The Cayley graph of ${\mathbb{Z}}^n$ for the standard basis is the integer lattice, and its word metric is the sum of coordinate differences

Given: The objects and hypotheses in the Example.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

A free abelian group on a set $X$ is an abelian group $A(X)$ together with a map $i:X\to A(X)$ such that, for every abelian group $B$ and every function $u:X\to B$, there is a unique group homomorphism $\widehat{u}:A(X)\to B$ satisfying (Free abelian group on a set).

The word length $\mathrm{\mid }g{\mathrm{\mid }}_S$ is the least $n$ such that $g$ is a product of $n$ elements of $S\cup S^{-1}$ (Word length of a group element with respect to a generating set).

The word metric of $G$ with respect to $S$ is $d_S(g,h)=\mathrm{\mid }{g}^{-1}h{\mathrm{\mid }}_S$ (The word metric of a group with respect to a generating set).

The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph (The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph).

$d_1(x,y):={\sum }_{k<n}\mathrm{\mid }{x}_k-y_k\mathrm{\mid },d_2(x,y):=\sqrt{{\sum }_{k<n}(x_k-y_k{)}^2},d_{\mathrm{\infty }}(x,y):=\max \{\mathrm{\mid }{x}_k-y_k\mathrm{\mid }:k<n\}\mathrm{.}$ (${\mathbb{R}}^n$ as the set of functions $n\to \mathbb{R}$, and $d_1$, $d_2$, $d_{\mathrm{\infty }}$ are metrics on it).