Pseudo-Anosov surfaces and Dynamics in 3-manifolds
Jason F. Manning, Christoforos Neofytidis
TL;DR
This work classifies closed orientable irreducible 3-manifolds that admit a self-map restricting to a pseudo-Anosov on an incompressible surface, showing the manifold must either be a pseudo-Anosov mapping torus or Seifert-fibered with a large base orbifold and a horizontal surface. It introduces and analyzes the notion of a partially pseudo-Anosov homeomorphism to extend surface dynamics to the ambient manifold, leveraging JSJ theory, maximal Seifert submanifolds, and McCarthy’s control over normalizers. The paper provides explicit constructions of pseudo-Anosov incompressible surfaces (mapping tori and Seifert-fibered examples), proves corollaries for reducible manifolds, and discusses the ubiquity of compressible pseudo-Anosov surfaces across all 3-manifolds. The results illuminate the relationship between surface dynamics and ambient 3-manifold topology, with connections to partially hyperbolic dynamics in the spirit of Anosov tori.
Abstract
We determine which closed orientable $3$-manifolds $M$ admit a self-homeomorphism restricting to a pseudo-Anosov map on an incompressible subsurface $Σ$, which we call a pseudo-Anosov surface. When $M$ is irreducible, we show that the self-homeomorphism of $M$ is isotopic rel $Σ$ to a "partially pseudo-Anosov" homeomorphism, a notion that we will introduce. This is motivated by the corresponding results for Anosov tori in irreducible $3$-manifolds, and the connection to partially hyperbolic diffeomorphisms, obtained by F. Rodriguez-Hertz, J. Rodriguez-Hertz and R. Ures.