Cubulating spaces with walls

By Bogdan Nica. 2003

105 citations recorded by the source corpus.

The elegant notion of a space with walls was introduced by Haglund and Paulin [6]. Prototypical examples of spaces with walls are CAT(0) cube complexes, introduced by Gromov in [5]. The purpose of this note is to observe that every space with walls has a canonical embedding in a CAT(0) cube complex and, consequently, a group action on a space with walls extends naturally to a group action on a CAT(0) cube complex. The usefulness of this result is that spaces with walls are often easily identifiable by geometric reasons.

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Citations in the corpus

The Poisson boundary of CAT(0) cube complex groups (2011, background): “group of real-rank r. 2. Basics on CAT(0) cube complexes 2.1. Hyperplanes and halfspaces. We recall basic terminology and facts about CAT(0) cube complexes, referring for more details to [ChN], [Gu], [N], [Ro], [S1]. Denition 2.1. A CAT(0) cube complex is a simply-connected combinatorial cell comple”
Isometries of CAT(0) cube complexes are semi-simple (2007, background): “Let us review some elementary properties of the CAT (0) cube complex X (see [2, 10] for details).”
Graph manifolds with boundary are virtually special (2011, unknown): “This gives M̃ the structure of a ‘space with walls’ (see [4] or [14]).”
Virtual cubulation of nonpositively curved graph manifolds (2011, methodology): “Generally speaking, from any ﬁnite collection of π1- injective surfaces, the Sageev construction allows one to obtain a CAT(0) cube complex on which the fundamental group of the 3-manifold acts, (cf. [Ni]). This will be a proper action if one can ﬁnd a ‘suﬃcient’ collection, and the virtual specialn”
Weak hyperbolicity of cube complexes and quasi‐arboreal groups (2011, background): “Discussions of Sageev’s construction in a general wallspace setting appear in [11, 24, 30].”
Quasiflats in hierarchically hyperbolic spaces (2017, methodology): “ns of W, with two 0{cubes joined by a 1{cube if the corresponding orientations dier on exactly one wall. The resulting graph is median, as was proven independently by Chatterji-Niblo [CN05] and Nica [Nic04], building on work of Sageev [Sag95]. Thus this graph is the 1{skeleton of a uniquely determin”
Rank Rigidity for Cat(0) Cube Complexes (2010, background): “Let a, b ∈ X be two points such that the geodesic segment [a, b] crosses ˆi−1, ˆi and ˆ hi+1 for some i ∈ Z .L etx ∈ [a, b] ∩ ˆi .W e shall show thatx is at distance at most N � away from ,w hereN � is some constant depending only on N .”
Coxeter groups are virtually special (2010, unknown): “MSC: 53C23; 20F36; 20F55; 20F67; 20F65; 20E26 Keywords: CAT(0) cube complexes; Coxeter group; Artin group; Subgroup separability”
Special Cube Complexes (2008, methodology): “it was shown in [CN], [ Ni ], that any group acting on a space with walls (see [HP]) gives rise to a group action on a CAT(0)-cube complex.”
Cubulating random groups at density less than 1/6 (2011, unknown): “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”.”