Topology of the isometry group of the Urysohn space

TL;DR

This paper proves that the isometry group of the Urysohn space is topologically equivalent to a separable Hilbert space, using classical infinite-dimensional geometry results and a key extension lemma.

Contribution

It establishes the homeomorphism between the Urysohn space's isometry group and a Hilbert space, and explores isometry subgroup relationships for finite subsets.

Findings

Isometry group is homeomorphic to a separable Hilbert space

Extension lemma for finite metric spaces

Relationships between isometry groups fixing finite subsets

Abstract

Using classical results of infinite-dimensional geometry, we show that the isometry group of the Urysohn space, endowed with its usual Polish group topology, is homeomorphic to the separable Hilbert space. The proof is basedon a lemma about extensions of metric spaces by finite metric spaces, which wealso use to investigate (answering a question of I. Goldbring) the relationship, when A,B are finite subsets of the Urysohn space, between the group of isometries fixing A pointwise, the group of isometries fixing B pointwise, and the group of isometries fixing the intersection of A and B pointwise.