On universal minimal compact G-spaces

TL;DR

This paper investigates the properties of universal minimal compact G-spaces for topological groups, showing limitations on their structure and symmetry, especially for groups acting on higher-dimensional manifolds.

Contribution

It proves that the universal minimal compact G-space cannot be a Hilbert cube or higher-dimensional manifolds, and that the G-action on M_G is never 3-transitive.

Findings

M_G is the circle S^1 for the group of orientation-preserving homeomorphisms of S^1

The circle cannot be replaced by higher-dimensional compact manifolds or the Hilbert cube

The action of G on M_G is not 3-transitive

Abstract

For every topological group G one can define the universal minimal compact G-space X=M_G characterized by the following properties: (1) X has no proper closed G-invariant subsets; (2) for every compact G-space Y there exists a G-map X-->Y. If G is the group of all orientation-preserving homeomorphisms of the circle S^1, then M_G can be identified with S^1 (V. Pestov). We show that the circle cannot be replaced by the Hilbert cube or a compact manifold of dimension >1. This answers a question of V. Pestov. Moreover, we prove that for every topological group G the action of G on M_G is not 3-transitive.