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Commensurability of groups quasi-isometric to RAAGs. (English) Zbl 1432.20029
Summary: Let $$G$$ be a right-angled Artin group with defining graph $$\Gamma$$ and let $$H$$ be a finitely generated group quasi-isometric to $$G$$. We show if $$G$$ satisfies that (1) its outer automorphism group is finite; (2) $$\Gamma$$ does not contain any induced 4-cycles; (3) $$\Gamma$$ is star-rigid; then $$H$$ is commensurable to $$G$$. We show condition (2) is sharp in the sense that if $$\Gamma$$ contains an induced 4-cycle, then there exists an $$H$$ quasi-isometric to $$G$$ but not commensurable to $$G$$. Moreover, one can drop condition (1) if $$H$$ is a uniform lattice acting on the universal cover of the Salvetti complex of $$G$$. As a consequence, we obtain a conjugation theorem for such uniform lattices. The ingredients of the proof include a blow-up building construction in [the author and B. Kleiner, Duke Math. J. 167, No. 3, 537–602 (2018; Zbl 1432.20030)] and a Haglund-Wise style combination theorem for certain class of special cube complexes. However, in most of our cases, relative hyperbolicity is absent, so we need new ingredients for the combination theorem.

##### MSC:
 20F65 Geometric group theory 20F36 Braid groups; Artin groups 20F67 Hyperbolic groups and nonpositively curved groups 20F69 Asymptotic properties of groups
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