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

TL;DR

This paper extends the understanding of the geometric structure of Teichmüller spaces by analyzing the $k$-multicurve graph and establishing its quasi-isometric relation to the space when electrified along thin parts, generalizing prior results.

Contribution

It introduces a new electrification approach for the $k$-multicurve graph, extending Masur and Minsky's results to a broader setting involving extremal length conditions.

Findings

Bound on $k$-multicurve graph distance via intersection number

Extension of quasi-isometry results to $k$-multicurve graphs

Teichmüller space is weakly relatively hyperbolic with respect to the thin part

Abstract

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.