Authors: Ian J Leary and Nansen Petrosyan 

Affiliation (both): University of Southampton 

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