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 \(n\ge 5\), 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 \(P\rightarrow \Theta (P)\) for infinite group models and obtain a long exact sequence for calculating the cohomology of a centric linking system with twisted coefficients.