Topological dynamics of kaleidoscopic groups

Kaleidoscopic groups are a class of permutation groups recently introduced by Duchesne, Monod, and Wesolek. Starting with a permutation group Gamma, the kaleidoscopic construction produces another permutation group X(Gamma) which acts on a Wazewski dendrite (a densely branching tree-like compact space). In this paper, we study how the topological dynamics of X(Gamma) can be expressed in terms of the one of Gamma, when the group Gamma is transitive. By proving a Ramsey theorem for decorated rooted trees, we show that the universal minimal flow (UMF) of X(Gamma) is metrizable iff Gamma is oligomorphic and the UMF of Gamma is metrizable. More generally, we give concrete calculations, in an appropriate model-theoretic framework, of the UMF of X(Gamma) when the UMF of a point stabilizer Gamma c has a comeager orbit. Our results also give a large class of examples of non-metrizable UMFs with a comeager orbit. These results extend previous work of Kwiatkowska and Duchesne about the full homeomorphism groups.(c) 2023 Elsevier Inc. All rights reserved.