Embeddability and quasi-isometric classification of partially commutative groups

Casals-Ruiz, Montserrat

doi:10.2140/agt.2016.16.597

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Embeddability and quasi-isometric classification of partially commutative groups

Montserrat Casals-Ruiz

Algebraic & Geometric Topology 16 (2016) 597–620

DOI: 10.2140/agt.2016.16.597

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 $G (Δ)$ and $G (Γ)$ are quasi-isometric, then $G (Δ)$ is a (nice) subgroup of $G (Γ)$ 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

References

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

Volume 16, issue 1 (2016)