Recognising perfect fits

TL;DR

The paper develops an algorithmic framework to decide whether a pseudo-Anosov flow on a closed 3-manifold exhibits perfect fits by bridging box decompositions with veering triangulations. It introduces two complementary routines: FindFit, which tests Fenley’s perfect-fit criterion via periodic-orbit itineraries and the conjugacy problem, and FindVeering, which constructs a veering triangulation from a box decomposition using the Agol-Guéritaud process. The core result is that the flow has perfect fits iff FindFit succeeds; conversely, absence of perfect fits yields a canonical veering triangulation that enables orbit-equivalence testing and suspension recognition. Together, these methods connect combinatorial models of flows with veering triangulations, enabling algorithmic and computational investigations of pseudo-Anosov dynamics and their ambient 3-manifolds.

Abstract

A pseudo-Anosov flow is said to have perfect fits if there are stable and unstable leaves that are asymptotic in the universal cover. We give an algorithm to decide, given a box decomposition of a pseudo-Anosov flow, if the flow has perfect fits. As a corollary, we obtain an algorithm to decide whether two flows without perfect fits are orbit equivalent.

Figures (18)

• Figure 1: The main goal of the paper: to decide when we can pass between a veering triangulation (left) and a box decomposition for a pseudo-Anosov flow (right).
• Figure 2:
• Figure 3: Left to right: a compact rectangle, a perfect fit rectangle, and a lozenge.
• Figure 4: A taut, veering tetrahedron. Co-orientations on the faces are induced by the direction pointing out of the page.
• Figure 5: A persistent tuple of walls.

Theorems & Definitions (64)

• Theorem 1.1
• Corollary 1.0
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Remark 2.6
• Definition 2.7
• Definition 2.8