On the 2-linearity of the free group

TL;DR

This paper constructs a faithful action of the free group on a homotopy category of modules over a zigzag algebra, linking algebraic, geometric, and homological concepts to analyze free group properties.

Contribution

It introduces a novel faithful action of the free group on a homotopy category and connects algebraic gradings with geometric monoids to study free group metrics.

Findings

Established a faithful action of $F_n$ on a homotopy category.

Connected homotopy classes with complexes of zigzag modules.

Provided homological constructions of free group metrics.

Abstract

We construct an action of the free group $F_n$ on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between homotopy classes of paths in the punctured disc and complexes of projective zigzag modules and explore the connection between gradings on the zigzag algebra and monoids in $F_n$. We use this connection to give homological constructions of the standard and Bessis dual word length metrics on the free group.