Median Sets of Isometries in CAT(0) Cube Complexes and Some Applications

Abstract

In this paper, we associate with isometries of $\mathrm{CAT}(0)$ cube complexes specific subspaces, referred to as median sets, which play a similar role as minimizing sets of semisimple isometries in $\mathrm{CAT}(0)$ 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 $\mathrm{CAT}(0)$ cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on $\mathrm{CAT}(0)$ cube complexes.