Title : Complexity and cohomology of cohomological Mackey functors

Author : Serge Bouc

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.

AMS Subject Classification : 16P90, 18G10, 18G15, 20J05.

Keywords : Cohomological, Mackey functor, complexity, growth.

Status : to appear in Advances in Mathematics.