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We show that for many natural topological groups G (including the group ℤ of integers) there exist compact metric G-spaces (cascades for G = ℤ) which are reflexively representable but not Hilbert representable. This answers a question of T. Downarowicz. The proof is based on a classical example of W. Rudin and its generalizations. A~crucial step in the proof is our recent result which states that every weakly almost periodic function on a compact G-flow X comes from a G-representation of X on reflexive...

An important consequence of a result of Katětov and Morita states that every metrizable space is contained in a complete metrizable space of the same dimension. We give an equivariant version of this fact in the case of a locally compact $\sigma$-compact acting group.

We study free topological groups defined over uniform spaces in some subclasses of the class $\mathrm{𝐍𝐀}$ of non-archimedean groups. Our descriptions of the corresponding topologies show that for metrizable uniformities the corresponding free balanced, free abelian and free Boolean $\mathrm{𝐍𝐀}$ groups are also metrizable. Graev type ultra-metrics determine the corresponding free topologies. Such results are in a striking contrast with free balanced and free abelian topological groups cases (in standard varieties). Another...

Let ⟨G,X,α⟩ be a G-space, where G is a non-Archimedean (having a local base at the identity consisting of open subgroups) and second countable topological group, and X is a zero-dimensional compact metrizable space. Let $⟨H\left({0,1}^{\aleph ₀}\right),{0,1}^{\aleph ₀},\tau ⟩$ be the natural (evaluation) action of the full group of autohomeomorphisms of the Cantor cube. Then (1) there exists a topological group embedding $\phi :G\hookrightarrow H\left({0,1}^{\aleph ₀}\right)$; (2) there exists an embedding $\psi :X\hookrightarrow {0,1}^{\aleph ₀}$, equivariant with respect to φ, such that ψ(X) is an equivariant retract of ${0,1}^{\aleph ₀}$ with respect to φ...