There are various interesting statements that boil down topassing froman action of a group G on a CAT(0) cube complex to anaction on a tree. These include Stallings' ends theorem, the Nielsenrealisation theorem forOut(Fn), and the Kropholler-Roller conjectureabout almost-invariant subsets. I will discuss a method for convertinga G-action on a CAT(0) cube complex X to a G-action on a "lowercomplexity" CAT(0) cube complex Y and describe conditions under whichthis can be used inductively to find a splitting of G. This leads tonew proofs of the first two of the preceding statements, as well as aspecial case of the Kropholler-Roller conjecture. Iwill also brieflydescribe some possible generalisations of the construction. Most ofthis talk is on joint work with Nicholas Touikan\; it will also touch onsome joint work with Henry Wilton.