The Polish topology of the isometry group of the infinite dimensional hyperbolic space

TL;DR

This paper explores the topological and dynamical properties of the isometry group of infinite dimensional hyperbolic space, revealing automatic continuity, minimality, and boundary identifications within the framework of Polish groups.

Contribution

It establishes automatic continuity, minimality of the topology, and characterizes the universal boundaries and flows of the isometry group of infinite dimensional hyperbolic space.

Findings

Automatic continuity of the group homomorphisms

Identification of the universal Furstenberg boundary

Description of the universal minimal flow

Abstract

We consider the isometry group of the infinite dimensional separable hyperbolic space with its Polish topology. This topology is given by the pointwise convergence. For non-locally compact Polish groups, some striking phenomena like automatic continuity or extreme amenability may happen. Our leading idea is to compare this topological group with usual Lie groups on one side and with non-Archimedean infinite dimensional groups like $\mathcal{S}_\infty$, the group of all permutations of a countable set on the other side. Our main results are Automatic continuity (any homomorphism to a separable group is continuous), minimality of the Polish topology, identification of its universal Furstenberg boundary as the closed unit ball of a separable Hilbert space with its weak topology, identification of its universal minimal flow as the completion of some suspension of the action of the additive group of the reals on its universal minimal flow. All along the text, we lead a parallel study with the sibling group of isometries of a separable Hilbert space.