Fixed point free actions on $Z$-acyclic 2-complexes
by Bob Oliver and Yoav Segev
AMS classification: primary 57S17, secondary 57M20, 20D05
Addresses:
Laboratoire de Mathematiques
Universite Paris-Nord
Av. J-B Clement
93430 Villetaneuse, France
Department of Mathematics
Ben Gurion University
Beer Sheva 84105, Israel
E-mail: bob@math.univ-paris13.fr, yoavs@math.bgu.ac.il
We show that a finite group has an "essential" fixed point free action on
an acyclic 2-complex if and only if it is one of the simple groups in the
following list:
- $PSL_2(2^k)$ for $k\ge2$,
- $PSL_2(q)$ for $q\equiv3,5$ (mod 8) and $q\ge5$,
- $Sz(2^k)$ for odd $k\ge3$.
More precisely, for any finite group $G$, and any 2-dimensional acyclic
$G$-CW complex $X$ without fixed points, there is a normal subgroup $H$ in
$G$ such that $G/H$ is in the above list, and such that the $G$-action on
$X$ looks "essentially" like the $G/H$-action which we construct.