Fixed point free actions on $Z$-acyclic 2-complexes
by Bob Oliver and Yoav Segev

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.