Complexity and cohomology of cohomological Mackey functors

TL;DR

This paper characterizes when finitely generated cohomological Mackey functors exhibit polynomial growth for finite groups over a field of characteristic p, linking this property to the structure of Sylow p-subgroups.

Contribution

It provides a complete characterization of poco groups over a field of characteristic p based on the structure of Sylow p-subgroups, including explicit calculations for elementary abelian 2-groups.

Findings

G is a poco group iff Sylow p-subgroups are cyclic for p>2 or have sectional rank ≤ 2 for p=2

Explicit description of extension groups between simple functors when p=2

Presentation of the graded algebra of self extensions of the simple functor S_1^G

Abstract

Let $k$ be a field of characteristic $p>0$. Call a finite group $G$ a poco group over $k$ if any finitely generated cohomological Mackey functor for $G$ over $k$ has polynomial growth. The main result of this paper is that $G$ is a poco group over $k$ if and only if the Sylow $p$-subgroups of $G$ are cyclic, when $p>2$, or have sectional rank at most 2, when $p=2$. A major step in the proof is the case where $G$ is an elementary abelian $p$-group. In particular, when $p=2$, all the extension groups between simple functors can be determined completely, using a presentation of the graded algebra of self extensions of the simple functor $S_1^G$, by explicit generators and relations.