The probabilistic nature of McShane's identity: planar tree coding of simple loops

TL;DR

This paper explores a probabilistic interpretation of McShane's identity, framing it as a finite measure on the space of embedded paths through a point, with implications for understanding simple loops.

Contribution

It introduces a novel probabilistic perspective on McShane's identity using planar tree coding of simple loops.

Findings

McShane's identity can be interpreted probabilistically.

Finite measure on embedded paths is characterized.

Planar tree coding effectively models simple loops.

Abstract

In this article, we discuss a probabilistic interpretation of McShane's identity as describing a finite measure on the space of embedded paths though a point.