A bijective quasi-isometry between word metric spaces of finitely generated groups is a bilipschitz equivalence

Given: The hypotheses of the Statement.

A subset is coarsely dense when every point of the space is within a fixed distance of it, and a quasi-isometry is a coarse Lipschitz map admitting a coarse Lipschitz quasi-inverse (Coarsely dense subsets, quasi-inverses and quasi-isometries).

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

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).

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).

• $f$ 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).