#### Volume 16, issue 1 (2016)

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Embeddability and quasi-isometric classification of partially commutative groups

### Montserrat Casals-Ruiz

Algebraic & Geometric Topology 16 (2016) 597–620
##### Abstract

The main goal of this note is to suggest an algebraic approach to the quasi-isometric classification of partially commutative groups (alias right-angled Artin groups). More precisely, we conjecture that if the partially commutative groups $\mathbb{G}\left(\Delta \right)$ and $\mathbb{G}\left(\Gamma \right)$ are quasi-isometric, then $\mathbb{G}\left(\Delta \right)$ is a (nice) subgroup of $\mathbb{G}\left(\Gamma \right)$ and vice-versa. We show that the conjecture holds for all known cases of quasi-isometric classification of partially commutative groups, namely for the classes of $n$–trees and atomic graphs. As in the classical Mostow rigidity theory for irreducible lattices, we relate the quasi-isometric rigidity of the class of atomic partially commutative groups with the algebraic rigidity, that is, with the co-Hopfian property of their $ℚ$–completions.

##### Keywords
partially commutative group, right-angled Artin group, embeddability, quasi-isometric classification
##### Mathematical Subject Classification 2010
Primary: 20A15, 20F36, 20F65, 20F69
##### Publication
Received: 4 March 2015
Revised: 9 June 2015
Accepted: 5 July 2015
Published: 23 February 2016
##### Authors
 Montserrat Casals-Ruiz Departamento de Matemáticas Universidad del País Vasco/Euskal Herriko Unibertsitatea Barrio Sarriena, s/n 48940 Leioa Spain