Cohomology of infinite groups realizing fusion systems

Abstract

Given a fusion system $\mathcal{F}$ defined on a $p$-group $S$, there exist infinite group models, constructed by Leary and Stancu, and Robinson, that realize $\mathcal{F}$. We study these models when $\mathcal{F}$ is a fusion system of a finite group $G$ and prove a theorem which relates the cohomology of an infinite group model $\pi$ to the cohomology of the group $G$. We show that for the groups $GL(n,2)$, where $n\geq 5$, the cohomology of the infinite group obtained using the Robinson model is different than the cohomology of the fusion system. We also discuss the signalizer functors $P\to \Theta(P)$ for infinite group models and obtain a long exact sequence for calculating the cohomology of a centric linking system with twisted coefficients.