Intersection numbers in the curve graph with a uniform constant

TL;DR

This paper establishes new inequalities relating intersection numbers of curves along geodesics in the curve graph, with a uniform constant depending solely on surface topology, extending previous results on tight geodesics.

Contribution

It introduces a method to derive inequalities for intersection numbers along geodesics, not just tight geodesics, with a uniform constant based on surface topology.

Findings

Inequalities involving intersection numbers along geodesics.

Uniform constant depending only on surface topology.

Extension of inequalities from tight geodesics to general geodesics.

Abstract

We derive various inequalities involving the intersection number of the curves contained in geodesics and tight geodesics in the curve graph. While there already exist such inequalities on tight geodesics, our method applies in the setting of geodesics. Furthermore, the method gives inequalities with a uniform constant depending only on the topology of the surface.