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

About this Title

William Goldman, Department of Mathematics , University of Maryland , College Park, MD 20742, USA, Greg McShane, Department of Mathematics , Institut Fourier , Grenoble, France, George Stantchev, University of Maryland and Ser Peow Tan, Department of Mathematics , National University of Singapore , 2, Science Drive 2, Singapore 117543, Singapore

Publication: Memoirs of the American Mathematical Society
Publication Year: 2019; Volume 259, Number 1249
ISBNs: 978-1-4704-3614-8 (print); 978-1-4704-5253-7 (online)
DOI: https://doi.org/10.1090/memo/1249
Published electronically: April 17, 2019
Keywords: character variety, free group, outer automorphism group, hyperbolic surface, nonorientable surface, hyperbolic surface, conical singularity, Nielsen transformation, ergodic equivalence relation, Fricke space, mapping class group, tree, Markoff map
MSC: Primary 57M05, (Low-dimensional, topology), 22D40, (Ergodic, theory, on, groups)

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Table of Contents

Chapters

• 1. Introduction
• 2. The rank two free group and its automorphisms
• 3. Character varieties and their automorphisms
• 4. Topology of the imaginary commutator trace
• 5. Generalized Fricke spaces
• 6. Bowditch theory
• 7. Imaginary trace labelings
• 8. Imaginary characters with $k>2$
• 9. Imaginary characters with $k<2$.
• 10. Imaginary characters with $k=2$.

Abstract

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 $\kappa _\Phi ^{-1}(k)$.

The Fricke space of marked hyperbolic structures on the 2-holed projective plane with funnels or cusps identifies with the subset $\mathfrak {F}(C_{0,2})\subset \mathbb {R}^3$ defined by \[ z\le -2, \quad xy + z \ge 2. \] The generalized Fricke space of marked hyperbolic structures on the 1-holed Klein bottle with a funnel, a cusp, or a conical singularity identifies with the subset $\mathfrak {F}’(C_{1,1})\subset \mathbb {R}^3$ defined by \[ z>2, \quad x y z \ge x^2 + y^2. \] We show that $\Gamma$ acts properly on the subsets $\Gamma \cdot \mathfrak {F}(C_{0,2})$ and $\Gamma \cdot \mathfrak {F}’(C_{1,1})$. Furthermore for each $k<2$, the action of $\Gamma$ is ergodic on the complement of $\Gamma \cdot \mathfrak {F}(C_{0,2})$ in $\kappa _\Phi ^{-1}(k)$ for $k < -14$. In particular, the action is ergodic on all of $\kappa _\Phi ^{-1}(k)$ for $-14\le k < 2$.

For $k>2$, the orbit $\Gamma \cdot \mathfrak {F}(C_{1,1})$ is open and dense in $\kappa _\Phi ^{-1}(k)$. We conjecture its complement has measure zero.