On finite groups acting on acyclic complexes of dimension two (Q1802367)

Using fixed point theorems of Smith type, it is shown that a cellular action of a finite solvable group on an acyclic CW-complex \(X\) of dimension at most 2 has a fixed point. The proof is short and elegant. Under the additional assumption that \(X\) is finite, the same result, with a different proof, is the subject of a paper by \textit{Y. Segev} [Isr. J. Math. 82, No. 1-3, 381-394 (1993; Zbl 0788.57024)]. This result, still for finite \(X\), has been generalized subsequently and shown to hold for `almost all' finite groups by \textit{M. Aschbacher} and \textit{Y. Segev} [Proc. Lond. Math. Soc., III. Ser. 67, No. 2, 329-354 (1993; Zbl 0834.57022)], in fact for every finite group which possesses no composition factors of Lie type and of Lie rank 1 or isomorphic to \(J_1\). A result in a similar vein for a finite acyclic complex \(X\) of arbitrary dimension and a finite simple group of Lie type and of Lie rank \(\geq \dim X + 1\) is proved by \textit{Y. Segev} in [Topology 32, No. 3, 665-675 (1993; Zbl 0798.57021)].