The automorphism group of the free group of rank two is a CAT(0) group

TL;DR

This paper demonstrates that the automorphism group of the free group of rank two acts on a CAT(0) 2-complex, highlighting a unique geometric property not shared by higher ranks.

Contribution

It establishes that the automorphism group of the free group of rank two is a CAT(0) group, a property not observed in higher ranks.

Findings

Automorphism group of free rank two acts on a CAT(0) 2-complex

Contrast with higher ranks where this property does not hold

Faithful and geometric action established

Abstract

We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.