On the isometry group of the Urysohn space

Abstract

We give a general criterion for the (bounded) simplicity of the automorphism groups of certain countable structures and apply it to show that the isometry group of the Urysohn space modulo the normal subgroup of bounded isometries is a simple group.

The isometry group of the bounded Urysohn space is simple

“…Note that we cannot expect bounded simplicity as in the results in [1] as there are isometries of U 1 with arbitrarily small displacement. The proof relies on the properties of an abstract independence relation.…”

“…The proof here follows the same lines as the proof in [1] and we will continue using notions from that paper. In the next section we will establish the following:…”

“…It was shown in [1] that the isometry group of the (general) Urysohn space modulo the subgroup of bounded isometries is a simple group and it was widely conjectured (in particular by M. Rubin and J. Melleray) that the isometry group of the bounded Urysohn space is a simple group. We here prove this conjecture using the approach from [1]: Theorem 1.1. The isometry group of U 1 is abstractly simple.…”

“…The proof relies on the properties of an abstract independence relation. We will continue to use the concepts introduced in [1], in particular the following notion of independence: Definition 1.2. We say that A and C are independent over B, written…”

The automorphism group of Hall’s universal group

“…Groups of automorphisms of such structures have received extensive attention in the literature (see e.g. [9], [10], [5] and [17]). Despite this, not much is known on Aut(H).…”