Stable homology of automorphism groups of free groups

TL;DR

This paper proves that the stable homology of automorphism groups of free groups matches that of symmetric groups, confirming they have no stable rational homology and resolving a conjecture by Hatcher and Vogtmann.

Contribution

It establishes the equivalence of stable homology between Aut(F_n) and symmetric groups using homotopy theory, confirming the absence of stable rational homology for Aut(F_n).

Findings

Stable homology of Aut(F_n) matches that of symmetric groups.

Aut(F_n) has no stable rational homology.

Settles a conjecture of Hatcher and Vogtmann.

Abstract

Homology of the group Aut(F_n) of automorphisms of a free group on n generators is known to be independent of n in a certain stable range. Using tools from homotopy theory, we prove that in this range it agrees with homology of symmetric groups. In particular, Aut(F_n) has no stable rational homology, settling a conjecture of Hatcher and Vogtmann.