When is Group Cohomology Finitary?

TL;DR

This paper studies when group cohomology functors are finitary, focusing on groups with cohomology finitary in high degrees, providing conditions and characterizations, especially for groups with finite virtual cohomological dimension.

Contribution

It establishes conditions under which groups have cohomology almost everywhere finitary and answers a question regarding groups of finite virtual cohomological dimension.

Findings

Groups with finite dimensional models often have cohomology almost everywhere finitary.

For groups of finite virtual cohomological dimension, specific conditions ensure cohomology is finitary in high degrees.

Characterization of locally polycyclic-by-finite groups with cohomology almost everywhere finitary.

Abstract

If $G$ is a group, then we say that the functor $H^n(G,-)$ is finitary if it commutes with all filtered colimit systems of coefficient modules. We investigate groups with cohomology almost everywhere finitary; that is, groups with $n$th cohomology functors finitary for all sufficiently large $n$. We establish sufficient conditions for a group $G$ possessing a finite dimensional model for $e.g.$ to have cohomology almost everywhere finitary. We also prove a stronger result for the subclass of groups of finite virtual cohomological dimension, and use this to answer a question of Leary and Nucinkis. Finally, we show that if $G$ is a locally (polycyclic-by-finite) group, then $G$ has cohomology almost everywhere finitary if and only if $G$ has finite virtual cohomological dimension and the normalizer of every non-trivial finite subgroup of $G$ is finitely generated.