Fusion systems and group actions with abelian isotropy subgroups

TL;DR

The paper demonstrates that finite groups with abelian isotropy subgroups of bounded rank can act freely on products of spheres, using fusion systems and recursive methods, with applications to solvable groups.

Contribution

It introduces a recursive method involving fusion systems to construct free smooth actions of finite groups on products of spheres, extending previous techniques.

Findings

Finite groups with abelian isotropy subgroups of rank ≤ k can act freely on certain sphere products.

Every finite solvable group admits a free smooth action on a product of spheres with trivial homology action.

The recursive construction utilizes abstract fusion systems to achieve these group actions.

Abstract

We prove that if a finite group $G$ acts smoothly on a manifold $M$ so that all the isotropy subgroups are abelian groups with rank $\leq k$, then $G$ acts freely and smoothly on $M \times \bbS^{n_1} \times...\times \bbS^{n_k}$ for some positive integers $n_1,...n_k$. We construct these actions using a recursive method, introduced in an earlier paper, that involves abstract fusion systems on finite groups. As another application of this method, we prove that every finite solvable group acts freely and smoothly on some product of spheres with trivial action on homology.