In this paper, we associate with isometries of cube complexes specific subspaces, referred to as median sets, which play a similar role as minimizing sets of semisimple isometries in spaces. Various applications are deduced, including a cubulation of centralizers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on cube complexes.
"Median Sets of Isometries in Cube Complexes and Some Applications." Michigan Math. J. Advance Publication 1 - 46, 2021. https://doi.org/10.1307/mmj/20195823