Quadratic Bounds on the Quasiconvexity of Nested Train Track Sequences

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.

arXiv comment24 pages, 5 figures. Version 2; minor typos fixed

References in corpus (2)

• Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs
• Uniform Hyperbolicity of the Graphs of Curves