[Paper Review] Hyperbolic groups with homeomorphic Gromov boundaries

Alexandre Martin, Jacek Świątkowski|arXiv (Cornell University)|Mar 27, 2013

Geometric and Algebraic Topology6 references3 citations

TL;DR

This paper establishes that the Gromov boundary of a hyperbolic group is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its 1-ended vertex groups in a terminal splitting over finite subgroups. It proves that for graphs of hyperbolic groups over finite edge groups, the boundary topology depends only on the boundary types of the non-elementary vertex groups, and provides a necessary and sufficient condition for two such groups to have homeomorphic boundaries.

ABSTRACT

We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condition for the Gromov boundaries of any two hyperbolic groups to be homeomorphic (in terms of the topology of the boundaries of factors in terminal splittings over finite subgroups).

Motivation & Objective

• To determine whether the Gromov boundary of a hyperbolic group is uniquely determined by the homeomorphism types of the boundaries of its factors in a free product.
• To generalize this result to graphs of hyperbolic groups over finite subgroups, particularly in the context of terminal splittings.
• To establish a necessary and sufficient condition for two hyperbolic groups to have homeomorphic Gromov boundaries, based on the boundary types of their 1-ended vertex groups in terminal splittings.
• To provide a topological model for the Gromov boundary of free products and graphs of groups using tree-like structures and equivariant actions.
• To prove that the boundary components of such groups correspond precisely to the boundaries of their 1-ended vertex groups, under appropriate topological conditions.

Proposed method

• Construct a geometric model $\Gamma = \Gamma(A,B)$ for the free product $G = A*B$ as a tree of spaces, where vertices are replaced by copies of the Cayley graphs of $A$ and $B$.
• Define a boundary $\delta\Gamma$ as the union of the boundary of the Bass-Serre tree $\partial T$ and the equivariant boundary components $\delta_{\text{Stab}}\Gamma$, which are images of $\partial A$ and $\partial B$.
• Equip $\delta\Gamma$ with a $G$-action and a projection $p: \delta\Gamma \to \mathcal{V}(T) \cup \partial T$, allowing identification of boundary components with preimages of vertices.
• Use natural halfspaces in $\delta\Gamma$, defined via edge decompositions of the Bass-Serre tree, to analyze topological separation and connectedness.
• Prove continuity of the equivariant maps $f_{g,j}: \partial A_j \to \delta\Gamma$, which embed the boundaries of vertex groups into the full boundary.
• Apply topological arguments—particularly the use of open-closed partitions and connectedness— to show that $\partial G_v \subset \delta\Gamma$ is a connected component iff the corresponding vertex group is 1-ended.

Experimental results

Research questions

• RQ1 Is the topology of the Gromov boundary of a free product $G_1 * G_2$ of two infinite hyperbolic groups uniquely determined by the homeomorphism types of $\partial G_1$ and $\partial G_2$?
• RQ2 Can the Gromov boundary of a graph of hyperbolic groups over finite edge groups be characterized solely by the homeomorphism types of the boundaries of its non-elementary vertex groups?
• RQ3 What is a necessary and sufficient condition for two hyperbolic groups to have homeomorphic Gromov boundaries, in terms of their terminal splittings over finite subgroups?
• RQ4 How do the connected components of the Gromov boundary of a hyperbolic group relate to the boundaries of its 1-ended vertex groups in a terminal splitting?
• RQ5 To what extent can the boundary topology of a hyperbolic group be reconstructed from its terminal splitting structure over finite subgroups?

Key findings

• The Gromov boundary of the free product $G_1 * G_2$ of two infinite hyperbolic groups is homeomorphic to a space constructed from the boundaries of $G_1$ and $G_2$, with the topology fully determined by the homeomorphism types of $\partial G_1$ and $\partial G_2$.
• For graphs of hyperbolic groups over finite subgroups, the Gromov boundary $\partial(\pi_1\mathcal{G})$ is homeomorphic if and only if the sets $h(\mathcal{G}_1)$ and $h(\mathcal{G}_2)$ of homeomorphism types of 1-ended vertex group boundaries are equal.
• The connected components of $\partial G$ for a hyperbolic group $G$ in a terminal splitting are precisely the singletons corresponding to ends of the Bass-Serre tree and the images of the boundaries of 1-ended vertex groups.
• A point $\eta \in \partial T$ is a connected component of $\delta\Gamma$ if and only if it is isolated in the boundary topology, which occurs when it lies in the boundary of a tree with trivial edge stabilizers.
• If a vertex group $A_j$ is 1-ended, then its boundary $\partial A_j$ embeds continuously into $\delta\Gamma$ as a connected component, and this component is homeomorphic to $\partial A_j$.
• The full Gromov boundary $\partial G$ is homeomorphic to the boundary of the geometric model $\delta\Gamma$, and this homeomorphism is $G$-equivariant.