The diameter of the thick part of moduli space and simultaneous Whitehead moves

Abstract

Let $S$ be a surface of genus $g$ with $p$ punctures with negative Euler characteristic. We study the diameter of the $ϵ$-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 $log\left(\frac{g+p}{ϵ}\right)$. The same result also holds for the $ϵ$-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 $log\left(n\right)$. 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.