Abstract 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.

中文翻译：

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生成组的几何，其度量由其 Cayley 颜色图诱导

摘要 设 G 为群，S 为 G 的生成集。本文介绍 G 上关于 S 的一个度量 dC，称为基数度量。然后比较 (G, dC) 和 ( G, dW)，其中 dW 表示词度量。特别地，我们证明如果 S 是有限的，那么 (G, dC) 和 (G, dW) 在 (G, dW) 具有无限直径的情况下不是拟等距的，否则它们是双李普希茨等价的。我们还使用 Cayley 颜色图给出了基数度量的替代描述。事实证明，凯莱有向图的颜色置换和颜色保持自同构是关于基数度量的等距。