Group Metrics for Graph Products of Cyclic Groups

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