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Full Description
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 $\kappa _{\Phi}^{-1}(k)$.
Contents
Introduction
The rank two free group and its automorphisms
Character varieties and their automorphisms
Topology of the imaginary commutator trace
Generalized Fricke spaces
Bowditch theory
Imaginary trace labelings
Imaginary characters with $k>2$
Imaginary characters with $k<2$
Imaginary characters with $k=2$
Bibliography
