The isometry group of the bounded Urysohn space is simple (Q2854230)

The bounded Urysohn space of diameter 1, \(\mathbb{U}_1\), is the unique complete homogeneous separable metric space of diameter \(1\) which embeds every finite metric space of diameter \(1\). In this note, the authors prove a fact that has been widely conjectured, namely that the isometry group of \(\mathbb{U}_1\) is (abstractly) simple.NEWLINENEWLINEThe proof of this result uses the same approach as an earlier work by the same authors [J. Lond. Math. Soc., II. Ser. 87, No. 1, 289--303 (2013; Zbl 1273.03136)], in which they proved that the isometry group of the general Urysohn space \(\mathbb{U}\), modulo the subgroup of bounded isometries, is a simple group. (The general Urysohn space \(\mathbb{U}\) is the unique complete homogeneous separable metric space which embeds every finite metric space.)