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

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