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Embeddability and quasi-isometric classification of partially commutative groups

Montserrat Casals-Ruiz

Algebraic & Geometric Topology 16 (2016) 597–620

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 –completions.

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