The Cayley graph of $\mathbb{Z}$ for the generating set $\{1\}$ is a line and its word metric is $\mathrm{\mid }m-n\mathrm{\mid }$

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).

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).

Define $d_{\mathbb{R}}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ by $d_{\mathbb{R}}(x,y):=\mathrm{\mid }x-y\mathrm{\mid }$. (The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded).

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).