## The geometry of Teichmüller space via geodesic currents.(English)Zbl 0653.32022

Let S be a compact orientable surface of genus $$g\geq 2$$. Denote by $${\mathcal T}(S)$$ its Teichmüller space, i.e., the space of isotopy classes of hyperbolic metrics on S. The author provides a very natural compactification of the Teichmüller space $${\mathcal T}(S)$$, which concludingly turns out to coincide with W. Thurston’s compactification via projective measured laminations [cf. papers of A. Fathi, F. Laudenbach and V. Poenaru in the book “Travaux de Thurston sur les surfaces” (1979; Zbl 0406.00016); e.g. F. Laudenbach, ibid., 209- 224 (1979; Zbl 0446.57018), A. Fathi and F. Laudenbach, ibid., 139-150 (1979; Zbl 0446.57015), and V. Poenaru, ibid., 5-20 (1979; Zbl 0446.57005)]. The authors construction is based upon the notion of geodesic currents, which has been introduced by himself in an earlier work [cf. the author, Ann. Math. 124, 71-158 (1986)]. The geodesic currents, i.e., the $$\pi_ 1(S)$$-invariant positive measures on the space $$G(\tilde S)$$ of (unoriented) geodesics on the universal covering $$\tilde S$$ of S, are shown to form a complete uniform space $${\mathcal C}(S)$$, whose projectivization $${\mathcal P}{\mathcal C}(S):=({\mathcal C}(S)- 0)/{\mathbb{R}}^+$$ is compact. It is then proved that $${\mathcal T}(S)$$ admits a proper topological embedding into $${\mathcal C}(S)$$, whose image is asymptotic to Thurston’s space $${\mathcal M}{\mathcal L}(S)$$ of measured laminations. This provides a compactification of $${\mathcal T}(S)$$ (within $${\mathcal P}{\mathcal C}(S))$$ by $${\mathcal P}{\mathcal M}{\mathcal L}(S)$$, so to speak a “unified” version of Thurston’s approach. Another advantage of the author’s construction is that it gives a representation of Teichmüller space $${\mathcal T}(S)$$ as a submanifold of an infinite-dimensional analog of the hyperbolic n-space $${\mathbb{H}}^ n$$. Then the metric on $${\mathcal T}(S)$$ induced by the hyperbolic metric on $${\mathbb{H}}^ n$$ is equal (up to a constant factor) to the celebrated Petersson-Weil metric.
Reviewer: W.Kleinert

### MSC:

 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds 32J05 Compactification of analytic spaces 58A25 Currents in global analysis 57R30 Foliations in differential topology; geometric theory

### Citations:

Zbl 0406.00016; Zbl 0446.57018; Zbl 0446.57015; Zbl 0446.57005
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### References:

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