Automorphisms of two-generator free groups and spaces of isometric actions on the hyperbolic plane

TL;DR

This paper studies the automorphisms of a two-generator free group acting on hyperbolic 3-space, revealing a dynamical system with invariant subsystems linked to geometric structures like hyperbolic surfaces and Klein bottles.

Contribution

It describes the dynamical decomposition of the automorphism action and characterizes the domain of discontinuity in terms of geometric structures and ergodic properties.

Findings

The domain of discontinuity corresponds to specific hyperbolic structures.

The action is ergodic outside the orbit of the Fricke space.

The orbit of the generalized Fricke space is open and dense.

Abstract

The automorphisms of a two-generator free group acting on the space of orientation-preserving isometric actions of 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 on R^3 by polynomial automorphisms preserving the cubic polynomial and an area form on the level surfaces. We describe the dynamical decomposition of this action. The domain of discontinuity of this action corresponds to geometric structures: either complete hyperbolic structures on the 2-holed cross-surface (projective plane) with cusps and funnels, or complete hyperbolic structures on a one-holed Klein bottle, or hyperbolic structures on a Klein bottle with one conical singularity. The action is ergodic on the complement of the orbit of the Fricke space of the 2-holed cross-surface, and we show that the orbit of the generalized Fricke space of the one-holed Klein bottle is open and dense.