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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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