The work of Kechris, Pestov and Todorcevic has revealed a close connection between Ramsey theory of model-theoretic structures and topological dynamics of their automorphism groups, providing a rich source of examples of so-called extremely amenable topological groups. Next to Ramsey theory, there is a second pathway to extreme amenability of topological groups: the phenomenon of measure concentration, which was exhibited in the 1970s by Milman (extending an idea going back to the work of Levy) and linked with extreme amenability in Milman's groundbreaking joint work with Gromov. In the late 1990s, Gromov offered a far-reaching generalization of the measure concentration phenomenon: the concentration topology on the space of metric measure spaces. Inspired by the striking applications of measure concentration in topological dynamics, Pestov suggested to study manifestations of Gromov's concentration to non-trivial spaces in the context of transformation groups.
In my talk, I will give an introduction to Gromov's metric measure geometry and report on recent progress concerning the case of topological groups, including automorphism group of both discrete structures and their metric counterparts.