For every topological group G one can define the universal minimal compact
G-space X=MG 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 S1, then MG can be identified with S1 (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 MG is not 3-transitive.
