Resolutions of CAT(0) cube complexes and accessibility properties

TL;DR

This paper extends the concept of resolutions from group actions on trees to actions on finite dimensional CAT(0) cube complexes, providing bounds on hyperplanes and applications to surfaces and 3-manifolds.

Contribution

It introduces a new framework for resolutions of group actions on CAT(0) cube complexes, generalizing Dunwoody's work from trees to higher dimensions.

Findings

Bound the number of hyperplanes in 2D resolutions.

Apply bounds to collections of codimension-1 submanifolds.

Extend resolution concepts to CAT(0) cube complexes.

Abstract

In [4], 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. He, moreover, 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.