The subgroup $2\mathbb{Z}\times \mathbb{Z}$ has index two in ${\mathbb{Z}}^2$ and its inclusion is a quasi-isometry

Given: The objects and hypotheses in the Example.

A subgroup of finite index in a finitely generated group is finitely generated and its inclusion is a quasi-isometry (A subgroup of finite index in a finitely generated group is finitely generated, and its inclusion is a quasi-isometry).

$[G:H]:=\mathrm{\mid }G\mathrm{/}H\mathrm{\mid }$ (The coset set $G\mathrm{/}H$ and the index $[G:H]$ of a subgroup).

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).

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 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).