# Cohomology of infinite groups realizing fusion systems

## TL;DR

This paper investigates the cohomology of infinite groups modeling fusion systems, especially those arising from finite groups, and establishes relationships and differences in their cohomological properties.

## Contribution

It relates the cohomology of infinite group models to that of finite groups and explores cohomological differences for specific groups like GL(n,2).

## Key findings

- Cohomology of Robinson models differs from that of the finite group for GL(n,2), n≥5.
- Established a long exact sequence for cohomology of centric linking systems with twisted coefficients.
- Analyzed signalizer functors and their role in the cohomology of infinite group models.

## 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.