Tree amalgamations and hyperbolic boundaries

TL;DR

This paper investigates how the hyperbolic boundary of a tree amalgamation of hyperbolic graphs relates to the boundaries of its components, establishing conditions for boundary homeomorphism based on factor boundaries.

Contribution

It proves that the hyperbolic boundary of a tree amalgamation depends solely on the boundaries of its factors and characterizes boundary homeomorphism via terminal factorisations.

Findings

Hyperbolic boundary homeomorphism depends only on factor boundaries.

Boundary homeomorphism characterized by terminal factorisations.

Results apply to locally finite quasi-transitive hyperbolic graphs.

Abstract

We look at tree amalgamations of locally finite quasi-transitive hyperbolic graphs and prove that the homeomorphism type of the hyperbolic boundary of such a tree amalgamation only depends on the homeomorphism types of the hyperbolic boundaries of their factors. Additionally, we show that two locally finite quasi-transitive hyperbolic graphs have homeomorphic hyperbolic boundaries if and only if the homeomorphism types of the hyperbolic boundaries of the factors of their terminal factorisations coincide.