The Tangle-Free Hypothesis on Random Hyperbolic Surfaces

Abstract

Abstract This article introduces the notion of $L$-tangle-free compact hyperbolic surfaces, inspired by the identically named property for regular graphs. Random surfaces of genus $g$, picked with the Weil–Petersson probability measure, are $(a \log g)$-tangle-free for any $a<1$. This is almost optimal, for any surface is $(4 \log g + O (1))$-tangled. We establish various geometric consequences of the tangle-free hypothesis at a scale $L$, among which the fact that closed geodesics of length $< \frac L 4$ are simple, disjoint, and embedded in disjoint hyperbolic cylinders of width $\geq \frac{L}{4}$.