Pseudo-Anosovs from the perspective of their mapping tori

TL;DR

The paper surveys how combinatorial dynamics on surface complexes (curve, arc, and pants) reflect the hyperbolic geometry of mapping tori of pseudo-Anosov diffeomorphisms, bridging dynamics, geometry, and topology. It develops and explains precise relations between monodromy actions on these complexes and geometric invariants such as volume, systole length, and cusp area in the associated 3-manifolds, including effective bounds and asymptotics. A central thread is the translation between curve/arc data and 3-manifold geometry via hierarchy paths, model manifolds, and ending laminations, with extensions to infinite-type surfaces through endperiodic maps and their mapping-torus dynamics. The work highlights the deep connections between Teichmüller theory, hyperbolic geometry, and combinatorial structures, and it culminates with fixed-point results (AFT) and a discussion of open problems and possible generalizations to outer-space dynamics and hierarchically hyperbolic group theory.

Abstract

In this chapter, we outline some of the many combinatorial tools developed over the past three decades for studying a pseudo-Anosov diffeomorphism of a surface by analyzing the geometry of its mapping torus. We begin with an overview of the various simplicial complexes associated with a surface (such as the curve, arc, and pants complexes) and explain how to relate the dynamics of the action of a given pseudo-Anosov on any one of these complexes to the dynamics of the diffeomorphism itself, or to the hyperbolic geometry of its mapping torus. We next cover some of the more modern features of the theory by discussing various analogs of pseudo-Anosov diffeomorphisms on surfaces of infinite type. We conclude with a description of original work-- due jointly to the author with Dave Futer and Sam Taylor-- that relates the action of a pseudo-Anosov on the curve complex to the minimum number of fixed points for any map in the corresponding isotopy class. The paper is written in as accessible a way as possible while assuming only the bare minimum in background. The hope is to informally convey to the reader some of the main ideas and strategies in the area.

Figures (7)

• Figure 2.1: Four curves on a genus $4$ surface and the corresponding vertices in the curve complex. We leave it as an exercise for the reader that the orange curve is indeed at a distance of $2$ to each of the black, red, and purple curves (implying the existence of the pictured green vertices).
• Figure 2.2: Several elementary moves are pictured.
• Figure 2.3: A curve together with its subsurface projection to a yellow subsurface. The top figure represents the cover associated with this subsurface, which can be compactified using the boundary at infinity in the universal cover, yielding the bottom picture.
• Figure 3.1: Each black or orange piecewise linear curve represents a pleated surface in the mapping torus. The red dots represent simple closed geodesics that lie within the pleating locus of a given pleated surface. Starting with the red geodesic $\alpha$ located highest on the page, there is a black pleated surface containing both it, and a curve $\alpha_{1}$ disjoint from $\alpha$. Then, one discovers a pleated surface (pictured in orange) containing both $\alpha_{1}$ and a curve $\alpha_{2}$ disjoint from $\alpha_{1}$. The colors alternate between black and orange in order to make it easier to see the "chain". We also draw some of the pleated surfaces in a way that emphasizes to the reader that the maps need not be embeddings.
• Figure 3.2: A picture of the negation of our simplifying assumption. In the general situation, it is possible that some of the pleated surfaces in the sweep-out might miss the image of $\omega$ completely.

Theorems & Definitions (13)

• Lemma 2.1
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Definition 4.1
• Definition 4.2
• Theorem 4.3
• Theorem 4.4