Trees of Graphs as Boundaries of Hyperbolic Groups

Abstract

We characterize those 1-ended word hyperbolic groups whose Gromov boundaries are homeomorphic to trees of graphs (i.e. to inverse limits of graphs that have particularly simple finitary descriptions). These are groups with the simplest connected Gromov boundaries of topological dimension 1. The characterization is expressed in terms of algebraic properties of the Bowditch JSJ splitting of the corresponding groups (i.e. the canonical JSJ splitting over 2-ended subgroups).