Fixed point free actions on \(\mathbb{Z}\)-acyclic 2-complexes.
In the paper under review, the authors completely describe the finite groups which admit fixed point free actions on 2-dimensional \(\mathbb{Z}\)-acyclic complexes. More specifically, the authors prove that for any finite group \(G\), there is an essential fixed point free 2-dimensional \(\mathbb{Z}\)-acyclic \(G\)-complex if and only if \(G\) is isomorphic to one of the simple groups \(\text{PSL}_2 (2^k)\) for \(k\geq 3\), \(\text{PSL}_2(q)\) for \(q\equiv \pm 3\pmod 8\) and \(q\geq 5\), or \(\text{Sz}(2^k)\) for odd \(k\geq 3\). As the authors show, the condition that the action is essential is imposed to ensure that for a normal subgroup \(N\) of \(G\), a prescribed action of the quotient group \(G/N\) does not automatically produce (via the quotient map \(G\to G/N\)) a similar action of \(G\). In order to prove the results, the authors apply the classification theorem for finite simple groups and they make use of their earlier works [\textit{R. Oliver}, Comment. Math. Helv. 50, 155--177 (1975; Zbl 0304.57020); Math. Z. 149, 79--96 (1976; Zbl 0334.57023); \textit{Y. Segev}, Isr. J. Math. 82, 381--394 (1993; Zbl 0788.57024) and \textit{M. Aschbacher} and \textit{Y. Segev}, Proc. Lond. Math. Soc., III. Ser. 67, 329--354 (1993; Zbl 0834.57022)].
\(\mathbb{Z}\)-acyclic 2-dimensional complexes
scientific article; zbMATH DE number 1930785
scientific article