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Inventiones mathematicae

, Volume 213, Issue 3, pp 1179–1247 | Cite as

Commensurability of groups quasi-isometric to RAAGs

  • Jingyin Huang
Article

Abstract

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 Huang and Kleiner (Duke Math. J. 167(3), 537-602 (2018).  https://doi.org/10.1215/00127094-2017-0042) 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.

Mathematics Subject Classification

20F65 20F67 20F69 

Notes

Acknowledgements

This paper would not be possible without the helpful discussions with B. Kleiner. In particular, he pointed out a serious gap in the author’s previous attempt to prove a special case of the main theorem. Also the author learned Lemma 4.11 from him. The author thanks D. T. Wise for pointing out the reference [35] and X. Xie for helpful comments and clarifications. The author thanks the referee for carefully reading the paper and providing many helpful comments and clarifications.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

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