Topological dynamics of kaleidoscopic groups

TL;DR

This paper investigates the topological dynamics of kaleidoscopic groups, showing how their universal minimal flows relate to the properties of the original permutation groups, with implications for metrizable flows and model theory.

Contribution

It establishes a connection between the dynamics of kaleidoscopic groups and their generating groups, providing new criteria for the metrizability of their universal minimal flows.

Findings

UMF of $ ext{K}( ext{Gamma})$ is metrizable iff $ ext{Gamma}$ is oligomorphic and its UMF is metrizable.

Provides concrete calculations of UMFs using model-theoretic frameworks.

Identifies classes of non-metrizable UMFs with a comeager orbit.

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.