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

Group Metrics for Graph Products of Cyclic Groups