Marked length spectrum rigidity for actions on CAT(0) cube complexes

Given an action on a metric space one can associate to each element of the group its translation length. This gives a function of the group to the reals called the marked length spectrum. Adding requirements for space and action, it is a natural question if the marked length spectrum already uniquely defines space and action. In this talk we want to show that this is the case when considering CAT(0) cube complexes (under some natural assumptions). The main tool to prove this will be a boundary rigidity using cross ratios. Joint workwith Elia Fioravanti.