Cubulating spaces with walls

Abstract

We describe a correspondence between spaces with walls and CAT(0) cube complexes.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-16.abs.html Version 3: minor misprint correcte

On some convex cocompact groups in real hyperbolic space

“…Let X be a CAT (0) even-gonal complex. Using the family of hyperplanes of X there is a naturally defined embedding of graphs X 1 → C 1 , where C is a CAT (0) cube complex (see [22,6]). Moreover the hyperplanes of X are in 1-1 correspondance with the hyperplanes of C, through the identification of parallelism classes of edges in both X and C. The cubes of C correspond to certain configurations of pairwise intersecting hyperplanes.…”

Cubulating random groups at density less than $1/6$

“…These are examples of Dunwoody's "tracks" and we expect they will be referred to as "walls" in future work on this subject. Indeed, subsequently, Nica [Nic04] and Chatterji and Niblo [CN05] have written out an explicit application of Sageev's construction to cubulate abstract "spaces with walls". Those were introduced by Haglund and Paulin [HP98] especially motivated by Coxeter groups and CAT(0) cube complexes.…”

A non-trivial example of a free-by-free group with the Haagerup property

“…In order to get the above statement, we prove the existence of a "space with walls" structure as introduced by Haglund and Paulin [11]. A theorem of Chatterji-Niblo [4] or Nica [14] (for similar constructions in other settings, see also [13] or [15]) gives the announced action on a CAT(0) cube complex.…”

“…In order to get Theorem 1 we will need the following (stronger) result (we refer to [14] for a similar statement):…”

“…Arzhantseva, M. Lustig and Y. Stalder for their remarks about preliminary versions of the paper. Last but not least, many thanks are due to the referee, who provided a great help to correct some mistakes and gave invaluable comments to make the paper better: among many other things he indicated to the author the results of Chatterji-Niblo [4] and Nica [14], and gave him the idea of looking for the dimension of the cube complex.…”