Speaker
Description
For a prime p, fusion systems over discrete p-toral groups are categories that model and generalise 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 generalise the p-local structure of finite groups. In the finite case, it is natural to say that a fusion system 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 talk I will discuss recent joint work with Carles Broto and Bob Oliver. We look at some of the different types of realizability for fusion systems over discrete p-toral groups, including realisability by linear torsion groups and sequential realisability, of which the latter is the most general. We show in particular that fusion systems of compact Lie groups are always realised by linear torsion groups (hence sequentially realisable). We also present two large families of examples of fusion systems that are not sequentially realisable.
I will proceed by comparing four different types of realisability 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 present a version of the Cartan-Eilenberg stable elements theorem for locally finite groups.
