Abstract
We complement the characterization of the graph products of cyclic groups G(Γ,p) admitting a Polish group topology of [9] with the following result. Let G=G(Γ,p), then the following are equivalent:(i)there is a metric on Γ which induces a separable topology in which EΓ is closed;(ii)G(Γ,p) is embeddable into a Polish group;(iii)G(Γ,p) is embeddable into a non-Archimedean Polish group. We also construct left-invariant separable group ultrametrics for G=G(Γ,p) and Γ a closed graph on the Baire space, which is of independent interest.
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