Hexagon decompositions and the Weil-Petersson metric

We give an explicit quasi-isometry from the flip graph of triangulations ${\mathcal{F}}_{g,n}$ to the $\epsilon$-thick part of Teichmüller space equipped with the Teichmüller metric, by mapping each ideal triangulation to the hyperbolic surface whose shearing coordinates along that triangulation are all equal to zero. Extending this construction, we obtain a quasi-isometry $Q$ from the hexagon graph ${\mathcal{H}}_{g,n}$ to the augmented Teichmüller space ${\stackrel{}{\mathrm{Teich}}}_{g,n}$ equipped with the Weil-Petersson metric, and we estimate its width, that is, the Hausdorff distance between $Q({\mathcal{H}}_{g,n})$ and ${\stackrel{}{\mathrm{Teich}}}_{g,n}$. By bounding the Weil-Petersson distance along grafting rays and the Teichmüller distance along shearing deformations, we show that the width is at most $\sqrt{g+n}\log (g+n)$. We also provide an explicit projection from any point in ${\stackrel{}{\mathrm{Teich}}}_{g,n}$ to the thick part of boundary strata, maintaining simultaneous control over the Weil-Petersson distance and the shearing coordinates.