On the action of finite simple groups of Lie type and characteristic \(p\) on finite \(\mathbb{Z}_ p\)-acyclic simplicial complexes
type: Item
instance of: scholarly article
title: On the action of finite simple groups of Lie type and characteristic \(p\) on finite \(\mathbb{Z}_ p\)-acyclic simplicial complexes
zbMATH Open document ID: 0798.57021
DOI: 10.1016/0040-9383(93)90015-N
author: Yoav Segev
published in: Topology
publication date: 1994-11-08T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 55M35
Mathematics Subject Classification ID: 55T99
zbMATH DE Number: 488121
zbMATH Keywords: \(\mathbb{Z}_ p\)-acyclic fixed point set
zbMATH Keywords: Smith theory
zbMATH Keywords: simplicial actions
zbMATH Keywords: finite \(p\)-group
zbMATH Keywords: finite simple group of Lie type
Wikidata QID: Q115364128
reviewed by: Krzysztof Pawałowski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 488121