Maximal equivariant compactification of the Urysohn spaces and other metric structures

We study isometric $G$-spaces and the question of when their maximal equivariant compactification is the Gromov compactification (meaning that it coincides with the compactification generated by the distance functions to points). Answering questions of Pestov, we show that this is the case for the Urysohn sphere and related spaces, but not for the unit sphere of the Gurarij space. We show that the maximal equivariant compactification of a separably categorical metric structure $M$ under the action of its automorphism group can be identified with the space $S_1(M)$ of 1-types over $M$, and is in particular metrizable. This provides a unified understanding of the previous and other examples. In particular, the maximal equivariant compactifications of the spheres of the Gurarij space and of the $L^p$ spaces are metrizable. We also prove a uniform version of Effros' Theorem for isometric actions of Roelcke precompact Polish groups.