The word metrics of $\mathbb{Z}$ for $\{1\}$ and for $\{2,3\}$ differ at $1$ and are bilipschitz equivalent

Given: The objects and hypotheses in the Example.

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 identity map between the word metrics of two finite generating sets of a group is a bilipschitz equivalence (The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence).

A map is a bilipschitz embedding when $c^{-1}d(x,x^{\mathrm{\prime }})\le d(f(x),f(x^{\mathrm{\prime }}))\le cd(x,x^{\mathrm{\prime }})$ for some $c>0$, and a bilipschitz equivalence when it is a bijective such map with bilipschitz inverse (Bilipschitz embeddings and bilipschitz equivalences of metric spaces).

• $d$ and $d^{\mathrm{\prime }}$ are topologically equivalent if they have the same metric topology: ${\mathcal{T}}_d={\mathcal{T}}_{d^{\mathrm{\prime }}}\mathrm{.}$ - $d$ and $d^{\mathrm{\prime }}$ are uniformly equivalent if for every real $\epsilon >0$ there are reals $\delta >0$ and ${\delta }^{\mathrm{\prime }}>0$ such that, for all $x,y\in X$, $d(x,y)<\delta ⟹d^{\mathrm{\prime }}(x,y)<\epsilon \text{and}{d}^{\mathrm{\prime }}(x,y)<{\delta }^{\mathrm{\prime }}⟹d(x,y)<\epsilon \mathrm{.}$ - $d$ and $d^{\mathrm{\prime }}$ are Lipschitz equivalent if there are reals $\alpha ,\beta >0$ with $\alpha d(x,y)\le d^{\mathrm{\prime }}(x,y)\le \beta d(x,y)\text{for all }x,y\in X\mathrm{.}$ (Topologically, uniformly and Lipschitz equivalent metrics on a set).