Complexity and cohomology of cohomological Mackey functors

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 , by explicit generators and relations. AMS Subject Classification : 16P90, 18G10, 18G15, 20J05.