Maximal equivariant compactification of the Urysohn spaces and other\n metric structures

2020/01/20 by Tomás Ibarlucía, Michael Megrelishvili, Ibarlucía, Tomás +1
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #General Topology (math.GN) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2001.07228

openalex publication_date 2020/01/20 · openalex created_date 2021/02/01 · openalex updated_date 2026/07/28

Abstract

We study isometric G-spaces and the question of when their maximal\nequivariant compactification is the Gromov compactification (meaning that it\ncoincides with the compactification generated by the distance functions to\npoints). Answering questions of Pestov, we show that this is the case for the\nUrysohn sphere and related spaces, but not for the unit sphere of the Gurarij\nspace.\n We show that the maximal equivariant compactification of a separably\ncategorical metric structure M under the action of its automorphism group can\nbe identified with the space S₁(M) of 1-types over M, and is in particular\nmetrizable. This provides a unified understanding of the previous and other\nexamples. In particular, the maximal equivariant compactifications of the\nspheres of the Gurarij space and of the L^p spaces are metrizable.\n We also prove a uniform version of Effros' Theorem for isometric actions of\nRoelcke precompact Polish groups.\n

Citations

• The dynamical hierarchy for Roelcke precompact Polish groups

Related

• Maximal equivariant compactifications
• A natural compactification of the Gromov-Hausdorff space
• The maximal G-compactifications of G-spaces with special actions
• Geometry in Urysohn's universal metric space
• Isometric Embeddings of Bounded Metric Spaces into the Gromov-Hausdorff Class