(Non-)vanishing results for extensions between simple outer functors on free groups

TL;DR

This paper investigates the cohomological Ext groups between simple polynomial outer functors on free groups, revealing that unlike in broader categories, these groups can be non-trivial outside a single degree, indicating richer structure.

Contribution

It demonstrates non-vanishing of Ext groups outside the known degree in the subcategory of polynomial outer functors, extending Vespa's results.

Findings

Ext groups between simple outer functors can be non-zero outside the expected degree

The structure of polynomial outer functors differs from the larger polynomial functor category

Provides new insights into the cohomological properties of outer automorphism categories

Abstract

In this article we study cohomological properties of the category of polynomial outer functors on free groups, which are the functors from the category of finitely generated free groups to the category of rational vector spaces which send all inner automorphisms to the identity morphism, and which satisfy a certain polynomiality property. More precisely, we prove vanishing and non-vanishing results for the Ext groups between simple polynomial outer functors. This work is inspired by an earlier result of Vespa for the category of all polynomial functors from finitely generated free groups to rational vector spaces; it follows in particular from her results that, in this larger category, the Ext groups between simple functors are always concentrated in a specific single degree. Our main results show that, when we pass to the full subcategory of polynomial outer functors, Ext groups between simple functors are sometimes non-trivial outside of this specific degree.