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

Hainaut, Louis

doi:10.2140/agt.2026.26.2421

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

Louis Hainaut

Algebraic & Geometric Topology 26 (2026) 2421–2441

DOI: 10.2140/agt.2026.26.2421

Abstract

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 nonvanishing 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 nontrivial outside of this specific degree.

Keywords

polynomial functor, outer automorphism, Ext group

Mathematical Subject Classification

Primary: 18A25, 18G15

References

Publication

Received: 26 February 2024

Revised: 13 June 2025

Accepted: 1 August 2025

Published: 1 September 2026

Authors

Louis Hainaut
Department of Mathematics University of Chicago Chicago, IL United States

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Volume 26, issue 7 (2026)