Degenerations of hyperbolic structures. III: Actions of 3-manifold groups on trees and Thurston's compactness theorem
type: Item
instance of: scholarly article
title: Degenerations of hyperbolic structures. III: Actions of 3-manifold groups on trees and Thurston's compactness theorem
zbMATH Open document ID: 0661.57004
DOI: 10.2307/2007003
author: John W. Morgan
author: Peter B. Shalen
published in: Annals of Mathematics. Second Series
publication date: 1988-01-01T00:00:00Z
review text: This paper is the last in a series of three papers [for parts I and II see ibid. 120, 401-476 (1984; Zbl 0583.57005) and ibid. 127, 403-456 (1988; Zbl 0656.57003)]. The main result of this paper is the following purely topological theorem. Theorem. Let \({\mathcal M}\) be a compact, orientable, connected, irreducible 3-manifold. Let \(\Gamma =\pi_ 1({\mathcal M})\). Suppose that \(\Gamma\) \(\times T\to T\) is an action of \(\Gamma\) on an \({\mathbb{R}}\)-tree such that for any non-degenerate segment \(A\subset T\) the subgroup of \(\Gamma\) fixing A does not contain a free group of rank 2. Let \(\Sigma\) \(\subset {\mathcal M}\) be the characteristic submanifold. Then for any component C of \({\mathcal M}\setminus \Sigma\) the subgroup \(\pi_ 1(C)\subset \Gamma\) has a fixed point in T. This theorem can be easily proved by standard 3-manifold topology in the case when T is a simplicial tree. To give a similar proof in the general case, one should replace surfaces by measured laminations in such a proof. This is the main idea of the proof, but its realisation requires a lot of work. It is based on the theory of measured laminations in 3- manifolds developed in the second paper in the series. As the main application of this theorem, the authors give a new proof of the following theorem of \textit{W. Thurston} [Ann. Math. II. Ser. 124, 203-246 (1986)]. Theorem. If \({\mathcal M}\) is a compact, orientable, irreducible 3-manifold, if \(\partial {\mathcal M}\) is incompressible and if (\({\mathcal M},\partial {\mathcal M})\) contains no essential annuli or tori, then the subspace of characters of discrete and faithful representations of \(\pi_ 1({\mathcal M})\) into \(SL_ 2({\mathbb{C}})\) or \(PGL_ 2({\mathbb{C}})\) is compact. To deduce this theorem from the previous one, the authors use an abstract theory of spaces of characters developed in the first part. This theorem is an important piece (but only a piece) of the yet to be published proof of Thurston's famous hy…
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57R30
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 20F99
zbMATH DE Number: 4080421
zbMATH Keywords: action on an \({\mathbb{R}}\)-tree
zbMATH Keywords: faithful representations of the fundamental group into \(SL_ 2({\mathbb{C}})\)
zbMATH Keywords: compact, orientable, connected, irreducible 3-manifold
zbMATH Keywords: characteristic submanifold
zbMATH Keywords: measured laminations
zbMATH Keywords: measured laminations in 3-manifolds
zbMATH Keywords: spaces of characters
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2007003
OpenAlex ID: W2331921018
description: scientific article; zbMATH DE number 4080421