### On geometric and algebraic dimensions for groups with torsion

### N. Brady, I. J. Leary and B. E. A. Nucinkis

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.