Anosov flows on Dehn surgeries on the figure-eight knot

TL;DR

This paper classifies Anosov flows on 3-manifolds obtained from Dehn surgeries on the figure-eight knot, showing their existence depends on the surgery ratio, and identifies infinitely many hyperbolic manifolds with taut foliations but no Anosov flows.

Contribution

It provides a complete classification of Anosov flows on these manifolds based on the surgery ratio, extending previous results and using branched surfaces for the analysis.

Findings

Unique Anosov flow for integer ratios

No Anosov flows for non-integer ratios

Existence of hyperbolic manifolds with taut foliations but no Anosov flows

Abstract

The purpose of this paper is to classify Anosov flows on the 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by M(r) (r is a ratioanl number), which contains the first class of hyperbolic 3-manifolds admitting Anosov flows in history, discovered by Goodman. Combining with the classification of Anosov flows on the sol-manifold M(0) due to Plante, we have: 1. if r is an integer, up to topological equivalence, M(r) exactly carries a unique Anosov flow, which is constructed by Goodman by doing a Dehn-Fried-Goodman surgery on a suspension Anosov flow; 2. if r is not an integer, M(r) does not carry any Anosov flow. As a consequence of the second result, we get infinitely many closed orientable hyperbolic 3-manifolds which carry taut foliations but does not carry any Anosov flow. The fundamental tool in the proofs is the set of branched surfaces built by Schwider, which is used to carry essential laminations on M(r).