Anosov flows on Dehn surgeries on the figure-eight knot (Q6177382)

In this article the author classifies Anosov flows on 3-manifolds obtained by Dehn surgeries on the figure-eight knot. This set of 3-manifolds is denoted by \(\{M(r)\mid r\in \mathbb Q\}\). Combining this with the classification of Anosov flows on the sol-manifold \(M(0)\) due to \textit{J. F. Plante} [J. Lond. Math. Soc., II. Ser. 23, 359--362 (1981; Zbl 0465.58020)], the author gets the following results: (1) If \(r\in \mathbb {Z}\), up to topological equivalence, \(M(r)\) carries a unique Anosov flow, as constructed by \textit{S. Goodman} [Lect. Notes Math. None, 300--307 (1983; Zbl 0532.58021)]. This is proved by performing a Dehn-Fried-Goodman surgery on a suspension Anosov flow; (2) If \(r\notin \mathbb {Z}\), \(M(r)\) does not carry any Anosov flow. As a consequence of the second result, the author gets infinitely many closed orientable hyperbolic 3-manifolds that carry taut foliations but do not carry any Anosov flow. The fundamental tool in the proofs is the set of branched surfaces built by \textit{ T. Schwider} [``The classification of essential laminations in Dehn sugeries on the figure-eight knot'', PhD dissertation, University of Michigan (2001)], which is used to carry essential laminations on \(M(r)\).