ORPP logo
Image from Google Jackets

Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical Society SeriesPublisher: Providence : American Mathematical Society, 2019Copyright date: ©2019Edition: 1st edDescription: 1 online resource (92 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470452537
Subject(s): Genre/Form: Additional physical formats: Print version:: Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic PlaneDDC classification:
  • 514/.34
LOC classification:
  • QA174.2 .G653 2019
Online resources:
Contents:
Cover -- Title page -- List of Figures -- Chapter 1. Introduction -- 1.1. Fricke spaces and Fricke orbits -- 1.2. The orientation-preserving case -- 1.3. The Main Theorem -- Acknowledgements -- Notation and terminology -- Chapter 2. The rank two free group and its automorphisms -- 2.1. The modular group and automorphisms -- 2.2. The tree of superbases -- 2.3. The tricoloring and the planar embedding -- 2.4. Paths and Alternating Geodesics -- 2.5. Relation to the one-holed torus -- 2.6. Effect of a {±1}-character -- Chapter 3. Character varieties and their automorphisms -- 3.1. The deformation space -- 3.2. Sign-changes -- 3.3. Action of automorphisms -- 3.4. Real forms of the character variety -- 3.5. Real and imaginary characters -- 3.6. Invariants of the action -- Chapter 4. Topology of the imaginary commutator trace -- 4.1. Preliminaries -- 4.2. Projection when &gt -- -2 -- 4.3. The invariant area form -- 4.4. The level set for \le-2 -- Chapter 5. Generalized Fricke spaces -- 5.1. Geometric structures on the two-holed cross-surface -- 5.2. Geometric structures on the one-holed Klein bottle -- 5.3. Lines on the Markoff surface -- Chapter 6. Bowditch theory -- 6.1. The trace labeling associated to a character -- 6.2. The flow associated to a character -- 6.3. Exceptional characters -- 6.4. The Fork Lemma -- 6.5. Alternating geodesics -- 6.6. Indecisive edges and orthogonality -- Chapter 7. Imaginary trace labelings -- 7.1. Well-directed trees -- 7.2. {±1}-characters on \Ft -- 7.3. Positive and negative vertices -- Chapter 8. Imaginary characters with &gt -- 2 -- 8.1. Alternating geodesics when &gt -- 2. -- 8.2. The Bowditch set -- 8.3. Planar projection of the Bowditch set -- 8.4. Density of the Bowditch set -- Chapter 9. Imaginary characters with &lt -- 2. -- 9.1. Existence of elliptics -- 9.2. Alternating geodesics for &lt -- 2.
9.3. Descending Paths -- 9.4. Ergodicity -- Chapter 10. Imaginary characters with =2. -- Bibliography -- Back Cover.
Summary: The automorphisms of a two-generator free group \mathsf F_2 acting on the space of orientation-preserving isometric actions of \mathsf F_2 on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action of a group \Gamma on \mathbb R ^3 by polynomial automorphisms preserving the cubic polynomial \kappa _\Phi (x,y,z) := -x^{2} -y^{2} + z^{2} + x y z -2 and an area form on the level surfaces \kappa _{\Phi}^{-1}(k).
Tags from this library: No tags from this library for this title. Log in to add tags.
Star ratings
    Average rating: 0.0 (0 votes)
No physical items for this record

Cover -- Title page -- List of Figures -- Chapter 1. Introduction -- 1.1. Fricke spaces and Fricke orbits -- 1.2. The orientation-preserving case -- 1.3. The Main Theorem -- Acknowledgements -- Notation and terminology -- Chapter 2. The rank two free group and its automorphisms -- 2.1. The modular group and automorphisms -- 2.2. The tree of superbases -- 2.3. The tricoloring and the planar embedding -- 2.4. Paths and Alternating Geodesics -- 2.5. Relation to the one-holed torus -- 2.6. Effect of a {±1}-character -- Chapter 3. Character varieties and their automorphisms -- 3.1. The deformation space -- 3.2. Sign-changes -- 3.3. Action of automorphisms -- 3.4. Real forms of the character variety -- 3.5. Real and imaginary characters -- 3.6. Invariants of the action -- Chapter 4. Topology of the imaginary commutator trace -- 4.1. Preliminaries -- 4.2. Projection when &gt -- -2 -- 4.3. The invariant area form -- 4.4. The level set for \le-2 -- Chapter 5. Generalized Fricke spaces -- 5.1. Geometric structures on the two-holed cross-surface -- 5.2. Geometric structures on the one-holed Klein bottle -- 5.3. Lines on the Markoff surface -- Chapter 6. Bowditch theory -- 6.1. The trace labeling associated to a character -- 6.2. The flow associated to a character -- 6.3. Exceptional characters -- 6.4. The Fork Lemma -- 6.5. Alternating geodesics -- 6.6. Indecisive edges and orthogonality -- Chapter 7. Imaginary trace labelings -- 7.1. Well-directed trees -- 7.2. {±1}-characters on \Ft -- 7.3. Positive and negative vertices -- Chapter 8. Imaginary characters with &gt -- 2 -- 8.1. Alternating geodesics when &gt -- 2. -- 8.2. The Bowditch set -- 8.3. Planar projection of the Bowditch set -- 8.4. Density of the Bowditch set -- Chapter 9. Imaginary characters with &lt -- 2. -- 9.1. Existence of elliptics -- 9.2. Alternating geodesics for &lt -- 2.

9.3. Descending Paths -- 9.4. Ergodicity -- Chapter 10. Imaginary characters with =2. -- Bibliography -- Back Cover.

The automorphisms of a two-generator free group \mathsf F_2 acting on the space of orientation-preserving isometric actions of \mathsf F_2 on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action of a group \Gamma on \mathbb R ^3 by polynomial automorphisms preserving the cubic polynomial \kappa _\Phi (x,y,z) := -x^{2} -y^{2} + z^{2} + x y z -2 and an area form on the level surfaces \kappa _{\Phi}^{-1}(k).

Description based on publisher supplied metadata and other sources.

Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

There are no comments on this title.

to post a comment.

© 2024 Resource Centre. All rights reserved.