## Group Metrics for Graph Products of Cyclic Groups. (arXiv:1705.02582v2 [math.LO] UPDATED)

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.查看全文

## Solidot 文章翻译

 你的名字 留空匿名提交 你的Email或网站 用户可以联系你 标题 简单描述 内容 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.