Topological dynamics of Polish groups has interesting aspects not present in dynamics of locally compact groups. For example, there exist Polish groups whose all continuous actions on compact spaces have fixed points. Groups of this type, called extremely amenable, were first constructed by Herer and Christensen using certain submeasures. Later, Gromov and Milman made a connection between extreme amenability and the concentration of measure phenomenon from probability theory.
I will describe the above developments. In this context, I will present a new concentration of measure theorem inspired by geometric ideas related to the Loomis-Whitney theorem. I will describe the dynamical consequences, in the spirit of Gromov and Milman, of our concentration of measure theorem. These consequences generalize the Herer-Christensen result mentioned above as well as related results of Glasner and Pestov. All this depends on a new geometric classification of submeasures, which I will outline. This is a joint work with F. Martin Schneider.