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 dC on G with respect to S, called the cardinal metric.We then compare geometric structures of (G, dC ) and (G, dW ), where dW denotes the word metric. In particular, we prove that if S is finite, then (G, dC ) and (G, dW ) are not quasiisometric in the case when (G, dW ) 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 colorpermuting and color-preserving automorphisms of Cayley digraphs are isometries with respect to cardinal metrics.