Topological dynamics of kaleidoscopic groups

Abstract

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 $\mathcal{K}(\Gamma)$ which acts on a Wa\.{z}ewski dendrite (a densely branching tree-like compact space). In this paper, we study how the topological dynamics of $\mathcal{K}(\Gamma)$ 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 $\mathcal{K}(\Gamma)$ 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 $\mathcal{K}(\Gamma)$ 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.