Tree amalgamations and hyperbolic boundaries

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.

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