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E-raamat: Axes in Outer Space

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The authors develop a notion of axis in the Culler-Vogtmann outer space $\mathcal{X}_r$ of a finite rank free group $F_r$, with respect to the action of a nongeometric, fully irreducible outer automorphism $\phi$. Unlike the situation of a loxodromic isometry acting on hyperbolic space, or a pseudo-Anosov mapping class acting on Teichmüller space, $\mathcal{X}_r$ has no natural metric, and $\phi$ seems not to have a single natural axis. Instead these axes for $\phi$, while not unique, fit into an ""axis bundle"" $\mathcal{A}_\phi$ with nice topological properties: $\mathcal{A}_\phi$ is a closed subset of $\mathcal{X}_r$ proper homotopy equivalent to a line, it is invariant under $\phi$, the two ends of $\mathcal{A}_\phi$ limit on the repeller and attractor of the source-sink action of $\phi$ on compactified outer space, and $\mathcal{A}_\phi$ depends naturally on the repeller and attractor.

The authors propose various definitions for $\mathcal{A}_\phi$, each motivated in different ways by train track theory or by properties of axes in Teichmüller space, and they prove their equivalence.
Chapter 1 Introduction 1(14)
1.1 Characterizations of the axis bundle
2(4)
1.2 The main theorems
6(1)
1.3 A question of Vogtmann
7(1)
1.4 Contents and proofs
7(3)
1.5 Problems and questions
10(5)
Chapter 2 Preliminaries 15(26)
2.1 Outer space and outer automorphisms
15(5)
2.2 Paths, circuits and edge paths
20(2)
2.3 Folds
22(2)
2.4 Train track maps
24(4)
2.5 The attracting tree T+
28(5)
2.6 Geodesic laminations in trees and marked graphs
33(2)
2.7 The expanding lamination Λ_
35(3)
2.8 Relating Λ_ to T_ and to T+
38(3)
Chapter 3 The ideal Whitehead graph 41(8)
3.1 Definition and structure of the ideal Whitehead graph
42(2)
3.2 Asymptotic leaves and the ideal Whitehead graph
44(1)
3.3 T+ and the ideal Whitehead graph
45(1)
3.4 An example of an ideal Whitehead graph
46(3)
Chapter 4 Cutting and pasting local stable Whitehead graphs 49(6)
4.1 Pasting local stable Whitehead graphs
49(2)
4.2 Cutting local stable Whitehead graphs
51(1)
4.3 The finest local decomposition
52(3)
Chapter 5 Weak train tracks 55(14)
5.1 Local decomposition of the ideal Whitehead graph
56(1)
5.2 Folding up to a weak train track
57(2)
5.3 Comparing train tracks to weak train tracks
59(3)
5.4 Rigidity and irrigidity of Λ_ isometries
62(3)
5.5 Examples of exceptional weak train tracks
65(4)
Chapter 6 Topology of the axis bundle 69(18)
6.1 Continuity properties of the normalized axis bundle
69(2)
6.2 The Gromov topology on weak train tracks
71(3)
6.3 Properness of the length map
74(3)
6.4 Applying Skora's method to the Properness Theorem 6.1
77(7)
6.5 Remarks on stable train tracks
84(3)
Chapter 7 Fold Lines 87(16)
7.1 Examples Of Fold Paths
87(6)
7.2 Characterizing Fold Lines
93(1)
7.3 Direct Limits Of Fold Rays
94(3)
7.4 Legal Laminations Of Split Rays
97(4)
7.5 Weak Train Tracks On Fold Lines
101(2)
Bibliography 103
Michael Handel is at CUNY, Herbert H. Lehman College, Bronx, NY