Geometry & Topology

Top-dimensional quasiflats in CAT(0) cube complexes

Jingyin Huang

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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
Received: 10 January 2016
Revised: 17 May 2016
Accepted: 25 July 2016
First available in Project Euclid: 19 October 2017

Permanent link to this document
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

Subjects
Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F69: Asymptotic properties of groups

Keywords
quasiflats CAT(0) cube complexes weakly special cube complexes

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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