Commensurability of groups quasi-isometric to RAAGs

• 成果类型: Article
• 署名作者: Huang, Jingyin
• 署名单位: McGill University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0803-3
• 发表日期: 2018
• 页码: 1179-1247
• 关键词: angled artin groups; special cube complexes; rank symmetric-spaces; residual finiteness; graph products; RIGIDITY; CLASSIFICATION; SUBGROUPS; MANIFOLDS; geometry

摘要

Let G be a right-angled Artin group with defining graph 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) does not contain any induced 4-cycles; (3) is star-rigid; then H is commensurable to G. We show condition (2) is sharp in the sense that if 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). 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.