A Combination Theorem for Trees of Metric Bundles

TL;DR

This paper establishes a new combination theorem for trees of metric bundles, showing that the total space is hyperbolic under specific conditions, extending previous work on hyperbolic spaces and complexes of groups.

Contribution

It introduces a unifying framework for trees of metric bundles and proves a hyperbolicity criterion that generalizes prior results by Bestvina-Feighn and Mj-Sardar.

Findings

Total space of a tree of metric bundles is hyperbolic under given conditions.

Provides a new combination theorem for complexes of groups.

Extends existing theories on hyperbolic metric spaces.

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 $1.3$).