Rigidity of the free factor complex

TL;DR

This paper proves that the automorphism group of the free-factor complex is isomorphic to the automorphism group of the free group, extending classical geometric theorems to a non-abelian setting.

Contribution

It establishes the isomorphism between automorphisms of the free-factor complex and automorphisms of the free group, a non-abelian analogue of a fundamental geometric theorem.

Findings

The natural map from Aut(F_n) to Aut(Free-factor complex) is an isomorphism.

The analogous theorem holds for Out(F_n) acting on conjugacy classes of free factors.

Provides a non-abelian extension of the Fundamental Theorem of Projective Geometry.

Abstract

We establish the following non-abelian analogue of the Fundamental Theorem of Projective Geometry: the natural map from ${\rm{Aut}}(F_n)$ to the automorphism group of the free-factor complex $\mathcal{AF}_n$ is an isomorphism. We also prove the corresponding theorem for the action of ${\rm{Out}}(F_n)$ on the complex of conjugacy classes of free factors.