## Geometry & Topology

### Top-dimensional quasiflats in CAT(0) cube complexes

Jingyin Huang

#### Abstract

We show that every $n$–quasiflat in an $n$–dimensional $CAT(0)$ 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 $CAT(0)$ 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.

#### Article information

Source
Geom. Topol., Volume 21, Number 4 (2017), 2281-2352.

Dates
Revised: 17 May 2016
Accepted: 25 July 2016
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.gt/1508437642

Digital Object Identifier
doi:10.2140/gt.2017.21.2281

Mathematical Reviews number (MathSciNet)
MR3654109

Zentralblatt MATH identifier
06726522

#### Citation

Huang, Jingyin. Top-dimensional quasiflats in CAT(0) cube complexes. Geom. Topol. 21 (2017), no. 4, 2281--2352. doi:10.2140/gt.2017.21.2281. https://projecteuclid.org/euclid.gt/1508437642

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