  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.
