# Group Metrics for Graph Products of Cyclic Groups

## TL;DR

This paper characterizes when graph products of cyclic groups can be embedded into Polish groups, providing conditions related to metrics on the underlying graph and constructing specific ultrametrics.

## Contribution

It establishes equivalences for embeddability of graph products into Polish groups and constructs new ultrametrics for certain graph products.

## Key findings

- Equivalence between graph metric properties and embeddability into Polish groups.
- Construction of left-invariant separable ultrametrics for specific graph products.
- Characterization of when graph products admit a Polish group topology.

## Abstract

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.