# Fundamental groups, slalom curves and extremal length

## TL;DR

This paper introduces a new extremal length invariant for elements of the fundamental group of the twice punctured complex plane, providing bounds that relate to 3-braid invariants and their applications.

## Contribution

It defines the extremal length for fundamental group elements and establishes bounds, connecting topological invariants with braid theory.

## Key findings

- Bounds differ by a multiplicative constant
- Extremal length relates to 3-braid invariants
- Provides tools for applications in braid theory

## Abstract

We define the extremal length of elements of the fundamental group of the twice punctured complex plane and give upper and lower bounds for this invariant. The bounds differ by a multiplicative constant. The main motivation comes from $3$-braid invariants and their application.