Abstract. We define and study the notion of ample metric generics for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal in [KR07]. Our work is based on the concept of a Polish topometric group, defined in this article. Using Kechris and Rosendal's work as a guide, we explore consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group Iso(U 1 ) of the bounded Urysohn space, the unitary group U (ℓ 2 ) of a separable Hilbert space, and the automorphism group Aut([0, 1], λ) of the Lebesgue measure algebra on [0, 1]. We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from Aut([0, 1], λ) into a separable topological group is continuous.

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 author's 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 Urysohn's universal metric space, Topology Appl. 154 (2007) 384-403]; a few results are new, most notably the fact that Iso(U) is not divisible.