[Paper Review] Small dilatation pseudo-Anosovs and 3-manifolds

Benson Farb, Christopher J. Leininger+1 more|ArXiv.org|May 2, 2009

Geometric and Algebraic Topology29 references16 citations

TL;DR

This paper establishes a universal finiteness theorem for small dilatation pseudo-Anosov homeomorphisms across all surfaces: when the product of surface Euler characteristic (in absolute value) and the logarithm of the dilatation is bounded, the resulting mapping tori of the punctured surfaces form a finite set. This implies all such pseudo-Anosovs arise as monodromies of Dehn fillings on a finite list of fibered hyperbolic 3-manifolds.

ABSTRACT

The main result of this paper is a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms, ranging over all surfaces. More precisely, we consider pseudo-Anosovs F:S to S with |chi(S)| log(lambda(F)) bounded above by some constant, and we prove that, after puncturing the surfaces at the singular points of the stable foliations, the resulting set of mapping tori is finite. Said differently, there is a finite set of fibered hyperbolic 3-manifolds so that all small dilatation pseudo-Anosovs occur as the monodromy of a Dehn filling on one of the 3-manifolds in the finite list, where the filling is on the boundary slope of a fiber.

Motivation & Objective

• To establish a universal finiteness result for all small dilatation pseudo-Anosov homeomorphisms across all surfaces.
• To show that the set of mapping tori of punctured surfaces (after removing singularities of stable/unstable foliations) is finite when $ |χ(S)|\log(\lambda(\phi)) $ is bounded.
• To demonstrate that all such pseudo-Anosovs arise as monodromies of Dehn fillings on a finite list of fibered hyperbolic 3-manifolds.
• To unify the dynamics of small dilatation pseudo-Anosovs through a finite set of 3-manifold templates.
• To extend McMullen's 3-manifold construction to a universal finiteness framework for all surfaces and small dilatations.

Proposed method

• Define $ \Psi_P^\circ $ as the set of restrictions of pseudo-Anosov maps $ \phi:S\to S $ to $ S^\circ $, the surface with singularities of stable/unstable foliations removed, where $ \lambda(\phi) \leq P^{1/|\chi(S)|} $.
• Construct the mapping torus $ M_\phi $ for each $ \phi \in \Psi_P^\circ $, which is a fibered hyperbolic 3-manifold.
• Use Thurston's theory of fibered 3-manifolds and the cone of fibered classes in cohomology to analyze the behavior of $ |\chi(\cdot)|\log(\lambda(\cdot)) $.
• Prove that the set $ \mathcal{T}(\Psi_P^\circ) $ of homeomorphism classes of such mapping tori is finite via covering space theory and Waldhausen's theorem.
• Construct a quotient map $ p: M^\circ \to N^\circ $, where $ N^\circ $ is a tame 3-manifold with compact core, and show $ N^\circ $ is homeomorphic to a Dehn filling of a finite list of 3-manifolds.
• Use the homeomorphism type of $ N^\circ $ and the monodromy action to show that $ \phi $ arises as the monodromy of a Dehn filling on one of finitely many 3-manifolds.

Experimental results

Research questions

• RQ1 Is there a finite set of 3-manifolds such that all small dilatation pseudo-Anosovs on any surface arise as monodromies of Dehn fillings on these manifolds?
• RQ2 Does the product $ |\chi(S)|\log(\lambda(\phi)) $ being bounded imply finiteness of the homeomorphism types of mapping tori?
• RQ3 Can the dynamics of all small dilatation pseudo-Anosovs be uniformly captured by a finite template of 3-manifolds?
• RQ4 How does the monodromy of a pseudo-Anosov map relate to the fundamental group and covering space structure of its mapping torus?
• RQ5 What is the role of the singularities of stable/unstable foliations in the finiteness of the resulting 3-manifold structures?

Key findings

• The set $ \mathcal{T}(\Psi_P^\circ) $ of homeomorphism classes of mapping tori for all $ \phi \in \Psi_P^\circ $ is finite for any fixed $ P \geq 1 $.
• All small dilatation pseudo-Anosovs arise as monodromies of Dehn fillings on a finite list of fibered hyperbolic 3-manifolds.
• The mapping torus $ M_\phi $ of a pseudo-Anosov $ \phi \in \Psi_P^\circ $ is homeomorphic to a Dehn filling of a compact core of a 3-manifold $ N^\circ $ with bounded complexity.
• The fundamental group of the resulting 3-manifold $ N^\circ $ is an extension of $ \mathbb{Z} $ by $ \pi_1(S^\circ) $ with monodromy $ \phi_* $, and the isomorphism type is determined by the monodromy.
• The construction ensures that $ N^\circ $ is tame and has a compact core, and the map $ p: M^\circ \to N^\circ $ induces an isomorphism on fundamental groups.
• Using Waldhausen’s theorem and strong deformation retractions, the map $ p $ is shown to be homotopic to a homeomorphism, proving the finiteness of the homeomorphism types of the mapping tori.