Top dimensional quasiflats in $CAT(0)$ cube complexes

TL;DR

This paper characterizes the structure of quasiflats in $CAT(0)$ cube complexes and introduces weakly special complexes, showing quasi-isometry invariants for right-angled Artin groups and extending some results to Euclidean buildings.

Contribution

It proves that $n$-quasiflats in $n$-dimensional $CAT(0)$ cube complexes are close to finite unions of orthants and introduces weakly special complexes to analyze quasi-isometries.

Findings

$n$-quasiflats are close to unions of orthants

Quasi-isometries preserve top-dimensional flats in weakly special complexes

Quasiflats in Euclidean buildings are close to Weyl cones

Abstract

We show that every $n$-quasiflat in a $n$-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 {\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 $CAT(0)$ spaces of finite geometric dimension. In particular, we give a short proof of the fact that a top dimensional quasiflat in a Euclidean buildings is Hausdorff close to finite union of Weyl cones, which was previously established in several other authors by different methods.