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Again as the title suggests is the above true? If not what examples are there. My only background regarding these spaces have to deals with Hadamard manifolds, which is why I am seeing if its true in greater generality.

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  • $\begingroup$ There are multiple definitions of Gromov hyperbolic spaces. Some of them require the space to be geodesic to begin with. Others, more general ('coarse") definitions, do not. $\endgroup$ – user357151 Jan 12 '18 at 5:41
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The point with Gromov hyperbolic spaces is that you have no control on their local behaviours. One way to translate this idea formally is that any bounded metric space is Gromov hyperbolic. Counterexamples to your question may be:

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