Top dimensional quasiflats in cube complexes
arXiv:1410.8195 · doi:10.2140/gt.2017.21.2281
Abstract
We show that every -quasiflat in a -dimensional cube complex is at finite Hausdorff distance from a finite union of -dimensional orthants. Then we introduce a class of cube complexes, called {\em weakly special} cube complexes and show that quasi-isometries between their universal coverings preserve top dimensional flats. We use this to establish several quasi-isometry invariants for right-angled Artin groups. 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 buildings is Hausdorff close to finite union of Weyl cones, which was previously established in several other authors by different methods.
Modifications and expansions according to referee's comments. 53 pages and 4 figures
References in corpus (2)
Cited by in corpus (16)
- Hierarchically hyperbolic spaces I: curve complexes for cubical groups
- On hierarchical hyperbolicity of cubical groups
- Quasi-isometric classification of right-angled Artin groups I: the finite out case
- Quasi-isometry classification of RAAGs that split over cyclic subgroups
- Quasi-Isometric Embeddings of Symmetric Spaces
- Groups quasi-isometric to RAAG's
- Measure equivalence classification of transvection-free right-angled Artin groups
- Large facing tuples and a strengthened sector lemma
- Quasi-isometry classification of right-angled Artin groups II: several infinite out cases
- Commensurability of groups quasi-isometric to RAAG's
- Boundary amenability and measure equivalence rigidity among two-dimensional Artin groups of hyperbolic type
- Cocompactly cubulated 2-dimensional Artin groups
- Homotopy equivalent boundaries of cube complexes
- Quasi-isometry invariants of weakly special square complexes
- Curvature bounds of subsets in dimension two
- Higher rank hyperbolicity