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On universal-homogeneous hyperbolic graphs and spaces and their isometry groups

The Urysohn space is the unique separable metric space that is universal and homogeneous for finite metric spaces, i.e., it embeds any finite metric space any isometry between finite subspaces extends to an isometry of the whole space. We here consider the existence of a universal-homogeneous hyperbolic space. We show that for $δ>0$ there is no $δ$-hyperbolic space which is universal and homogeneous in the above sense for all finite $δ$-hpyerbolic spaces. We then show that for any $δ\geq 0$ and any countable class $\mathcal{C}$ of $δ$-hyperbolic spaces with countably many distinguished $δ$-closed subspaces there exists a $δ$-hyperbolic metric space $\mathbb{H}_\mathcal{C}$ such that every $X\in \mathcal{C}$ can be embedded into $\mathbb{H}_\mathcal{C}$ as a $δ$-closed subspace and any isometry between distinguished $δ$-closed subspaces extends to an isometry of $\mathbb{H}_\mathcal{C}$. If $\mathcal{C}$ consists of $δ$-hyperbolic geodesic spaces, then $\mathbb{H}_\mathcal{C}$ contains the quasi-tree of spaces as defined by Bestvina et al.. For $\mathcal{C}_δ$ the class of all finite $δ$-hyperbolic spaces with rational distances or the class of finite $δ$-hyperbolic graphs, the limit $\mathbb{H}_δ$ is a $δ$-hyperbolic space (or graph, respectively) universal for all finite $δ$-hyperbolic spaces with rational distances (or finite $δ$-hyperbolic graphs) and such that any isometry between $δ$-closed subspaces extends to an isometry of $\mathbb{H}_δ$. We show that the isometry group of $\mathbb{H}_δ$ does not contain elements of bounded displacement and has no dense conjugacy class.