Quadratic Bounds on the Quasiconvexity of Nested Train Track Sequences

TL;DR

This paper establishes quadratic bounds on the quasiconvexity of nested train track sequences in the curve graph of surfaces, providing effective versions of key theorems in geometric topology.

Contribution

It proves that nested train track sequences are $O((g,p)^2)$-quasiconvex, improving previous qualitative results with explicit bounds, and extends these bounds to splitting and sliding sequences.

Findings

Nested train track sequences are $O((g,p)^2)$-quasiconvex.

Genus $g$ disk set is $O(g^2)$-quasiconvex.

Splitting and sliding sequences project to $O((g,p)^2)$-quasi-geodesics.

Abstract

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p$ punctures. We show that nested train track sequences constitute $O((g,p)^{2})$-quasiconvex subsets of the curve graph, effectivizing a theorem of Masur and Minsky. As a consequence, the genus $g$ disk set is $O(g^{2})$-quasiconvex. We also show that splitting and sliding sequences of birecurrent train tracks project to $O((g,p)^{2})$-unparameterized quasi-geodesics in the curve graph of any essential subsurface, an effective version of a theorem of Masur, Mosher, and Schleimer.