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

TL;DR

This paper introduces a new metric called the cardinal metric on groups with generating sets, compares it to the word metric, and explores their geometric relationships and automorphisms via Cayley color graphs.

Contribution

It defines the cardinal metric, compares its geometric properties with the word metric, and characterizes automorphisms of Cayley color graphs as isometries under this metric.

Findings

$(G, d_C)$ and $(G, d_W)$ are not quasi-isometric if $(G, d_W)$ has infinite diameter.

They are bi-Lipschitz equivalent if $(G, d_W)$ has finite diameter.

Color automorphisms of Cayley digraphs are isometries with respect to the cardinal metric.

Abstract

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.