Geometry of generated groups with metrics induced by their Cayley color graphs

Let $G$ be a group and let $S$ be a generating set of $G$. In this article, we introduce a metric $d_C$ on $G$ with respect to $S$, called the cardinal metric. We then compare geometric structures of $(G,d_C)$ and $(G,d_W)$, where $d_W$ denotes the word metric. In particular, we prove that if $S$ is finite, then $(G,d_C)$ and $(G,d_W)$ are not quasi-isometric in the case when $(G,d_W)$ has infinite diameter and they are bi-Lipschitz equivalent otherwise. We also give an alternative description of cardinal metrics by using Cayley color graphs. It turns out that color-permuting and color-preserving automorphisms of Cayley digraphs are isometries with respect to cardinal metrics.