Extension between functors from groups

Abstract: Motivated in part by the study of the stable homology of automorphism groups of free groups, we consider cohomological calculations in the category $\mathcal{F}(\textbf{gr})$ of functors from finitely generated free groups to abelian groups.In particular, we compute the groups $Ext^*_{\mathcal{F}(\textbf{gr})}(T^n \circ \mathfrak{a}, T^m \circ \mathfrak{a})$ where $\mathfrak{a}$ is the abelianization functor and $T^n$ is the n-th tensor power functor for abelian groups. These groups are shown to be non-zero if and only if $*=m-n \geq 0$ and $Ext^{m-n}_{\mathcal{F}(\textbf{gr})}(T^n \circ \mathfrak{a}, T^m \circ \mathfrak{a})=\mathbb{Z}[Surj(m,n)]$ where $Surj(m,n)$ is the set of surjections from a set having $m$ elements to a set having $n$ elements. We make explicit the action of symmetric groups on these groups and the Yoneda and external products. We deduce from these computations those of rational Ext-groups for functors of the form $F \circ \mathfrak{a}$ where $F$ is a symmetric or an exterior power functor. Combining these computations with a recent result of Djament we obtain explicit computations of stable homology of automorphism groups of free groups with coefficients given by particular contravariant functors.