We prove that, up to homotopy equivalence, every connected CW-complex is the quotient of a contractible complex by a proper action of a discrete group, and that every CW-complex is the quotient of an aspherical complex by an action of a group of order two.

These results may be viewed as analogues of the Kan-Thurston theorem, in which the universal free G-space has been replaced by the universal proper G-space. The fact that our result concerns homotopy type (whereas the Kan-Thurston theorem concerns homology) is a reflection of the existence of `contractible groups', i.e., groups for which the quotient of the universal proper G-space by G is contractible.

Topology 40 (2001) 539-550.

The Mathematical Review of this paper mis-states one of our results as `every CW-complex is the quotient of an acyclic complex by an action of a group of order two'. I don't know whether this result is true or not, but it certainly can't be true that `every finite-dimensional CW-complex is the quotient of a finite-dimensional acyclic complex by an action of a group of order two', since any such space would have to be mod-2 acyclic by Smith theory.