Separability properties of extended admissible groups
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Hoang Thanh Nguyen
Abstract
Extended admissible groups belong to a particular class of graph of groups which possess a graph of groups decomposition generalizing that of any non-geometric 3-manifold and Croke–Kleiner admissible groups. In this paper, under mild conditions on vertex groups of an extended admissible group G, we show that finitely generated abelian subgroups of G are separable (in particular G is residually finite and has solvable word problem), cohomologically good, and the profinite topology on G is efficient. These results extend those in [E. Hamilton, Abelian subgroup separability of Haken 3-manifolds and closed hyperbolic n-orbifolds, Proc. Lond. Math. Soc. (3) 83 2001, 3, 626–646] and [H. Wilton and P. Zalesskii, Profinite properties of graph manifolds, Geom. Dedicata 147 2010, 29–45] from 3-manifold groups to a wider class of graphs of groups. Finally, we establish that G has property (QT) in the sense of Bestvina–Bromberg–Fujiwara.
Funding statement: The author was partially supported by the National Key Program for the development of Mathematics in the period from 2021 to 2030 under grant number B2024-CTT-04.
Acknowledgements
We thank Alex Margolis and Wenyuan Yang for useful conversations. The author is grateful for the insightful and detailed critiques of the referee that have helped improve the exposition of this paper.
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