Let be a surface of genus with punctures with negative Euler characteristic. We study the diameter of the -thick part of moduli space of equipped with the Teichmüller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order . The same result also holds for the -thick part of the moduli space of metric graphs of rank equipped with the Lipschitz metric. The proof involves a sorting algorithm that sorts an arbitrarily labeled tree with labels using simultaneous Whitehead moves, where the number of steps is of order . As a related combinatorial problem, we also compute, in the appendix of this paper, the asymptotic diameter of the moduli space of pants decompositions on in the metric of elementary moves.
Kasra Rafi. Jing Tao. "The diameter of the thick part of moduli space and simultaneous Whitehead moves." Duke Math. J. 162 (10) 1833 - 1876, 15 July 2013. https://doi.org/10.1215/00127094-2323128