The pants graph as a combinatorial model for the Teichmüller space of a non-orientable surface

We study the relation between the pants graph of a non-orientable surface and that of its orientable double cover. Given a non-orientable surface $N$ with orientable double cover $\pi\colon S\to N$, we construct a natural map between pants graphs induced by lifting pants decompositions. We prove that this map defines a quasi-isometric embedding of $\mathrm{Pants}(N)$ into $\mathrm{Pants}(S)$. Using Brock's quasi-isometry between $\mathrm{Pants}(S)$ and Teichm\"uller space $\mathrm{Teich}(S)$ endowed with the Weil-Petersson metric, together with the identification of the Teichm\"uller space of $N$ with the fixed-point locus of the deck involution on $\mathrm{Teich}(S)$, we prove that $\mathrm{Pants}(N)$ is quasi-isometric to the Teichm\"uller space of $N$ equipped with the induced Weil-Petersson metric.