Direct products of free groups in ${{\rm{Aut}}}(F_N)$

TL;DR

This paper characterizes how maximal direct products of free groups embed into automorphism groups of free groups, revealing fixed points in the boundary of Outer space that describe these embeddings.

Contribution

It provides a complete classification of maximal direct product embeddings into Aut(F_N) and Out(F_N), including their fixed point properties in Outer space boundary.

Findings

Embeddings have canonical fixed points in Outer space boundary.

Complete description of maximal direct product embeddings.

Insights into the structure of Aut(F_N) and Out(F_N) embeddings.

Abstract

We give a complete description of the embeddings of direct products of nonabelian free groups into ${{\rm{Aut}}}(F_N)$ and ${{\rm{Out}}}(F_N)$ when the number of direct factors is maximal. To achieve this, we prove that the image of each such embedding has a canonical fixed point of a particular type in the boundary of Outer space.