Goldman bracket and length equivalent filling curves

TL;DR

The paper constructs infinitely many pairs of length equivalent curves on a surface using Goldman brackets, especially focusing on self-intersecting and filling curves, revealing new relationships between curve intersections and length spectra.

Contribution

It introduces a method to generate infinite pairs of length equivalent curves from Goldman brackets, including for self-intersecting and filling curves, expanding understanding of length spectra.

Findings

Infinite pairs of length equivalent curves are constructed from Goldman brackets.

Self-intersecting geodesics yield sequences of length equivalent pairs.

Filling curves produce filling pairs of length equivalent curves.

Abstract

A pair of distinct free homotopy classes of closed curves in an orientable surface $F$ with negative Euler characteristic is said to be length equivalent if for any hyperbolic structure on $F$, the length of the geodesic representative of one class is equal to the length of the geodesic representative of the other class. Suppose $\alpha$ and $\beta$ are two intersecting oriented closed curves on $F$ and $P$ and $Q$ are any two intersection points between them. If the two terms $\langle\alpha *_P\beta\rangle$ and $\langle\alpha*_Q\beta\rangle$ in $[\langle\alpha\rangle,\langle\beta\rangle]$, the Goldman bracket between them, are the same, then we construct infinitely many pairs of length equivalent curves in $F.$ These pairs correspond to the terms of the Goldman bracket between a power of $\alpha$ and $\beta$. As a special case, our construction shows that given a self-intersecting geodesic $\alpha$ of $F$ and any self-intersection point $P$ of $\alpha$, we get a sequence of such pairs. Furthermore if $\alpha$ is a filling curve then these pairs are also filling.