Oriented graphs on curve complex I: hyperbolic and extremal length

TL;DR

This paper studies oriented graphs derived from the curve complex of a surface, introducing Dehn quasi-homothetic functions, and establishes rigidity results linking length functions to hyperbolic surface metrics.

Contribution

It introduces Dehn quasi-homothetic functions and proves their induced graphs are rigid, leading to a new surface metric determination method based on length comparisons.

Findings

Different positive functions induce different graphs unless proportional.

Automorphisms of the induced graphs correspond to surface homeomorphisms.

Length comparisons of disjoint curves determine the hyperbolic metric.

Abstract

We investigate oriented graphs based on the curve complex $C(S)$ of a closed surface $S$ and induced by functions on the vertex set of $C(S)$. In particular, we introduce the Dehn quasi-homothetic functions, which behave similarly to homotheties under repeated Dehn twists. We prove that any two positive such functions of the same type induce different oriented graphs unless they are proportional. This leads to a new rigidity result for closed hyperbolic surfaces -- distinct from the $9g-9$ theorem and length spectrum rigidity -- knowing only for any two disjoint simple closed curves which one is longer (in terms of hyperbolic or extremal length) suffices to determine the hyperbolic metric on the surface. We also prove that each automorphism of the oriented graph induced by a function with sublevel sets finite is induced by a self-homeomorphism of $S$.