Let $\mathcal{T}(S)$ denotes the Teichmuller space of a finite type surface $S$ equipped with Teichmuller metric and $\mathcal{C}(S)$ denotes the curve complex. Define a map $$\phi:\mathcal{T}(S)\rightarrow \mathcal{C}(S),$$ by taking a hyperbolic metric to one of its systols. I read somewhere that this map is coarsely Lipschitz. Can anyone please give a proper reference of this result.

PS: Please mention the theorem not just the paper.


The proper reference is Lemma 2.4 of the paper "Geometry of the complex of curves I: Hyperbolicity" by Masur and Minsky. They use extremal length to determine the systole set. They remark on the same page that hyperbolic length would also work.

I haven't found an explicit reference for the latter. You can prove it following the proof in Masur and Minsky. You will need:

  • a bound on the hyperbolic length of a systole curve (the Bers constant),
  • a bound on how much the hyperbolic length of a systole can change when moving distance one in the Teichmüller metric, and
  • a bound on the intersection number between a pair of simple closed hyperbolic geodesics in terms of their length (collar lemma).

You can deduce the second bullet from, say, the bounds in Maskit's paper "Comparison of hyperbolic and extremal lengths" and then applying Kerckhoff's characterization (see Equation 2.1 of the paper of Masur and Minsky).

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