an:01545120
Zbl 0968.57011
Brock, J. F.
Continuity of Thurston's length function
EN
Geom. Funct. Anal. 10, No. 4, 741-797 (2000).
1016-443X 1420-8970
2000
j
57M50 57N10
measured lamination; average length; pleated surfaces; hyperbolic 3-manifolds
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.
Michele Mulazzani (Bologna)