The geometric dimension of a discrete group 
is the minimum dimension of a CW-complex
that is a classifying space 
for 
(Brown 1982). If no finite-dimensional
model exists, its geometric dimension is infinite. The
model has fundamental group 
and contractible universal
cover. The definition minimizes dimension, not
the number of cells.
For a nontrivial free group, a one-dimensional classifying graph gives geometric dimension 1. Integral cohomological dimension provides a lower bound. The Eilenberg-Ganea conjecture asks whether a group of integral cohomological dimension 2 always has geometric dimension 2.
