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

Given: The hypotheses of the Statement.

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 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).

Word length is defined on every element and satisfies $\mathrm{\mid }gh{\mathrm{\mid }}_S\le \mathrm{\mid }g{\mathrm{\mid }}_S+\mathrm{\mid }h{\mathrm{\mid }}_S$, $\mathrm{\mid }{g}^{-1}{\mathrm{\mid }}_S=\mathrm{\mid }g{\mathrm{\mid }}_S$, and $\mathrm{\mid }g{\mathrm{\mid }}_S=0$ exactly when $g$ is the identity (Word length is defined on every element and satisfies the subadditivity, inversion and vanishing laws).

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).

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).

A group is finitely generated when some finite subset generates it (Finitely generated groups).

A set $A$ is finite when $A\approx n$ for some $n\in \mathbb{N}$. (The cardinality $\mid A\mid$ of a finite set).