An Alternative Approach to Extending pseudo-Anosovs Over Compression Bodies

TL;DR

This paper offers an alternative proof that certain pseudo-Anosov homeomorphisms extend over minimal compression bodies, using disk collections related to stable and unstable laminations, and establishes finiteness of such bodies.

Contribution

It provides a new proof method for extending pseudo-Anosovs over compression bodies and demonstrates the finiteness of minimal compression bodies for such extensions.

Findings

Alternative proof using Long and Casson techniques

Existence of disk collections from laminations

Finiteness of minimal compression bodies

Abstract

A recent paper (\cite{BJM}) by Biringer, Johnson, and Minsky prove that any pseudo-Anosov whose stable lamination is the limit of disks in a compression body has a power which extends over some non-trivial minimal compression body. This paper presents an alternative proof of their theorem, using techniques of Long and Casson which first appeared in \cite{CL} and \cite{L}. The key ingredient is the existence of a certain collection of disks whose boundaries are formed from an arc of the stable lamination and an arc of the unstable lamination. Furthermore, the proof here shows that there are only finitely many minimal compression bodies over which a power of a pseudo-Anosov can extend.