From branching quasiflats to flats in CAT(0) cube complexes

TL;DR

This paper investigates quasiisometric embeddings of finite-dimensional CAT(0) cube complexes, introducing geometric conditions that relate flats in the domain to flats in the codomain, with applications to right-angled Artin groups.

Contribution

It introduces new geometric branching conditions that ensure flats are mapped close to flats, advancing understanding of embeddings in CAT(0) cube complexes and related spaces.

Findings

Flats in the domain are mapped within finite Hausdorff distance of flats under certain conditions.

Embeddings between graphs associated with Tits boundaries are established.

Methods recover rigidity results for symmetric spaces and Euclidean buildings.

Abstract

We study quasiisometric embeddings between finite-dimensional CAT(0) cube complexes. More specifically, we introduce geometric branching conditions under which flats in the domain, not necessarily of top rank, are mapped within finite Hausdorff distance of flats. As a consequence, one obtains embeddings between natural graphs associated with the Tits boundaries of those cube complexes. These results form a key step in understanding quasiisometric embeddings between right-angled Artin groups. In an appendix, we also explain how the same methods recover previously established rigidity results for quasiisometric embeddings of symmetric spaces and Euclidean buildings of the same spherical type.