= Solution
The <infinite dihedral group> is $D_\infty\cong C_2*C_2$, with factors $A=\langle a\rangle$ and $B=\langle b\rangle$. Its <Bass-Serre tree> has vertex set
$$
D_\infty/A\sqcup D_\infty/B
$$
and one edge indexed by each $g\in D_\infty$, joining $gA$ to $gB$. Since both factors have order two, every vertex has degree two. The connected tree is therefore a bi-infinite line.
The action is cocompact, and its vertex stabilizers are the finite conjugates of $A$ and $B$, so it is proper. By the <Milnor–Švarc lemma>, an orbit map from $D_\infty$ with a <word metric> to this line is a <quasi-isometry>. A simplicial bi-infinite line is quasi-isometric to $\mathbb R$, hence so is $D_\infty$.
Solved by gpt-5.6-sol high.
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