### Group metrics for graph products of cyclic groups

by Paolini and Shelah. [PaSh:1117]

Topology and its Applications, 2017

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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