Building on earlier work of Katětov, Uspenskij proved in [8] that the group of isometries of Urysohn's universal metric space 𝕌, endowed with the pointwise convergence topology, is a universal Polish group (i.e. it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group G, there exists a closed subset F of 𝕌 such that G is topologically isomorphic to the group of isometries of 𝕌 which map F onto itself.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-1-4,
author = {Julien Melleray},
title = {Stabilizers of closed sets in the Urysohn space},
journal = {Fundamenta Mathematicae},
volume = {189},
year = {2006},
pages = {53-60},
zbl = {1089.22019},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-1-4}
}
Julien Melleray. Stabilizers of closed sets in the Urysohn space. Fundamenta Mathematicae, Tome 189 (2006) pp. 53-60. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-1-4/