Group actions on spheres with rank one isotropy

TL;DR

This paper constructs new examples of finite G-CW-complexes with rank two finite groups acting with rank one p-subgroup isotropy, where fixed sets are homotopy spheres, expanding the known non-linear actions.

Contribution

It introduces a method to build finite G-CW-complexes with specified isotropy and fixed set properties for rank two finite groups, providing new non-linear examples.

Findings

Constructed finite G-CW-complexes with homotopy sphere fixed sets.

Provided an infinite family of non-linear G-CW-complex examples.

Demonstrated existence under certain group-theoretic conditions.

Abstract

Let G be a rank two finite group, and let $\cH$ denote the family of rank one p-subgroups of G, at all primes where G has p-rank two. We show that a rank two finite group G which satisfies certain group-theoretic conditions admits a finite G-CW-complex X with isotropy in $\cH$, whose fixed sets are homotopy spheres. Our construction provides an infinite family of new non-linear G-CW-complex examples.