Extension between functors from groups

TL;DR

This paper computes Ext groups between certain functors from free groups to abelian groups, revealing their structure and applications to the stable homology of automorphism groups of free groups.

Contribution

It provides explicit calculations of Ext groups in the functor category from free groups, including the action of symmetric groups and applications to automorphism group homology.

Findings

Ext groups are non-zero only when * = m - n ≥ 0

Explicit description of Ext groups as free abelian groups on surjections

Applications to computing stable homology of automorphism groups of free 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.