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Groups and polytopes

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