#### Volume 21, issue 4 (2017)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
Top-dimensional quasiflats in CAT(0) cube complexes

### Jingyin Huang

Geometry & Topology 21 (2017) 2281–2352
##### Abstract

We show that every $n$–quasiflat in an $n$–dimensional $CAT\left(0\right)$ cube complex is at finite Hausdorff distance from a finite union of $n$–dimensional orthants. Then we introduce a class of cube complexes, called weakly special cube complexes, and show that quasi-isometries between their universal covers preserve top-dimensional flats. This is the foundational result towards the quasi-isometric classification of right-angled Artin groups with finite outer automorphism group.

Some of our arguments also extend to $CAT\left(0\right)$ spaces of finite geometric dimension. In particular, we give a short proof of the fact that a top-dimensional quasiflat in a Euclidean building is Hausdorff close to a finite union of Weyl cones, which was previously established by Kleiner and Leeb (1997), Eskin and Farb (1997) and Wortman (2006) by different methods.

##### Keywords
quasiflats, CAT(0) cube complexes, weakly special cube complexes
##### Mathematical Subject Classification 2010
Primary: 20F67
Secondary: 20F65, 20F69