Realizability of fusion systems by discrete groups

TL;DR

The paper develops a framework for realizability of fusion systems over discrete $p$-toral groups by introducing notions such as sequential realizability and LT-realizability. It proves that fusion systems from compact Lie groups are LT-realizable and hence sequentially realizable, and establishes tools to show nonrealizability in broad families of infinite/discrete cases. Using linear torsion groups and $p$-compact group theory, it connects the p-local structure of infinite groups to classifying-space techniques, with CFSG-based results sharpening when LT-realizability can occur. The work clarifies which fusion systems can be realized by linear torsion groups and by finite or locally finite groups, impacting the construction and classification of exotic fusion systems and guiding further study of their homotopical realizations.

Abstract

For a prime $p$, fusion systems over discrete $p$-toral groups are categories that model and generalize the $p$-local structure of Lie groups and certain other infinite groups in the same way that fusion systems over finite $p$-groups model and generalize the $p$-local structure of finite groups. In the finite case, it is natural to say that a fusion system $\mathcal{F}$ is realizable if it is isomorphic to the fusion system of a finite group, but it is less clear what realizability should mean in the discrete $p$-toral case. In this paper, we look at some of the different types of realizability for fusion systems over discrete $p$-toral groups, including realizability by linear torsion groups and sequential realizability, of which the latter is the most general. After showing that fusion systems of compact Lie groups are always realized by linear torsion groups (hence sequentially realizable), we give some new tools for showing that certain fusion systems are not sequentially realizable, and illustrate it with two large families of examples.

Theorems & Definitions (126)

• Theorem 1: Theorem \ref{['t:fin.real.']}
• Theorem 2: Theorem \ref{['t:cpt.Lie']}
• Theorem 3: Theorem \ref{['t:seq-exotic']}
• Theorem 4: Theorem \ref{['t:cn.p-cpt']}(b)
• Definition 1.1
• Definition 1.2
• Proposition 1.3: BLO3
• Lemma 1.4
• proof
• Lemma 1.5