On dimensions of groups with cocompact classifying spaces for proper actions

We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for E_G, the classifying space for proper actions of G. They are the first examples that have these properties and also admit cocompact models for E_G. 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.