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

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

2011-08-21

arxiv.org/abs/1108.4150v2

Mathematics

Geometric Topology

30 pages, 10 figures. Minor revisions; changed the title

Scientific paper

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).

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