Group metrics for graph products of cyclic groups

by Paolini and Shelah. [PaSh:1117]

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


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