Shadows of Teichm\"uller discs in the curve graph

TL;DR

This paper investigates the geometric properties of specific curve sets related to Teichmüller discs within the curve graph, establishing their quasiconvexity, bounded Hausdorff distance, and approximation of projections, with applications to bounded geodesic image theorems.

Contribution

It introduces new results on the coarse geometry of curve sets associated with Teichmüller discs, including quasiconvexity and projection approximations, advancing understanding of their structure.

Findings

Curve sets are quasiconvex in the curve graph.

These sets agree up to bounded Hausdorff distance.

Operations on curves approximate nearest point projections.

Abstract

We consider several natural sets of curves associated to a given Teichm\"uller disc, such as the systole set or cylinder set, and study their coarse geometry inside the curve graph. We prove that these sets are quasiconvex and agree up to uniformly bounded Hausdorff distance. Furthermore, we describe two operations on curves and show that they approximate nearest point projections to their respective targets. Our techniques can be used to prove a bounded geodesic image theorem for a natural map from the curve graph to the filling multi-arc graph associated to a Teichm\"uller disc.