Continuity of Thurston's length function
type: Item
instance of: scholarly article
title: Continuity of Thurston's length function
zbMATH Open document ID: 0968.57011
DOI: 10.1007/PL00001637
published in: Geometric and Functional Analysis. GAFA
publication date: 2001-09-19T00:00:00Z
review text: 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.
reviewed by: Michele Mulazzani
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 1545120
zbMATH Keywords: measured lamination
zbMATH Keywords: average length
zbMATH Keywords: pleated surfaces
zbMATH Keywords: hyperbolic 3-manifolds
author: Jeffrey F. Brock
MaRDI profile type: Publication
full work available at URL: pl00001637
OpenAlex ID: W2036491324
description: scientific article; zbMATH DE number 1545120