Embeddability and quasi-isometric classification of partially commutative groups

Casals-Ruiz, Montserrat¹

¹Departamento de Matemáticas, Universidad del País Vasco/Euskal Herriko Unibertsitatea, Barrio Sarriena, s/n, 48940 Leioa, Spain

Algebraic and Geometric Topology, Tome 16 (2016) no. 1, pp. 597-620

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.597

Classification : 20A15, 20F36, 20F65, 20F69
Keywords: partially commutative group, right-angled Artin group, embeddability, quasi-isometric classification

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