On Graev type ultra-metrics

Abstract: We study Graev ultra-metrics which were introduced by Gao. We show that the free non-archimedean balanced topological group defined over an ultra-metric space is metrizable by a Graev ultra-metric. We prove that the Graev ultra-metric has a maximal property. Using this property, among others, we show that the Graev ultra-metric associated with an ultra-metric space $(X,d)$ with diameter$\leq 1$ coincides with the ultra-metric $\hat{d}$ of Savchenko and Zarichnyi.