Trees of Graphs as Boundaries of Hyperbolic Groups

TL;DR

This work provides a complete characterization of when a 1-ended word hyperbolic group G has a Gromov boundary ∂G homeomorphic to a tree of graphs. The authors develop a robust framework of tree systems and their limits to realize ∂G as a limit object, and connect this with the reduced Bowditch JSJ splitting, introducing rigid cluster factors and abstract/flexible factors via Whitehead graphs. They prove the equivalence between boundary type (tree of graphs or regular tree of 2-connected graphs) and algebraic data (virtually free rigid cluster factors) and give an explicit description of the boundary as a regular tree of graphs arising from a graphical connecting system R_G associated to the Bowditch JSJ splitting. The results yield concrete structural descriptions, including detailed cases with θ-graphs and Whitehead graphs, and identify a broad class of examples (e.g., right-angled Coxeter groups) where the boundary can be realized as explicit regular trees of graphs. Overall, the paper links topological boundaries to JSJ-based decompositions, enabling explicit, computable descriptions of ∂G in dimension 1 with broad implications for understanding hyperbolic group boundaries.

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).

Figures (7)

• Figure 1: "Graph of surfaces" $M_1$ and JSJ splitting $\mathcal{G}_1$ of its fundamental group $G_1=\pi_1M_1$.
• Figure 2: "Graph of surfaces" $M_2$ and JSJ splitting $\mathcal{G}_2$ of its fundamental group $G_2=\pi_1M_2$.
• Figure 3: "Graph of surfaces" $M_3$ and splittings $\mathcal{G}_3$ and $\mathcal{G}_3^r$ of its fundamental group $G_3=\pi_1M_3$.
• Figure 4: "Graph of surfaces" $M_4$ and splittings $\mathcal{G}_4$ and $\mathcal{G}_4^r$ of its fundamental group $G_4=\pi_1M_4$.
• Figure 5:

Theorems & Definitions (185)

• Theorem 1
• Theorem 2
• Definition 2.1
• Example 2.2
• Definition 2.3
• Proposition 2.4
• Lemma 2.7
• Proposition 2.8
• Theorem 2.9
• Theorem 3.2