On stability of ultrametrically injective hulls

The injective hull, or known tight span, of its object of a concrete category usually has many nice geometric or algebraic properties. In this paper, we first investigate the stability of the injective hulls of ultrametric spaces by making use of isometric embeddings. To that end, we prove that there exists an isometric embedding between their two injective hulls of an ultrametric space and its subspace, and further present an extension result for rough nets via isometric embeddings. This result yields a sharp stability estimate: the Gromov-Hausdorff ultrametric of the injective hulls of two ultrametric spaces is at most twice the Gromov-Hausdorff ultrametric between themselves. As a direct consequence, we obtain that two injective hulls are strongly roughly isometric with respect to the Gromov-Hausdorff ultrametric if so are the original spaces. In addition, we give a characterization of an ultrametric space that is a rough net in its injective hull.