A Combination Theorem for Trees of Metric Bundles

Abstract

Motivated by the work of Bestvina-Feighn ([BF92]) and Mj-Sardar ([MS12]), we define trees of metric bundles subsuming both the trees of metric spaces and the metric bundles. Then we prove a combination theorem for these spaces. More precisely, we prove that the total space of a tree of metric bundles is hyperbolic if the following hold (see Theorem $1.5$). $(1)$ The fibers are uniformly hyperbolic metric spaces and the base is also hyperbolic metric space, $(2)$ barycenter maps for the fibers are uniformly coarsely surjective, $(3)$ the edge spaces are uniformly qi embedded in the corresponding fibers and $(4)$ the Bestvina-Feighn hallway flaring condition is satisfied. As an application, we provide a combination theorem for certain complexes of groups over finite simplicial complex (see Theorem $1.3$).