Groups and polytopes

In a series of papers the authors associated to an $L^2$-acyclic group $\mathrm{\Gamma }$ an invariant $\mathcal{P}(\mathrm{\Gamma })$ that is a formal difference of polytopes in the vector space $H_1(\mathrm{\Gamma };\mathbb{R})$. 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 $\mathcal{P}(\mathrm{\Gamma })$ and we state some of the main properties. We conclude with a list of open problems.