Automorphism groups of universal diversities

Abstract: We prove that the automorphism group of the Urysohn diversity is a universal Polish group. Furthermore we show that the automorphism group of the rational Urysohn diversity has ample generics, a dense conjugacy class and that it embeds densely into the automorphism group of the (full) Urysohn diversity. It follows that this latter group also has a dense conjugacy class.