Resolutions of CAT(0) cube complexes and accessibility properties

In 1985, Dunwoody defined resolutions for finitely presented group actions on simplicial trees, that is, an action of the group on a tree with smaller edge and vertex stabilizers. Moreover, he proved that the size of the resolution is bounded by a constant depending only on the group. Extending Dunwoody’s definition of patterns, we construct resolutions for group actions on a general finite-dimensional CAT(0) cube complex. In dimension two, we bound the number of hyperplanes of this resolution. We apply this result for surfaces and 3-manifolds to bound collections of codimension-1 submanifolds.