Group actions on spheres with rank one isotropy

Let $G$ be a rank two finite group, and let $\mathcal{H}$ denote the family of all rank one $p$ -subgroups of $G$ for which ${\mathrm{rank}}_p(G)=2$ . We show that a rank two finite group $G$ which satisfies certain explicit group-theoretic conditions admits a finite $G$ -CW-complex $X\simeq S^n$ with isotropy in $\mathcal{H}$ , whose fixed sets are homotopy spheres. Our construction provides an infinite family of new non-linear $G$ -CW-complex examples.