Complexity and cohomology of cohomological Mackey functors
Serge Bouc (LAMFA)

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 be a field of characteristic . Call a finite group a poco group over if any finitely generated cohomological Mackey functor for over has polynomial growth. The main result of this paper is that is a poco group over if and only if the Sylow -subgroups of are cyclic, when , or have sectional rank at most 2, when . A major step in the proof is the case where is an elementary abelian -group. In particular, when , 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 , by explicit generators and relations.
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