedman's celebrated proof of Alon's conjecture. We further prove that our tangle-free hypothesis significantly reduces the number of local topological types of short geodesics, replacing the exponential proliferation observed on tangled surfaces by a polynomial growth.