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Representations of 3-Manifold Groups CHAPTER 3 Representations of 3-manifold groups Peter B. Shalen University of Illinois at Chicago Contents 1. Some basic concepts and examples : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.1. Representations and actions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 1.2. A word about base points : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 1.3. The universal covering : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.4. The tree associated with a hypersurface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 1.5. The three-dimensional case: essential surfaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 10 1.6. The action associated to a hyperbolic structure : : : : : : : : : : : : : : : : : : : : : : : : : : : : 12 2. Actions of 3-manifold groups on trees : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 2.1. Constructing an equivariant map : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 2.2. Constructing a surface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15 2.3. Remarks on dual surfaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 2.4. Making a dual surface essential : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 2.5. Applications, I: Nonseparating surfaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 21 2.6. Applications, II: Free products and stu® : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 25 3. The tree for SL2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 3.1. Valuations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 27 3.2. The p-adic valuation : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 3.3. Fields of meromorphic functions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 28 3.4. The valuation ring : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 29 3.5. The p-adics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 30 3.6. De¯ning the tree for SL2 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31 3.7. The action : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 38 3.8. Getting to know the tree, I: the link of a vertex : : : : : : : : : : : : : : : : : : : : : : : : : : : 38 3.9. Getting to know the tree, II: axes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 40 3.10. Application: Ihara's theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41 4. Varieties of representations and varieties of characters : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42 April 23, 1999 { Draft version; Typeset by LaTEX Supported by an NSF grant and a U. of I. University Scholar award Handbook of Geometric Topology { Style ¯les Created by F.D. Mesman and W.j. Maas °c 1996 Elsevier Science B.V. 1 2 P. Shalen 4.1. The variety of representations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 42 4.2. A little algebraic geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 43 4.3. More on varieties of representations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 45 4.4. Varieties of characters : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 46 4.5. The irreducible component of a discrete faithful character : : : : : : : : : : : : : : : : : : : 49 5. Ideal points and trees : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 52 5.1. Some more algebraic geometry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 52 5.2. Projective varieties : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 54 5.3. Canonical completions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 56 5.4. Associating an action on a tree with an ideal point : : : : : : : : : : : : : : : : : : : : : : : : : 57 5.5. More about the action : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 59 5.6. Separating surfaces in knot exteriors : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 61 6. The proof of the weak Neuwirth Conjecture : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 63 7. The Smith Conjecture : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 65 7.1. Preliminary observations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 66 7.2. The character variety appears : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 67 7.3. The character variety argument completed : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 69 7.4. The equivariant loop theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 70 7.5. The root-of-unity phenomenon : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71 7.6. Extensions of the theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 71 8. Degrees and Norms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 72 8.1. Degrees of trace functions; de¯ning the norm : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 73 8.2. A word about norms : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 76 8.3. Further properties of the norm : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 77 9. Applications to Dehn surgery : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 80 9.1. The Cyclic Surgery Theorem : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 80 9.2. The \no-closed-surface" case : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 80 9.3. A little geometry of numbers: a theorem of Minkowski's : : : : : : : : : : : : : : : : : : : : : 84 9.4. The \no-closed-surface" case, concluded : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 9.5. The case where there are closed surfaces : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86 9.6. Other applications to surgery : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 87 10. Boundary slopes and genera: Neuwirth and Poincar¶e Conjectures : : : : : : : : : : : : : : : : : : : 89 10.1. A lower bound for the diameter of the boundary slopes : : : : : : : : : : : : : : : : : : : : : 90 10.2. Some related results : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91 11. R-trees and degenerations : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 93 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 94 Representations of 3-manifold groups 3 The phrase \representations of 3-manifold groups" is likely to suggest di®erent things to di®erent people. When I was asked to write a chapter for this handbook on this subject, I thought it would be wise to focus on something I knew about, and I therefore decided to concentrate on the interaction between two kinds of repre- sentations of fundamental groups of 3-manifolds: representations by 2 2 matrices, and representations by automorphisms of trees. Representations of the£ ¯rst kind are related to hyperbolic structures on 3-manifolds, while those of the second kind are related to surfaces in 3-manifolds. The interaction between these two kinds of representations therefore provides a link between what are probably the two most useful kinds of structures on 3-manifolds, whence the utility of this theory in studying group actions, Dehn surgery, surfaces in knot exteriors, and degeneration of hyperbolic structures. The core of this chapter is an attempt to present|with the necessary background|some of the content of a series of joint papers [17], [18], [19], |by Marc Culler and myself, and of the ¯rst chapter of our paper [15] with Cameron Gordon and John Luecke. I have been able to give only the briefest hints about the remarkable developments in the area that have been made by Steven Boyer and Xingru Zhang in [5], [6] and [7], by Daryl Cooper and Darren Long and their co-authors in [11], [13], and other papers|see [14] for a survey|and by Nathan Dun¯eld in [22] and [23]. I have been even briefer about my closely related joint work with John
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