We argue that the geometric dimension of a discrete group G ought to be defined to be the minimal dimension of a model for the universal proper G-space rather than the minimal dimension of a model for the universal free G-space. For torsion-free groups, these two quantities are equal, but the new quantity can be finite for groups containing torsion whereas the old one cannot. There is an analogue of cohomological dimension (defined in terms of Bredon cohomology) for which analogues of the Eilenberg-Ganea and Stalling-Swan theorems (due to W. Lueck and M. J. Dunwoody respectively) hold. We show that some groups constructed by M. Bestvina and M. Davis provide counterexamples to the analogue of the Eilenberg-Ganea conjecture.

J. London Math. Soc. 64 (2001) 489-500.