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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.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57N10 Topology of general $$3$$-manifolds (MSC2010)
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