Some geometric and dynamical properties of the Urysohn space

This is a survey article about the geometry and dynamical properties of the Urysohn space. Most of the results presented here are part of the authorʹs Ph.D. thesis and were published in the articles [J. Melleray, Stabilizers of closed sets in the Urysohn space, Fund. Math. 189 (1) (2006) 53–60; J. Melleray, Compact metrizable groups are isometry groups of compact metric spaces, Proc. Amer. Math. Soc. 136 (4) (2008) 1451; J. Melleray, On the geometry of Urysohnʹs universal metric space, Topology Appl. 154 (2007) 384–403]; a few results are new, most notably the fact that Iso ( U ) is not divisible.