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 $S_1^G$, by explicit generators and relations.

💡 Research Summary

The paper investigates the homological complexity of cohomological Mackey functors over a field k of characteristic p > 0. A finite group G is called a “poco group” (polynomially‑controlled) if every finitely generated cohomological Mackey functor for G over k has polynomial growth, i.e. the dimensions of its projective resolutions grow at most like a polynomial in the resolution degree. The authors’ main achievement is a complete classification of poco groups:

If p > 2, G is a poco group precisely when its Sylow p‑subgroups are cyclic.
If p = 2, G is a poco group precisely when each Sylow 2‑subgroup has sectional rank at most 2 (equivalently, every elementary abelian 2‑subgroup has rank ≤ 2).

The proof proceeds in two major phases. First, the authors reduce the problem to the case where G is a p‑group, because induction and restriction for Mackey functors allow one to control the growth of functors on G by the growth on its Sylow p‑subgroup. Within the p‑group case, the crucial step is the analysis of elementary abelian p‑groups E ≅ (Cₚ)ʳ. For such E, they compute the Ext‑algebra \