Maximal equivariant compactification of the Urysohn spaces and other metric structures

Tomás Ibarlucía, Michael Megrelishvili

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3 Scopus citations


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 S1(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 Lp spaces are metrizable. We also prove a uniform version of Effros' Theorem for isometric actions of Roelcke precompact Polish groups.

Original languageEnglish
Article number107599
JournalAdvances in Mathematics
StatePublished - 26 Mar 2021

Bibliographical note

Publisher Copyright:
© 2021 Elsevier Inc.


The first author's research was partially supported by the ANR contract AGRUME ( ANR-17-CE40-0026 ). The second author's research was supported by a grant of the Israel Science Foundation 1194/19 .

FundersFunder number
Israel Science Foundation1194/19


    • Effros' theorem
    • Equivariant compactification
    • Gromov compactification
    • Katětov functions
    • Separably categorical structures
    • Urysohn and Gurarij spaces


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