The Diameter of the Thick Part of Moduli Space and Simultaneous Whitehead Moves

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the -thick part of moduli space of S equipped with the Teichmüller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order log.gCp /. The same result also holds for the -thick part of the moduli space of metric graphs of rank n equipped with the Lipschitz metric. The proof involves a sorting algorithm that sorts an arbitrarily labeled tree with n labels using simultaneous Whitehead moves, where the number of steps is of order log.n/. 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 S in the metric of elementary moves.