Small dilatation pseudo-Anosovs and 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.

40 pages, 6 color figures

References in corpus (5)

• A family of pseudo-Anosov braids with small dilatation
• The asymptotic behavior of least pseudo-Anosov dilatations
• On the minimum dilatation of pseudo-Anosov homeomorphisms on surfaces of small genus
• Pseudo-Anosov braids with small entropy and the magic 3-manifold
• Entropy vs volume for pseudo-Anosov maps

Cited by in corpus (10)

• Small dilatation pseudo-Anosov mapping classes coming from the simplest hyperbolic braid
• On the zeroes of the Alexander polynomial of a Lorenz knot
• Minimal dilatations of pseudo-Anosovs generated by the magic 3-manifold and their asymptotic behavior
• Mapping classes associated to mixed-sign Coxeter graphs
• Quotient families of mapping classes
• The entropy efficiency of point-push mapping classes on the punctured disk
• Achievable spectral radii of symplectic Perron-Frobenius matrices
• Homological eigenvalues of mapping classes and torsion homology growth for fibered 3--manifolds
• On pseudo-Anosov mapping classes with minimum dilatation and Lanneau-Thiffeault numbers
• Small dilatation pseudo-Anosov mapping classes and short circuits on train track automata