Concentration of measure and whirly actions of Polish groups

A weakly continuous near-action of a Polish group G on a standard Lebesgue measure space (X,μ) is whirly if for every A ⊆ X of strictly positive measure and every neighbourhood V of identity in G the set V A has full measure. This is a strong version of ergodicity, and locally compact groups never admit whirly actions. On the contrary, every ergodic near-action by a Polish Lévy group in the sense of Gromov and Milman, such as U(l), is whirly (Glasner–Tsirelson–Weiss). We give examples of closed subgroups of the group Aut (X,μ) of measure preserving automorphisms of a standard Lebesgue measure space (with the weak topology) whose tautological action on (X,μ) is whirly, and which are not Lévy groups, thus answering a question of Glasner and Weiss. §