The intersection dual of geodesic currents

TL;DR

This paper characterizes curve functionals dual to geodesic currents on hyperbolic surfaces using axiomatic and combinatorial methods, providing new insights and unifying existing frameworks.

Contribution

It offers a new axiomatic and combinatorial characterization of geodesic currents as curve functionals, enabling a measure-free perspective.

Findings

Characterization of dual geodesic currents via additivity and smoothing properties.

New axiomatic descriptions of measured laminations and hyperbolic length functions.

Unified framework for dual currents from metric structures and cross-ratios.

Abstract

Geodesic currents on closed hyperbolic surfaces are measures on the unit tangent bundle invariant under geodesic flow and orientation reversal. Every geodesic current induces a dual function on curves via the geometric intersection pairing. It is natural to ask which curve functions are dual to geodesic currents, that is, which arise as intersection functionals of a geodesic current. In this paper we give a purely axiomatic and combinatorial characterization of curve functionals dual to geodesic currents. This yields a new definition of geodesic currents as curve functionals or, equivalently, as functions on surface groups, without reference to measures or flows. More precisely, we show that a function on curves arises as the geometric intersection pairing with a geodesic current if and only if it is additive under disjoint union and satisfies a simple \emph{smoothing} property: it is non-increasing under surgery of essential crossings. As applications, we obtain new axiomatic characterizations of measured laminations and hyperbolic length functions, and new descriptions of small surface group actions on real trees, including a concise proof of a classical theorem of Skora. We also provide a unified framework for dual geodesic currents arising from metric structures and generalized cross-ratios, including those associated with certain Anosov representations. Our approach subsumes all previously known constructions of dual geodesic currents and yields broad new families of examples.