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Top-dimensional quasiflats in CAT(0) cube complexes

Jingyin Huang

Geometry & Topology 21 (2017) 2281–2352

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.

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quasiflats, CAT(0) cube complexes, weakly special cube complexes
Mathematical Subject Classification 2010
Primary: 20F67
Secondary: 20F65, 20F69
Received: 10 January 2016
Revised: 17 May 2016
Accepted: 25 July 2016
Published: 19 May 2017
Proposed: Walter Neumann
Seconded: Bruce Kleiner, Dmitri Burago
Jingyin Huang
The Department of Mathematics and Statistics
McGill University
Burnside Hall, Room 1242
805 Sherbrooke W.
Montreal QC H3A 0B9