Nonreduction of relations in the Gromov space to Polish actions. (English) Zbl 1455.03061

Summary: We show that in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov-Hausdorff distance cannot be reduced to the equivalence relation defined by any Polish action.


03E15 Descriptive set theory
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
54E50 Complete metric spaces
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