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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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