The center of the Goldman Lie algebra of a surface of infinite genus

TL;DR

This paper proves that the center of the Goldman Lie algebra for an infinite genus surface is generated solely by the constant loop, extending previous conjectures and results to a new class of surfaces.

Contribution

It establishes the structure of the center of the Goldman Lie algebra for infinite genus surfaces, connecting it to the center of the Lie algebra of oriented chord diagrams.

Findings

Center of Goldman Lie algebra is spanned by the constant loop.

Result extends the conjecture for closed surfaces to infinite genus surfaces.

Connection made between Goldman Lie algebra and chord diagram Lie algebra.

Abstract

Let $\Sigma_{\infty, 1}$ be the inductive limit of compact oriented surfaces with one boundary component. We prove the center of the Goldman Lie algebra of the surface $\Sigma_{\infty,1}$ is spanned by the constant loop. A similar statement for a closed oriented surface was conjectured by Chas and Sullivan, and proved by Etingof. Our result is deduced from a computation of the center of the Lie algebra of oriented chord diagrams.