The diameter of the thick part of moduli space and simultaneous Whitehead moves. (English) Zbl 1277.32013
Let \(\mathcal{M}_{g,p}\) be the moduli space of the complete, finite volume, hyperbolic surfaces of genus \(g\) with \(p\) labeled punctures. The \(\epsilon\)-thick part \(\mathcal{M}_{g,p}^{\epsilon}\) of \(\mathcal{M}_{g,p}\) is the space of surfaces in \(\mathcal{M}_{g,p}\) the systole of which has length at least \(\epsilon\), where \(\epsilon \leq \epsilon_M\), the Margulis constant.
The authors study the diameter of \(\mathcal{M}_{g,p}^{\epsilon}\) equipped with the Teichmüller metric or the Lipschitz metric (both are \(L^{\infty}\)-metrics) and show that it is of order \(\log((g+p)/\epsilon)\) in both metrics. Also the width and the height of \(\mathcal{M}_{g,p}^{\epsilon}\) are obtained asymptotically. Here the width of \(\mathcal{M}_{g,p}^{\epsilon}\) is the diameter of the set \(\mathcal{B}_{g,p}\) of surfaces in \(\mathcal{M}_{g,p}^{\epsilon}\) which admit a pants decomposition consisting of curves of length \(\epsilon_M\); the height of \(\mathcal{M}_{g,p}^{\epsilon}\) is the Hausdorff distance between \(\mathcal{M}_{g,p}^{\epsilon}\) and \(\mathcal{B}_{g,p}\).
The authors consider the space Graph\((g,p)\) of graphs of rank \(g\) with \(p\) marked valence 1 vertices and \((2g-2+p)\) valence 3 vertices associated to the surfaces in \(\mathcal{B}_{g,p}\). It is shown that the diameter of Graph\((g,p)\) equipped with the metric of simultaneous Whitehead moves is of the same order as that of \(\mathcal{B}_{g,p}\). For the proof the authors introduce two algorithms for transforming trees by simultaneous Whitehead moves to reduce their heights and to get fully sorted trees. The result that expander graphs are not coarsely equidistributed in Graph\((g)\) is implied by the order of the length of dividing curves on a surface in \(\mathcal{B}_g\) associated to an expander graph.
The asymptotic diameter of the \(\epsilon\)-thick part of the moduli space of metric graphs is also studied in the Lipschitz metric. Furthermore the asymptotic diameters of Graph\((g,p)\) and of Graph\((g,p)\)/Sym\(_p\) (unlabeled punctures) in the metric of Whitehead moves are established.
The authors study the diameter of \(\mathcal{M}_{g,p}^{\epsilon}\) equipped with the Teichmüller metric or the Lipschitz metric (both are \(L^{\infty}\)-metrics) and show that it is of order \(\log((g+p)/\epsilon)\) in both metrics. Also the width and the height of \(\mathcal{M}_{g,p}^{\epsilon}\) are obtained asymptotically. Here the width of \(\mathcal{M}_{g,p}^{\epsilon}\) is the diameter of the set \(\mathcal{B}_{g,p}\) of surfaces in \(\mathcal{M}_{g,p}^{\epsilon}\) which admit a pants decomposition consisting of curves of length \(\epsilon_M\); the height of \(\mathcal{M}_{g,p}^{\epsilon}\) is the Hausdorff distance between \(\mathcal{M}_{g,p}^{\epsilon}\) and \(\mathcal{B}_{g,p}\).
The authors consider the space Graph\((g,p)\) of graphs of rank \(g\) with \(p\) marked valence 1 vertices and \((2g-2+p)\) valence 3 vertices associated to the surfaces in \(\mathcal{B}_{g,p}\). It is shown that the diameter of Graph\((g,p)\) equipped with the metric of simultaneous Whitehead moves is of the same order as that of \(\mathcal{B}_{g,p}\). For the proof the authors introduce two algorithms for transforming trees by simultaneous Whitehead moves to reduce their heights and to get fully sorted trees. The result that expander graphs are not coarsely equidistributed in Graph\((g)\) is implied by the order of the length of dividing curves on a surface in \(\mathcal{B}_g\) associated to an expander graph.
The asymptotic diameter of the \(\epsilon\)-thick part of the moduli space of metric graphs is also studied in the Lipschitz metric. Furthermore the asymptotic diameters of Graph\((g,p)\) and of Graph\((g,p)\)/Sym\(_p\) (unlabeled punctures) in the metric of Whitehead moves are established.
Reviewer: Gou Nakamura (Toyota)
MSC:
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 05C85 | Graph algorithms (graph-theoretic aspects) |
| 20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |
Keywords:
thick part of moduli space; simultaneous Whitehead move; Teichmüller metric; Lipschitz metric; pants decomposition; metric graphReferences:
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