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Continuity of Thurston’s length function. (English) Zbl 0968.57011
This paper concerns measured laminations \(\mu\) geodesically realized in marked hyperbolic 3-manifolds \(M\). If \(M\cong S\times\mathbb{R}\), where \(S\) is a closed oriented surface, the author proves Thurston’s claim that the function measuring the average length of the maximal realizable sublamination of \(\mu\) is bicontinuous in \(M\) and \(\mu\), by developing a new uniform estimate on the geometry of pleated surfaces. While the length of \(\mu\) in \(M\) is not well defined when \(\mu\) is not realizable in \(M\), the realizable measured laminations are dense in the set of all measured laminations \(\mu\) of marked hyperbolic 3-manifolds \(M\). This property allows the author to extend the length function to this larger set and, using his proof of Thurston’s claim together with a train track shortening technique of F. Bonahon, to prove the continuity of such extension. Since connected positive non-realizable measured laminations arise as zeros of this function, this continuity suggests new behavioral features of quasi-isometry invariants under limits of hyperbolic 3-manifolds.

57M50 General geometric structures on low-dimensional manifolds
57N10 Topology of general \(3\)-manifolds (MSC2010)
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