Croke–Kleiner Admissible Groups: Property (QT) and Quasiconvexity

Abstract

Croke–Kleiner admissible groups first introduced by Croke and Kleiner [CK02] belong to a particular class of graphs of groups, which generalizes fundamental groups of three-dimensional graph manifolds. In this paper, we show that if G is a Croke–Kleiner admissible group, then a finitely generated subgroup of G has finite height if and only if it is strongly quasiconvex. We also show that if G↷X is a flip CKA action, then G is quasiisometrically embedded into a finite product of quasitrees. With further assumption on the vertex groups of the flip CKA action G↷X, we show that G satisfies property (QT) introduced by Bestvina, Bromberg, and Fujiwara [BBF21].