The infinite dihedral group is quasi-isometric to $\mathbb{Z}$, and to $\mathbb{Z}\times \mathbb{Z}\mathrm{/}2$

Given: The objects and hypotheses in the Example.

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set $\{\{g,gs\}:g\in G,s\in (S\cup S^{-1})\setminus \{e\}\}$ (The Cayley graph of a group with respect to a subset).

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 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).

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 finitely generated group is quasi-isometric to a metric space when its word metric for some, equivalently every, finite generating set is (The quasi-isometry type of a finitely generated group).

The group with presentation (Group presentation by generators and relations).

If the evaluation of every $r\in R$ under $u$ is $e_H$, then there is a unique homomorphism (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group).

Let $u,v\in F(X)$ and put $N=⟨\text{\hspace{0.17em}⁣}⟨R⟩\text{\hspace{0.17em}⁣}{⟩}_{F(X)}$. The words $u$ and $v$ represent the same element of $⟨X\mid R⟩$ if and only if (In $⟨X\mid R⟩$, the words $u$ and $v$ represent the same element if and only if $u^{-1}v\in ⟨\text{\hspace{0.17em}⁣}⟨R⟩\text{\hspace{0.17em}⁣}⟩$).

$\mathrm{Dih}(A)=A⋊C_2\mathrm{.}$ ( The generalized dihedral group $\mathrm{Dih}(A)=A⋊C_2$ for an abelian group $A$).