Groups and polytopes
  In a series of papers the authors associated to an $L^2$-acyclic group $\Gamma$ an invariant $\mathcal{P}(\Gamma)$ that is a formal difference of polytopes in the vector space $H_1(\Gamma;\Bbb{R})$. This invariant is in particular defined for most 3-manifold groups, for most 2-generator 1-relator groups and for all free-by-cyclic groups. In most of the above cases the invariant can be viewed as an actual polytope.   In this survey paper we will recall the definition of the polytope invariant $\mathcal{P}(\Gamma)$ and we state some of the main properties. We conclude with a list of open problems. 
Stefan Friedl, Wolfgang L\"uck and Stephan Tillmann
math.GT math.GR