Separability properties of extended admissible groups

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.