Electric Teichm\"uller spaces and $k$-multicurve graphs

Masur and Minsky showed that the curve graph is quasi-isometric to the Teichm\"uller space electrified along its thin part, and hence the Teichm\"uller space is weakly relatively hyperbolic with respect to the thin part. In this paper, we extend this result to the $k$-multicurve graph by electrifying the Teichm\"uller space along the thin part where the extremal length of $k$ curves is sufficiently small. A key ingredient is a bound on the $k$-multicurve graph distance in terms of the intersection number, which is obtained by adapting the upper bound for the pants graph due to Lackenby and Yazdi.