On the action of finite simple groups of Lie type and characteristic \(p\) on finite \(\mathbb{Z}_ p\)-acyclic simplicial complexes
Assume a finite \(p\)-group \(G\) acts on a finite simplicial complex \(D\) via automorphisms of \(D\) in such a way that whenever \(g\in G\) fixes a simplex of \(D\), it also fixes all points of the simplex. Then, according to the theory of P. A. Smith, if \(D\) is \(\mathbb{Z}_ p\)-acyclic, so is the set \(D^ G\) of points left fixed under the action of \(G\) on \(D\). It is well known that this result is no more true when \(G\) is a finite group not of prime power order [\textit{R. Oliver}, Comment. Math. Helv. 50, 155-177 (1975; Zbl 0304.57020)]. However, as the author of the paper under review shows in Theorem 1, \(D^ G\) is \(\mathbb{Z}_ p\)-acyclic provided \(G\) is a finite simple group of Lie type, of Lie rank \(\ell\) and characteristic \(p\), and \(\dim D\leq \ell-1\). The result follows from the theory of P. A. Smith and the theory of finite simple groups of Lie type, as a corollary of a more general assertion presented in Theorem 2. The latter theorem is proven by using simplicial methods and results from another paper of the author [Isr. J. Math. 82, No. 1-3, 381-394 (1993; Zbl 0788.57024)]. The author also adds a second proof of Theorem 2, using a spectral sequences argument, which is due to the reviewer.
\(\mathbb{Z}_ p\)-acyclic fixed point set
scientific article; zbMATH DE number 488121