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The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores. (English) Zbl 1059.30036
Let \(S\) be a closed oriented surface of negative Euler characteristic. The pant’s graph \({\mathbf P}(S)\) of \(S\) is a graph whose vertices are the pants decompositions of \(S\), and where a pant’s decomposition \(P\) is related to a pant’s decomposition \(P'\) by an edge if \(P'\) is obtained from \(P\) by an elementary move, that is, a move that consists in replacing a curve \(\alpha\) of \(P\) by another curve \(\beta\) intersecting \(\alpha\) minimally. Setting the length each edge equal to 1 and taking the corresponding length metric, the graph \({\mathbf P}(S)\) becomes a metric space. The pant’s graph has been first defined by Hatcher and Thurston.
The author proves the following Theorem 1: The graph \({\mathbf P}(S)\) is naturally quasi-isometric to the Teichmüller space \({\mathcal T}(S)\) equipped with the Weil-Petersson metric. The author connects then this result to the theory of hyperbolic 3-manifolds. To describe this relation, we recall first that by the work of Bers, a pair \((X,Y)\in {\mathcal T}(S) \times {\mathcal T}(S)\) naturally determines a quasi-Fuchsian hyperbolic 3-manifold \(Q(X,Y)\simeq S\times {\mathbb R}\) with \(X\) and \(Y\) in its conformal boundary at infinity. The convex core \((Q(X,Y))\) of the quasi-Fuchsian manifold is then the smallest convex subset of \(Q(X,Y)\) carrying its fundamental group.
The author proves the following Theorem 2: Let \(d_{WP}(X,Y)\) denote the Weil-Petersson distance between \(X\) and \(Y\) and let \(\text{vol}(\text{core}(Q(X,Y)))\) denote the volume of the convex core of \(Q(X,Y)\). Then, there exist constants \(K_1>1\) and \(K_2>0\) depending only on \(S\) such that for any \((X,Y)\in {\mathcal T}(S) \times {\mathcal T}(S)\), we have \[ {d_{WP}(X,Y)\over K_1} -K_2\leq \text{vol}(\text{core}(Q(X,Y)))\leq K_1 d_{WP}(X,Y) +K_2. \] Theorem 2 was cojectured by Thurston. The author deduces from Theorem 2 the following : Theorem 3: Let \(\lambda_0(X,Y)\) denote the lowest eigenvalue of the Laplacian on the quasi-Fuchsian manifold \(Q(X,Y)={\mathbf H}^3/\Gamma\) and let \(D(X,Y)\) denote the Hausdorff dimension of the limit set of \(\Gamma\). Then, there are constants \(K>0\), \(C_1\), \(C_2\), \(C_3\) and \(C_4>1\) such that if \(d_{WP}(X,Y)>K\), then \[ {C_1\over d_{WP}(X,Y)^2}\leq \lambda_0(X,Y)\leq {C_2\over d_{WP}(X,Y)^2} \] and \[ 2-{C_3\over d_{WP}(X,Y)^2}\leq D(X,Y)\leq 2-{C_4\over d_{WP}(X,Y)^2} . \]

MSC:
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
30F60 Teichmüller theory for Riemann surfaces
37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
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