Median sets of isometries in CAT(0) cube complexes and some applications
Abstract
In this article, we associate to isometries of CAT(0) cube complexes specific subspaces, referred to as median sets, which play a similar role as minimising sets of semisimple isometries in CAT(0) spaces. Various applications are deduced, including a cubulation of centralisers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a CAT(0) cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on CAT(0) cube complexes.
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