The universal minimal flow of the homeomorphism group of the Menger curve

The Menger universal curve M is a well known compact connected metric fractal, usually described as a decreasing intersection of cubical complexes. By adapting techniques developed for manifolds by Gutman-Tsankov-Zucker, we prove that the group G of self-homeomorphisms of M has a nonmetrizable universal minimal flow M(G). This is a compact space on which G acts minimally, and such that for any other compact space Y on which G acts minimally, there is a continuous equivariant surjection from M(G) onto Y. Much work has been done in recent years to study M(G) for a variety of Polish groups G, as this provides a natural dividing line in the class of Polish groups: groups G for which M(G) is nonmetrizable are considered to have wild dynamics, while groups G for which M(G) is metrizable are considered to have tame dynamics. This is joint work with Gianluca Basso and Andrea Vaccaro.