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

Kasra Rafi, Jing Tao
2013 Duke mathematical journal
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 g +p . 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 arbitrary labeled tree with n
more » ... bels using simultaneous Whitehead moves, where the number of steps is of order log(n). * Partially supported by NSF Research Grant, DMS-1007811. is either (k − 1)-pre-sorted or (k − 1)-sorted. We argue in five cases depending on how the local picture around e changes under σ. In each case, the lemma essentially follows from the definition. Case 1: e is pre-sorted and the sorting move is of type 1. In this case, the children of σ(e) are images of the children of e and the grandchildren of e are mapped to the grandchildren of σ(e). Since σ(e) is k-sorted, the digits (k + 1) to d of the ends of T σ(e) match and the k-th digits are as depicted in Figure 5 . This means e r and e l are k-sorted. But e r and e l are ripe, hence the grandchildren of e are (k − 1)-sorted or (k − 1)-pre-sorted. Thus, the grandchildren of σ(e) are (k − 1)-sorted. Hence, the children of σ(e) are either (k − 1)-sorted or (k − 1)-pre-sorted (the conditions (b1) and (b2) holds but (b3) may or may not hold). That is, σ(e) is ripe in T . Case 2: e is pre-sorted and the sorting move is of type 2 or 3. By symmetry, we may assume type 2. In this case, a child of σ(e) is an image of either e r or e r r . First consider σ(e r ) = σ(e) l . As before, e l is k-sorted and its children are at least (k − 1)-presorted. But since the k-th digit of labels at the ends of T e l match, e l l and e l r are in fact at l east (k − 1)-sorted. Hence, e l is either (k − 1)-sorted or (k − 1)-presorted and σ(e l ) is at least (k − 1)-sorted. (see Figure 5 ). Note also that, σ(e) l r is an image of a grandchild of e and, as argued in previous case, it is at least (k − 1)-sorted. Thus, the children if σ(e) l are both at least (k − 1)-sorted and hence σ(e) l is either (k − 1)pre-sorted or (k − 1)-sorted. The argument is easier for σ(e r r ) = σ(e) r since σ(e) r is an image of a grandchild of e and hence it is (k − 1)-sorted. Therefore, σ(e) is ripe. Case 3: e is not pre-sorted and the children of e are mapped to the children of σ(e). In this case, e is as sorted as σ(e). Hence, e r and e l are at least k-sorted and, since they are ripe, the grandchildren are (k − 1)-sorted or (k − 1)-pre-sorted. That is, the children of σ(e) are either (k − 1)-presorted or (k − 1)-sorted. This implies that σ(e) is ripe.