Automorphisms of Twogenerator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane 1st Edition by William Goldman, Greg Mcshane, George Stantchev, Ser Peow Tan ISBN 1470436140 9781470436148

Automorphisms of Two-generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane 1st Edition by William Goldman, Greg Mcshane, George Stantchev, Ser Peow Tan – Ebook PDF Instant Download/Delivery: 1470436140, 978-1470436148

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Product details:

ISBN 10: 1470436140

ISBN 13: 978-1470436148

Author: William Goldman, Greg Mcshane, George Stantchev, Ser Peow Tan

Automorphisms of Two-generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane 1st Edition: The automorphisms of a two-generator free group $mathsf F2$ acting on the space of orientation-preserving isometric actions of $mathsf F2$ 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) := -x2 -y2 + z2 + x y z -2 $ and an area form on the level surfaces $kappa Phi-1(k)$.

 

Automorphisms of Two-generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane 1st Edition Table of contents:

1. Introduction

   • Fricke spaces and Fricke orbits
   • The orientation-preserving case
   • The Main Theorem
   • Acknowledgements
   • Notation and terminology

2. The rank two free group and its automorphisms

   • The modular group and automorphisms
   • The tree of superbases
   • The tricoloring and the planar embedding
   • Paths and alternating geodesics
   • Relation to the one-holed torus
   • Effect of a {1}-character

3. Character varieties and their automorphisms

   • The deformation space
   • Sign-changes
   • Action of automorphisms
   • Real forms of the character variety
   • Real and imaginary characters
   • Invariants of the action

4. Topology of the imaginary commutator trace

   • Preliminaries
   • Projection when > -2
   • The invariant area form
   • The level set for k < -2

5. Generalized Fricke spaces

   • Geometric structures on the two-holed cross-surface
   • Geometric structures on the one-holed Klein bottle
   • Lines on the Markoff surface

6. Bowditch theory

   • The trace labeling associated with a character
   • The flow associated with a character
   • Exceptional characters
   • The Fork Lemma
   • Alternating geodesics
   • Indecisive edges and orthogonality

7. Imaginary trace labelings

   • Well-directed trees
   • {1}-characters on F2

 

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William Goldman,Greg Mcshane,George Stantchev,Ser Peow,Automorphisms,Twogenerator,Free Groups,Spaces,Isometric Actions,Hyperbolic Plane