Panel collapse and its applications

  • Mark F. Hagen

    University of Bristol, UK
  • Nicholas W. M. Touikan

    University of New Brunswick, Fredericton, Canada
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We describe a procedure called panel collapse for replacing a CAT(0) cube complex by a "lower complexity" CAT(0) cube complex whenever contains a codimension-2 hyperplane that is extremal in one of the codimension-1 hyperplanes containing it. Although is not in general a subcomplex of , it is a subspace consisting of a subcomplex together with some cubes that sit inside "diagonally." The hyperplanes of extend to hyperplanes of . Applying this procedure, we prove: if a group acts cocompactly on a CAT(0) cube complex , then there is a CAT(0) cube complex so that acts cocompactly on and for each hyperplane of , the stabiliser in of acts on essentially.

Using panel collapse, we obtain a new proof of Stallings's theorem on groups with more than one end. As another illustrative example, we show that panel collapse applies to the exotic cubulations of free groups constructed by Wise in [44]. Next, we show that the CAT(0) cube complexes constructed by Cashen and Macura in [7] can be collapsed to trees while preserving all of the necessary group actions. (It also illustrates that our result applies to actions of some non-discrete groups.) We also discuss possible applications to quasi-isometric rigidity for certain classes of graphs of free groups with cyclic edge groups. Panel collapse is also used in forthcoming work of the first-named author and Wilton to study fixed-point sets of finite subgroups of Out on the free splitting complex. Finally, we apply panel collapse to a conjecture of Kropholler, obtaining a short proof under a natural extra hypothesis.

Cite this article

Mark F. Hagen, Nicholas W. M. Touikan, Panel collapse and its applications. Groups Geom. Dyn. 13 (2019), no. 4, pp. 1285–1334

DOI 10.4171/GGD/524