Bounded geodesic image theorem via bicorn curves

TL;DR

This paper establishes a uniform bound for the bounded geodesic image theorem on closed surfaces using bicorn curves, improving understanding of geodesic behavior in surface complexes.

Contribution

It introduces a new proof technique utilizing bicorn curves to obtain explicit bounds, including a smaller bound under specific distance conditions.

Findings

Bound of 44 for non-annular and annular subsurfaces.

Reduced bound of 3 when geodesic is sufficiently distant from boundary.

Comparable bounds to previous results by Masur-Minsky and Webb.

Abstract

We give a uniform bound of the bounded geodesic image theorem for the closed oriented surfaces. The proof utilizes the bicorn curves introduced by Przytycki and Sisto (see arXiv:1502.02176). With the uniformly bounded Hausdorff distance of the bicorn paths and 1-slimness of the bicorn curve triangles, we are able to show the bound is 44 for both non-annular and annular subsurfaces. In a particular case when the distance between a geodesic and an essential boundary component of subsurface (or core if it is annular) is $\geq 18$, then the bound can be as small as 3, which is comparable to the bound 4 in the motivating examples by Masur and Minsky (see arXiv:9807150), and is same as the bound given by Webb for non-annular subsurfaces.