Diameter of the thick part of moduli space and simultaneous Whitehead moves

TL;DR

This paper investigates the diameter of the thick part of moduli space for surfaces and metric graphs, showing it grows logarithmically with genus, punctures, and rank, using a sorting algorithm based on Whitehead moves.

Contribution

It establishes the asymptotic logarithmic growth of the diameter in both Teichmüller and Lipschitz metrics, introducing a novel sorting algorithm for labeled trees with Whitehead moves.

Findings

Diameter grows as log(g+p/ε) in moduli space.

Diameter of metric graph moduli space also grows logarithmically.

A new sorting algorithm for labeled trees with Whitehead moves is developed.

Abstract

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the $\epsilon$-thick part of moduli space of S equipped with the Teichm\"uller or Thurston's Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order $\log \frac{g+p}{\epsilon}$. The same result also holds for the $\epsilon$-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 arbitrary labeled tree with n labels with simultaneous Whitehead moves, where the number of steps is of order log(n).