Realizability of fusion systems by discrete groups: II

TL;DR

The paper develops a framework connecting saturated fusion systems over discrete $p$-toral groups with realizations by locally finite groups. It defines four realizability notions (LT-, LFS-, LF-, and sequential realizability) and shows their hierarchical implications, with a core result that finite subgroups determine saturation; it also proves a Cartan–Eilenberg–type stable elements theorem for locally finite groups. By constructing fusion and linking systems arising from locally finite $p$-artinian groups, the authors establish when a fusion system can be realized and how classifying spaces relate to $p$-completed spaces. The work extends realizability beyond finite and linear-torsion cases, providing a broader algebraic model for $p$-local structures in locally finite and strongly $p$-artinian groups.

Abstract

We compare four different types of realizability for saturated fusion systems over discrete $p$-toral groups. For example, when $G$ is a locally finite group all of whose $p$-subgroups are artinian (hence discrete $p$-toral), we show that it has ``weakly Sylow'' $p$-subgroups and give explicit constructions of saturated fusion systems and associated linking systems associated to $G$. We also show that a fusion system over a discrete $p$-toral group $S$ is saturated if its set of morphisms is closed under a certain topology and the finite subgroups of $S$ satisfy the saturation axioms, and prove a version of the Cartan-Eilenberg stable elements theorem for locally finite groups.

Theorems & Definitions (50)

• Theorem A
• Theorem B
• Theorem C
• Definition 1.1
• Proposition 1.2
• proof
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Proposition 1.6