Sur l'homologie des groupes d'automorphismes des groupes libres \`a coefficients polynomiaux

TL;DR

This paper investigates the stable homology of automorphism groups of free groups with polynomial functor coefficients, showing null results for covariant functors and explicit calculations for contravariant ones.

Contribution

It provides new results on the vanishing and computation of stable homology for automorphism groups with polynomial coefficients using functor categories.

Findings

Homology is zero for reduced covariant polynomial functors.

First homology group computed for reduced contravariant polynomial functors.

Methods rely on functor categories.

Abstract

We study in this article stable homology of automorphism groups of free groups with coefficients twisted by a poynomial functor. We show that this homology is zero for a reduced covariant polynomial functor. For a reduced contravariant functor, we compute the first homology group, which is in general non zero. Our methods relie on the use of functor categories. ---On \'etudie dans cet article l'homologie stable des groupes d'automorphismes des groupes libres \`a coefficients tordus par un foncteur polynomial. On montre que cette homologie est nulle pour un foncteur polynomial covariant r\'eduit. Dans le cas d'un foncteur polynomial r\'eduit contravariant, on calcule le premier groupe d'homologie, qui n'est g\'en\'eralement pas nul. Nos m\'ethodes reposent sur l'utilisation de cat\'egories de foncteurs.