Quasiisometric embeddings between right-angled Artin groups: rigidity

TL;DR

This paper establishes new rigidity results for quasiisometric embeddings between right-angled Artin groups, using branching conditions on defining graphs to derive structural and embedding obstructions.

Contribution

It introduces branching conditions that imply embeddings between extension graphs, leading to several classification and obstruction results for quasiisometric embeddings of RAAGs.

Findings

Obstructions to embeddings into products of trees.

Classification of self-embeddings of RAAGs on cycles.

No universal RAAG for quasiisometric embeddings of the same dimension.

Abstract

By introducing branching conditions on the defining graph, we prove a range of rigidity results for quasiisometric embeddings between right-angled Artin groups. The starting point for these is that, under mild conditions on the codomain, the branching conditions imply that a quasiisometric embedding induces an embedding between the associated extension graphs. Among other things, we: (1) provide obstructions to the existence of quasiisometric embeddings into products of trees; (2) prove that if the direct product $F_2^n\times A_{C_5}^m$ can be quasiisometrically embedded in a RAAG of the same dimension, then this can be seen from its defining graph; (3) classify all self--quasiisometric-embeddings of RAAGs defined on cycles; (4) show that no $n$--dimensional RAAG is a universal receiver for quasiisometric embeddings of $n$--dimensional RAAGs. We also establish a strong rigidity theorem for the quasiisometric images of 2--flats in RAAGs defined by triangle-free graphs that are not stars, generalising a theorem of Bestvina--Kleiner--Sageev.