Kaleidoscopic groups and the generic point property

Duchesne, Monod and Wesolek described how to associate a group acting on a certain one-dimensional space to each permutation group of countable degree. This is called its kaleidoscopic group. We study such groups under the lens of their continuous actions on compact spaces, and determine which dynamical properties of are preserved in the kaleidoscopic construction. Doing so requires a novel structural Ramsey theorem and produces a new class of examples exhibiting a poorly understood phenomenon. Indeed, a Polish group has the generic point property if every minimal flow has a comeager orbit. It has been shown that any Polish group with metrizable universal minimal flow has the generic point property, but the converse does not hold, as shown by Kwiatkowska. Until recently, Kwiatkowska’s counterexample was the only known instance of this phenomenon, but we find a large class of new counterexamples among kaleidoscopic groups.
This is joint work with Todor Tsankov.