Equal angles of intersecting geodesics for every hyperbolic metric

TL;DR

This paper investigates the geometric properties of Goldman brackets on hyperbolic surfaces, establishing conditions under which intersection angles are equal and providing an alternative proof of a related theorem.

Contribution

It introduces an obstruction criterion for the equality of Goldman bracket terms and offers a new proof of Chas's theorem linking simple representatives to intersection counts.

Findings

Equal angles at intersection points imply Goldman bracket term equality.

An obstruction criterion for Goldman bracket term equality is established.

Provides an alternative proof of Chas's theorem relating simple curves and intersection numbers.

Abstract

We study the geometric properties of the terms of the Goldman bracket between two free homotopy classes of oriented closed curves in a hyperbolic surface. We provide an obstruction for the equality of two terms in the Goldman bracket, namely if two terms in the Goldman bracket are equal to each other then for every hyperbolic metric, the angles corresponding to the intersection points are equal to each other. As a consequence, we obtain an alternative proof of a theorem of Chas, i.e. if one of the free homotopy classes contains a simple representative then the geometric intersection number and the number of terms (counted with multiplicity) in the Goldman bracket are the same.