  We construct groups G that are virtually torsion-free and have virtual
cohomological dimension strictly less than the minimal dimension for any model
for the classifying space for proper actions of G. They are the first examples
that have these properties and also admit cocompact models for this classifying
space. We exhibit groups G whose virtual cohomological dimension and Bredon
cohomological dimension are two that do not admit any 2-dimensional
contractible proper G-CW-complex.
