The tangle-free hypothesis on random hyperbolic surfaces

TL;DR

This paper introduces the concept of L-tangle-free hyperbolic surfaces, showing that random surfaces are typically tangle-free at certain scales, which has implications for the geometry of geodesics.

Contribution

It defines the L-tangle-free property for hyperbolic surfaces and proves that random surfaces are almost surely tangle-free at a scale proportional to log g, with geometric consequences.

Findings

Random hyperbolic surfaces are (a log g)-tangle-free for any a<1.

Surfaces are at most (4 log g + O(1))-tangled, nearly optimal.

Closed geodesics of length < L/4 are simple, disjoint, and embedded in disjoint cylinders.

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, amongst which the fact that closed geodesics of length < L/4 are simple, disjoint and embedded in disjoint hyperbolic cylinders of width $\ge$ L/4.