The curves not carried

Article, Preprint English OPEN
Gadre, Vaibhav ; Schleimer, Saul (2014)
  • Publisher: European Mathematical Society
  • Related identifiers: doi: 10.4171/GGD/382
  • Subject: QA | 57M99 (Primary), 30F60, 20F65 (Secondary) | Mathematics - Geometric Topology
    arxiv: Mathematics::Geometric Topology

Suppose $\tau$ is a train track on a surface $S$. Let $C(\tau)$ be the set of isotopy classes of simple closed curves carried by $\tau$. Masur and Minsky [2004] prove $C(\tau)$ is quasi-convex inside the curve complex $C(S)$. We prove the complement, $C(S) - C(\tau)$, is quasi-convex.
  • References (7)

    [1] Vaibhav Gadre and Chia-Yen Tsai. Minimal pseudo-Anosov translation lengths on the complex of curves. Geom. Topol., 15(3):1297-1312, 2011, arXiv:1101.2692. [4]

    [2] Howard Masur, Lee Mosher, and Saul Schleimer. On train-track splitting sequences. Duke Math. J., 161(9):1613-1656, 2012, arXiv:1004.4564v1. [6]

    [3] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math., 138(1):103-149, 1999, arXiv:math/9804098v2. [1, 2, 4]

    [4] Howard A. Masur and Yair N. Minsky. Quasiconvexity in the curve complex. In In the tradition of Ahlfors and Bers, III, volume 355 of Contemp. Math., pages 309-320. Amer. Math. Soc., Providence, RI, 2004, arXiv:math/0307083v1. [1, 4]

    [5] Lee Mosher. Train track expansions of measured foliations. 2003. http://math.newark.rutgers.edu/»mosher/. [4]

    [6] Saul Schleimer. Notes on the complex of curves. http://www.warwick.ac.uk/»masgar/Maths/notes.pdf. [4]

    [7] Itaru Takarajima. A combinatorial representation of curves using train tracks. Topology Appl., 106(2):169-198, 2000. [6]

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