Geometry & Topology
- Geom. Topol.
- Volume 21, Number 4 (2017), 2281-2352.
Top-dimensional quasiflats in CAT(0) cube complexes
We show that every –quasiflat in an –dimensional cube complex is at finite Hausdorff distance from a finite union of –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 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.
Geom. Topol., Volume 21, Number 4 (2017), 2281-2352.
Received: 10 January 2016
Revised: 17 May 2016
Accepted: 25 July 2016
First available in Project Euclid: 19 October 2017
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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