A coarse Lipschitz map between word metric spaces of finitely generated groups is Lipschitz

Given: The hypotheses of the Statement.

A map is $(L,C)$-coarse Lipschitz when $d(f(x),f(x^{\mathrm{\prime }}))\le Ld(x,x^{\mathrm{\prime }})+C$, and an $(L,C)$-quasi-isometric embedding when in addition $L^{-1}d(x,x^{\mathrm{\prime }})-C\le d(f(x),f(x^{\mathrm{\prime }}))$ (Coarse Lipschitz maps and quasi-isometric embeddings).

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).

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

• $f$ is Lipschitz with constant $L$, where $L\in \mathbb{R}$ and $L\ge 0$, if $d_Y(f(x),f(x^{\mathrm{\prime }}))\le L{d}_X(x,x^{\mathrm{\prime }})\text{for all }x,x^{\mathrm{\prime }}\in X\mathrm{.}$ $f$ is Lipschitz if it is Lipschitz with some such constant. (Lipschitz map, $\alpha$-Hölder map for rational $0<\alpha \le 1$, and contraction).