Group actions on spheres with rank one isotropy (Q2790719)

Let \(G\) be a finite group. Recall that the \textit{\(p\)-rank} of \(G\), written \(\operatorname{rank}_p(G)\), is the largest rank of an elementary abelian \(p\)-subgroup of \(G\) and the \textit{rank} of G is the maximum of \(\operatorname{rank}_p(G)\) taken over all primes \(p\). A \textit{\(G\)-homotopy representation} is a finite \(G\)-CW-complex \(X\) such that the \(H\)-fixed-point set \(X^H\) is homotopy equivalent to a sphere for any subgroup \(H \subseteq G\). The main result of the paper, Theorem A, contains explicit group-theoretic conditions for a rank-two finite group to admit a \(G\)-homotopy representation with all isotropy subgroups of rank one. An easy-to-state consequence of Theorem A is the following: if \(G\) is such that \(\operatorname{rank}_p(G)=2\) for a fixed prime \(p>2\), and \(\operatorname{rank}_q(G)\leq 1\) for any prime \(q\neq p\), and \(G\) is \(\mathrm{Qd}(p)\)-free, then there exists a \(G\)-homotopy representation with rank one \(p\)-group isotropy. Here, \(\mathrm{Qd}(p)\) denotes \((\mathbb{Z}/p \times \mathbb{Z}/p) \rtimes \mathrm{SL}_2(p)\) with the obvious action of \(\mathrm{SL}_2(p)\) on \(\mathbb{Z}/p \times \mathbb{Z}/p\), and \(G\) is \textit{\(\mathrm{Qd}(p)\)-free} if \(N_G(H)/H\) does not contain a subgroup isomorphic to \(\mathrm{Qd}(p)\) for any subgroup \(H \subseteq G\) of order prime to \(p\). The proof of Theorem A involves building suitable algebraic models (the so called algebraic homotopy representations) and then realizing them geometrically by means of the previous work of the authors [Homology Homotopy Appl. 16, No. 2, 345--369 (2014; Zbl 1325.18005)].NEWLINENEWLINEUsing Theorem A, the authors provide many new examples of non-linear homotopy representations. In particular, they recover, via a more systematic method, an earlier result from their joint work with \textit{S. Pamuk} [Comment. Math. Helv. 88, No. 2, 369--425 (2013; Zbl 1272.57027)], where it was shown that there exists an \(S_5\)-homotopy representation with rank-one \(2\)-power order isotropy. They also determine in Theorem C that the only rank two finite simple groups which admit homotopy representations with rank one prime-power order isotropy are: (i) \(\mathrm{PSL}_2(p)\), \(p\geq 5\), (ii) \(\mathrm{PSL}_2(p^2)\), \(p \geq 3\), (iii) \(\mathrm{PSU}_3(3)\), or (iv) \(\mathrm{PSU}_3(4)\).